adjsym - Hilbert Space Explorer

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Theorem adjsym 10039

Description: Symmetry property of an adjoint.

**Assertion**
Ref	Expression
adjsym	⊢ ((S: ℋ –→ ℋ ⋀ T: ℋ –→ ℋ ) → (∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y) ↔ ∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (T ‘y)) = ((S ‘x) ·_ih y)))

Distinct variable groups: x,y,S x,T,y

**Proof of Theorem adjsym**
Step	Hyp	Ref	Expression
1		ax-his1 9225	. . . . . . . . . . . 12 ⊢ (((T ‘y) ∈ ℋ ⋀ x ∈ ℋ ) → ((T ‘y) ·_ih x) = (∗ ‘(x ·_ih (T ‘y))))
2		ffvelrn 3928	. . . . . . . . . . . 12 ⊢ ((T: ℋ –→ ℋ ⋀ y ∈ ℋ ) → (T ‘y) ∈ ℋ )
3	1, 2	sylan 450	. . . . . . . . . . 11 ⊢ (((T: ℋ –→ ℋ ⋀ y ∈ ℋ ) ⋀ x ∈ ℋ ) → ((T ‘y) ·_ih x) = (∗ ‘(x ·_ih (T ‘y))))
4	3	adantrl 394	. . . . . . . . . 10 ⊢ (((T: ℋ –→ ℋ ⋀ y ∈ ℋ ) ⋀ (S: ℋ –→ ℋ ⋀ x ∈ ℋ )) → ((T ‘y) ·_ih x) = (∗ ‘(x ·_ih (T ‘y))))
5		ax-his1 9225	. . . . . . . . . . . 12 ⊢ ((y ∈ ℋ ⋀ (S ‘x) ∈ ℋ ) → (y ·_ih (S ‘x)) = (∗ ‘((S ‘x) ·_ih y)))
6		ffvelrn 3928	. . . . . . . . . . . 12 ⊢ ((S: ℋ –→ ℋ ⋀ x ∈ ℋ ) → (S ‘x) ∈ ℋ )
7	5, 6	sylan2 453	. . . . . . . . . . 11 ⊢ ((y ∈ ℋ ⋀ (S: ℋ –→ ℋ ⋀ x ∈ ℋ )) → (y ·_ih (S ‘x)) = (∗ ‘((S ‘x) ·_ih y)))
8	7	adantll 392	. . . . . . . . . 10 ⊢ (((T: ℋ –→ ℋ ⋀ y ∈ ℋ ) ⋀ (S: ℋ –→ ℋ ⋀ x ∈ ℋ )) → (y ·_ih (S ‘x)) = (∗ ‘((S ‘x) ·_ih y)))
9	4, 8	eqeq12d 1532	. . . . . . . . 9 ⊢ (((T: ℋ –→ ℋ ⋀ y ∈ ℋ ) ⋀ (S: ℋ –→ ℋ ⋀ x ∈ ℋ )) → (((T ‘y) ·_ih x) = (y ·_ih (S ‘x)) ↔ (∗ ‘(x ·_ih (T ‘y))) = (∗ ‘((S ‘x) ·_ih y))))
10	9	ancoms 438	. . . . . . . 8 ⊢ (((S: ℋ –→ ℋ ⋀ x ∈ ℋ ) ⋀ (T: ℋ –→ ℋ ⋀ y ∈ ℋ )) → (((T ‘y) ·_ih x) = (y ·_ih (S ‘x)) ↔ (∗ ‘(x ·_ih (T ‘y))) = (∗ ‘((S ‘x) ·_ih y))))
11		cj11 7031	. . . . . . . . 9 ⊢ (((x ·_ih (T ‘y)) ∈ ℂ ⋀ ((S ‘x) ·_ih y) ∈ ℂ) → ((∗ ‘(x ·_ih (T ‘y))) = (∗ ‘((S ‘x) ·_ih y)) ↔ (x ·_ih (T ‘y)) = ((S ‘x) ·_ih y)))
12		hicl 9223	. . . . . . . . . . 11 ⊢ ((x ∈ ℋ ⋀ (T ‘y) ∈ ℋ ) → (x ·_ih (T ‘y)) ∈ ℂ)
13	12, 2	sylan2 453	. . . . . . . . . 10 ⊢ ((x ∈ ℋ ⋀ (T: ℋ –→ ℋ ⋀ y ∈ ℋ )) → (x ·_ih (T ‘y)) ∈ ℂ)
14	13	adantll 392	. . . . . . . . 9 ⊢ (((S: ℋ –→ ℋ ⋀ x ∈ ℋ ) ⋀ (T: ℋ –→ ℋ ⋀ y ∈ ℋ )) → (x ·_ih (T ‘y)) ∈ ℂ)
