Theorem hfsmvalt 9514

Description: Value of the sum of two Hilbert space functionals.

Assertion

Ref	Expression
hfsmvalt	|- ((S:H~-->CC /\ T:H~-->CC) -> (S +fn T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))})

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

Proof of Theorem hfsmvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8869	. . . 4 |- H~ e. V
2	1	opabex2 3610	. . 3 |- {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))} e. V
3		fveq1 3723	. . . . . . 7 |- (f = S -> (f` x) = (S` x))
4	3	opreq1d 3975	. . . . . 6 |- (f = S -> ((f` x) + (g` x)) = ((S` x) + (g` x)))
5	4	eqeq2d 1486	. . . . 5 |- (f = S -> (y = ((f` x) + (g` x)) <-> y = ((S` x) + (g` x))))
6	5	anbi2d 616	. . . 4 |- (f = S -> ((x e. H~ /\ y = ((f` x) + (g` x))) <-> (x e. H~ /\ y = ((S` x) + (g` x)))))
7	6	opabbidv 2670	. . 3 |- (f = S -> {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))} = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (g` x)))})
8		fveq1 3723	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
9	8	opreq2d 3976	. . . . . 6 |- (g = T -> ((S` x) + (g` x)) = ((S` x) + (T` x)))
10	9	eqeq2d 1486	. . . . 5 |- (g = T -> (y = ((S` x) + (g` x)) <-> y = ((S` x) + (T` x))))
11	10	anbi2d 616	. . . 4 |- (g = T -> ((x e. H~ /\ y = ((S` x) + (g` x))) <-> (x e. H~ /\ y = ((S` x) + (T` x)))))
12	11	opabbidv 2670	. . 3 |- (g = T -> {<.x, y>. | (x e. H~ /\ y = ((S` x) + (g` x)))} = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))})
13		df-hfsum 9509	. . . 4 |- +fn = {<.<.f, g>., h>. | ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}
14		axcnex 5267	. . . . . . . 8 |- CC e. V
15	14, 1	elmap 4334	. . . . . . 7 |- (f e. (CC ^m H~) <-> f:H~-->CC)
16	14, 1	elmap 4334	. . . . . . 7 |- (g e. (CC ^m H~) <-> g:H~-->CC)
17	15, 16	anbi12i 482	. . . . . 6 |- ((f e. (CC ^m H~) /\ g e. (CC ^m H~)) <-> (f:H~-->CC /\ g:H~-->CC))
18	17	anbi1i 481	. . . . 5 |- (((f e. (CC ^m H~) /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))}) <-> ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))}))
19	18	oprabbii 3997	. . . 4 |- {<.<.f, g>., h>. | ((f e. (CC ^m H~) /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})} = {<.<.f, g>., h>. | ((f:H~-->CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}
20	13, 19	eqtr4 1498	. . 3 |- +fn = {<.<.f, g>., h>. | ((f e. (CC ^m H~) /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = ((f` x) + (g` x)))})}
21	2, 7, 12, 20	oprabval2 4028	. 2 |- ((S e. (CC ^m H~) /\ T e. (CC ^m H~)) -> (S +fn T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))})
22	14, 1	elmap 4334	. 2 |- (S e. (CC ^m H~) <-> S:H~-->CC)
23	14, 1	elmap 4334	. 2 |- (T e. (CC ^m H~) <-> T:H~-->CC)
24	21, 22, 23	syl2anbr 456	1 |- ((S:H~-->CC /\ T:H~-->CC) -> (S +fn T) = {<.x, y>. | (x e. H~ /\ y = ((S` x) + (T` x)))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {copab 2666  -->wf 3178  ` cfv 3182  (class class class)co 3963  {copab2 3964   ^m cm 4322  CCcc 5232   + caddc 5237  H~chil 8788   +fn chfs 8810

This theorem is referenced by:  hfsvalt 9521

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-qs 4266  df-map 4324  df-ni 5000  df-nq 5038  df-np 5086  df-nr 5167  df-c 5240  df-hfsum 9509

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