15		hicl 9223	. . . . . . . . . . 11 ⊢ (((S ‘x) ∈ ℋ ⋀ y ∈ ℋ ) → ((S ‘x) ·_ih y) ∈ ℂ)
16	15, 6	sylan 450	. . . . . . . . . 10 ⊢ (((S: ℋ –→ ℋ ⋀ x ∈ ℋ ) ⋀ y ∈ ℋ ) → ((S ‘x) ·_ih y) ∈ ℂ)
17	16	adantrl 394	. . . . . . . . 9 ⊢ (((S: ℋ –→ ℋ ⋀ x ∈ ℋ ) ⋀ (T: ℋ –→ ℋ ⋀ y ∈ ℋ )) → ((S ‘x) ·_ih y) ∈ ℂ)
18	11, 14, 17	sylanc 473	. . . . . . . 8 ⊢ (((S: ℋ –→ ℋ ⋀ x ∈ ℋ ) ⋀ (T: ℋ –→ ℋ ⋀ y ∈ ℋ )) → ((∗ ‘(x ·_ih (T ‘y))) = (∗ ‘((S ‘x) ·_ih y)) ↔ (x ·_ih (T ‘y)) = ((S ‘x) ·_ih y)))
19	10, 18	bitr2d 532	. . . . . . 7 ⊢ (((S: ℋ –→ ℋ ⋀ x ∈ ℋ ) ⋀ (T: ℋ –→ ℋ ⋀ y ∈ ℋ )) → ((x ·_ih (T ‘y)) = ((S ‘x) ·_ih y) ↔ ((T ‘y) ·_ih x) = (y ·_ih (S ‘x))))
20	19	an4s 511	. . . . . 6 ⊢ (((S: ℋ –→ ℋ ⋀ T: ℋ –→ ℋ ) ⋀ (x ∈ ℋ ⋀ y ∈ ℋ )) → ((x ·_ih (T ‘y)) = ((S ‘x) ·_ih y) ↔ ((T ‘y) ·_ih x) = (y ·_ih (S ‘x))))
21	20	anassrs 443	. . . . 5 ⊢ ((((S: ℋ –→ ℋ ⋀ T: ℋ –→ ℋ ) ⋀ x ∈ ℋ ) ⋀ y ∈ ℋ ) → ((x ·_ih (T ‘y)) = ((S ‘x) ·_ih y) ↔ ((T ‘y) ·_ih x) = (y ·_ih (S ‘x))))
22		eqcom 1520	. . . . 5 ⊢ (((T ‘y) ·_ih x) = (y ·_ih (S ‘x)) ↔ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x))
23	21, 22	syl6bb 539	. . . 4 ⊢ ((((S: ℋ –→ ℋ ⋀ T: ℋ –→ ℋ ) ⋀ x ∈ ℋ ) ⋀ y ∈ ℋ ) → ((x ·_ih (T ‘y)) = ((S ‘x) ·_ih y) ↔ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x)))
24	23	ralbidva 1705	. . 3 ⊢ (((S: ℋ –→ ℋ ⋀ T: ℋ –→ ℋ ) ⋀ x ∈ ℋ ) → (∀y ∈ ℋ (x ·_ih (T ‘y)) = ((S ‘x) ·_ih y) ↔ ∀y ∈ ℋ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x)))
25	24	ralbidva 1705	. 2 ⊢ ((S: ℋ –→ ℋ ⋀ T: ℋ –→ ℋ ) → (∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (T ‘y)) = ((S ‘x) ·_ih y) ↔ ∀x ∈ ℋ ∀y ∈ ℋ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x)))
26		ralcom 1820	. . . 4 ⊢ (∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y) ↔ ∀y ∈ ℋ ∀x ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y))
27		fveq2 3835	. . . . . . . 8 ⊢ (z = y → (S ‘z) = (S ‘y))
28	27	opreq2d 4034	. . . . . . 7 ⊢ (z = y → (x ·_ih (S ‘z)) = (x ·_ih (S ‘y)))
29		opreq2 4027	. . . . . . 7 ⊢ (z = y → ((T ‘x) ·_ih z) = ((T ‘x) ·_ih y))
30	28, 29	eqeq12d 1532	. . . . . 6 ⊢ (z = y → ((x ·_ih (S ‘z)) = ((T ‘x) ·_ih z) ↔ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y)))
31	30	ralbidv 1709	. . . . 5 ⊢ (z = y → (∀x ∈ ℋ (x ·_ih (S ‘z)) = ((T ‘x) ·_ih z) ↔ ∀x ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y)))
32	31	cbvralv 1846	. . . 4 ⊢ (∀z ∈ ℋ ∀x ∈ ℋ (x ·_ih (S ‘z)) = ((T ‘x) ·_ih z) ↔ ∀y ∈ ℋ ∀x ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y))
33	26, 32	bitr4i 174	. . 3 ⊢ (∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y) ↔ ∀z ∈ ℋ ∀x ∈ ℋ (x ·_ih (S ‘z)) = ((T ‘x) ·_ih z))
34		opreq1 4026	. . . . . 6 ⊢ (x = y → (x ·_ih (S ‘z)) = (y ·_ih (S ‘z)))
35		fveq2 3835	. . . . . . 7 ⊢ (x = y → (T ‘x) = (T ‘y))
36	35	opreq1d 4033	. . . . . 6 ⊢ (x = y → ((T ‘x) ·_ih z) = ((T ‘y) ·_ih z))
37	34, 36	eqeq12d 1532	. . . . 5 ⊢ (x = y → ((x ·_ih (S ‘z)) = ((T ‘x) ·_ih z) ↔ (y ·_ih (S ‘z)) = ((T ‘y) ·_ih z)))
38	37	cbvralv 1846	. . . 4 ⊢ (∀x ∈ ℋ (x ·_ih (S ‘z)) = ((T ‘x) ·_ih z) ↔ ∀y ∈ ℋ (y ·_ih (S ‘z)) = ((T ‘y) ·_ih z))
39	38	ralbii 1713	. . 3 ⊢ (∀z ∈ ℋ ∀x ∈ ℋ (x ·_ih (S ‘z)) = ((T ‘x) ·_ih z) ↔ ∀z ∈ ℋ ∀y ∈ ℋ (y ·_ih (S ‘z)) = ((T ‘y) ·_ih z))
40		fveq2 3835	. . . . . . 7 ⊢ (z = x → (S ‘z) = (S ‘x))
41	40	opreq2d 4034	. . . . . 6 ⊢ (z = x → (y ·_ih (S ‘z)) = (y ·_ih (S ‘x)))
42		opreq2 4027	. . . . . 6 ⊢ (z = x → ((T ‘y) ·_ih z) = ((T ‘y) ·_ih x))
43	41, 42	eqeq12d 1532	. . . . 5 ⊢ (z = x → ((y ·_ih (S ‘z)) = ((T ‘y) ·_ih z) ↔ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x)))
44	43	ralbidv 1709	. . . 4 ⊢ (z = x → (∀y ∈ ℋ (y ·_ih (S ‘z)) = ((T ‘y) ·_ih z) ↔ ∀y ∈ ℋ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x)))
45	44	cbvralv 1846	. . 3 ⊢ (∀z ∈ ℋ ∀y ∈ ℋ (y ·_ih (S ‘z)) = ((T ‘y) ·_ih z) ↔ ∀x ∈ ℋ ∀y ∈ ℋ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x))
46	33, 39, 45	3bitri 175	. 2 ⊢ (∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y) ↔ ∀x ∈ ℋ ∀y ∈ ℋ (y ·_ih (S ‘x)) = ((T ‘y) ·_ih x))
47	25, 46	syl6rbbr 542	1 ⊢ ((S: ℋ –→ ℋ ⋀ T: ℋ –→ ℋ ) → (∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (S ‘y)) = ((T ‘x) ·_ih y) ↔ ∀x ∈ ℋ ∀y ∈ ℋ (x ·_ih (T ‘y)) = ((S ‘x) ·_ih y)))

Colors of variables: wff set class

Syntax hints: → wi 3 ↔ wb 144 ⋀ wa 221 = wceq 992 ∈ wcel 994 ∀wral 1691 –→wf 3259 ‘cfv 3263 (class class class)co 4021 ℂcc 5386 ∗ccj 6950 ℋ chil 9063 ·_ih csp 9068

This theorem is referenced by: dfadj2 10092 adjval2 10095 cnlnadjeui 10289 cnlnssadj 10292 adjbdln 10295

This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 998 ax-gen 999 ax-8 1000 ax-9 1001 ax-10 1002 ax-11 1003 ax-12 1004 ax-13 1005 ax-14 1006 ax-17 1007 ax-4 1009 ax-5o 1011 ax-6o 1014 ax-9o 1159 ax-10o 1177 ax-16 1247 ax-11o 1255 ax-ext 1500 ax-rep 2767 ax-sep 2777 ax-nul 2784 ax-pow 2818 ax-pr 2855 ax-un 3089 ax-inf2 4770 ax-hfi 9222 ax-his1 9225

This theorem depends on definitions: df-bi 145 df-or 222 df-an 223 df-3or 782 df-3an 783 df-ex 1017 df-sb 1209 df-eu 1421 df-mo 1422 df-clab 1506 df-cleq 1511 df-clel 1514 df-ne 1630 df-nel 1631 df-ral 1695 df-rex 1696 df-reu 1697 df-rab 1698 df-v 1858 df-sbc 1987 df-csb 2052 df-dif 2101 df-un 2102 df-in 2103 df-ss 2105 df-pss 2107 df-nul 2333 df-if 2416 df-pw 2459 df-sn 2470 df-pr 2471 df-tp 2473 df-op 2474 df-uni 2570 df-int 2601 df-iun 2635 df-br 2693 df-opab 2741 df-tr 2755 df-eprel 2910 df-id 2913 df-po 2918 df-so 2929 df-fr 2947 df-we 2962 df-ord 2978 df-on 2979 df-lim 2980 df-suc 2981 df-om 3219 df-xp 3265 df-rel 3266 df-cnv 3267 df-co 3268 df-dm 3269 df-rn 3270 df-res 3271 df-ima 3272 df-fun 3273 df-fn 3274 df-f 3275 df-f1 3276 df-fo 3277 df-f1o 3278 df-fv 3279 df-opr 4023 df-oprab 4024 df-1st 4140 df-2nd 4141 df-rdg 4233 df-1o 4269 df-oadd 4271 df-omul 4272 df-er 4401 df-ec 4403 df-qs 4406 df-en 4509 df-dom 4510 df-sdom 4511 df-ni 5154 df-pli 5155 df-mi 5156 df-lti 5157 df-plpq 5189 df-mpq 5190 df-enq 5191 df-nq 5192 df-plq 5193 df-mq 5194 df-rq 5195 df-ltq 5196 df-1q 5197 df-np 5240 df-1p 5241 df-plp 5242 df-mp 5243 df-ltp 5244 df-plpr 5318 df-mpr 5319 df-enr 5320 df-nr 5321 df-plr 5322 df-mr 5323 df-ltr 5324 df-0r 5325 df-1r 5326 df-m1r 5327 df-c 5394 df-0 5395 df-1 5396 df-i 5397 df-r 5398 df-plus 5399 df-mul 5400 df-lt 5401 df-sub 5510 df-neg 5512 df-pnf 5641 df-mnf 5642 df-xr 5643 df-ltxr 5644 df-le 5645 df-div 5855 df-re 6952 df-im 6953 df-cj 6954