Theorem ruclem35 7577

Description: Lemma for ruc 7582. The supremum we have constructed lies between all values of the G and H functions. Compare ruclem29 7571, which states the opposite for the input function F.

Hypotheses

Ref	Expression
ruclem.0	|- F:NN-->RR
ruclem.1	|- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
ruclem.2	|- D = {<.<.x, y>., z>. | ((x e. (RR X. RR) /\ y e. RR) /\ z = if(((1st` x) < y /\ y < (2nd` x)), <.(((2 x. y) + (2nd` x)) / 3), ((y + (2 x. (2nd` x))) / 3)>., <.(((2 x. (1st` x)) + (2nd` x)) / 3), (((1st` x) + (2 x. (2nd` x))) / 3)>.))}
ruclem.3	|- G = (1st o. (D seq1 C))
ruclem.4	|- H = (2nd o. (D seq1 C))
ruclem.5	|- S = sup(ran G, RR, < )
ruclem.a	|- A e. NN

Assertion

Ref	Expression
ruclem35	|- ((G` A) < S /\ S < (H` A))

Distinct variable groups:   x,y,z   z,F

Proof of Theorem ruclem35

Step	Hyp	Ref	Expression
1		ruclem.0	. . . 4 |- F:NN-->RR
2		ruclem.1	. . . 4 |- C = ({<.1, <.((F` 1) + 1), ((F` 1) + 2)>.>.} u. (F |` (NN \ {1})))
3		ruclem.2	. . . 4 |- D = {<.<.x, y>., z>. | ((x e. (RR X. RR) /\ y e. RR) /\ z = if(((1st` x) < y /\ y < (2nd` x)), <.(((2 x. y) + (2nd` x)) / 3), ((y + (2 x. (2nd` x))) / 3)>., <.(((2 x. (1st` x)) + (2nd` x)) / 3), (((1st` x) + (2 x. (2nd` x))) / 3)>.))}
4		ruclem.3	. . . 4 |- G = (1st o. (D seq1 C))
5		ruclem.4	. . . 4 |- H = (2nd o. (D seq1 C))
6		ruclem.a	. . . 4 |- A e. NN
7	1, 2, 3, 4, 5, 6	ruclem26 7568	. . 3 |- (G` A) < (G` (A + 1))
8	1, 2, 3, 4, 5	ruclem17 7559	. . . . . . 7 |- G:NN-->RR
9		ffn 3643	. . . . . . 7 |- (G:NN-->RR -> G Fn NN)
10	8, 9	ax-mp 7	. . . . . 6 |- G Fn NN
11		peano2nn 5949	. . . . . . 7 |- (A e. NN -> (A + 1) e. NN)
12	6, 11	ax-mp 7	. . . . . 6 |- (A + 1) e. NN
13		fnfvelrn 3829	. . . . . 6 |- ((G Fn NN /\ (A + 1) e. NN) -> (G` (A + 1)) e. ran G)
14	10, 12, 13	mp2an 701	. . . . 5 |- (G` (A + 1)) e. ran G
15	1, 2, 3, 4, 5	ruclem33 7575	. . . . . 6 |- (ran G (_ RR /\ ran G =/= (/) /\ E.w e. RR A.v e. ran G v <_ w)
16	15	suprubii 6094	. . . . 5 |- ((G` (A + 1)) e. ran G -> (G` (A + 1)) <_ sup(ran G, RR, < ))
17	14, 16	ax-mp 7	. . . 4 |- (G` (A + 1)) <_ sup(ran G, RR, < )
18		ruclem.5	. . . 4 |- S = sup(ran G, RR, < )
19	17, 18	breqtrri 2655	. . 3 |- (G` (A + 1)) <_ S
20	1, 2, 3, 4, 5, 6	ruclem22 7564	. . . 4 |- (G` A) e. RR
21	1, 2, 3, 4, 5, 12	ruclem22 7564	. . . 4 |- (G` (A + 1)) e. RR
22	1, 2, 3, 4, 5, 18	ruclem34 7576	. . . 4 |- S e. RR
23	20, 21, 22	ltletri 5607	. . 3 |- (((G` A) < (G` (A + 1)) /\ (G` (A + 1)) <_ S) -> (G` A) < S)
24	7, 19, 23	mp2an 701	. 2 |- (G` A) < S
25	1, 2, 3, 4, 5, 12	ruclem23 7565	. . . . . 6 |- (H` (A + 1)) e. RR
26		fvelrnb 3776	. . . . . . . . 9 |- (G Fn NN -> (u e. ran G <-> E.w e. NN (G` w) = u))
27	10, 26	ax-mp 7	. . . . . . . 8 |- (u e. ran G <-> E.w e. NN (G` w) = u)
28		breq2 2638	. . . . . . . . . . 11 |- ((G` w) = u -> ((H` (A + 1)) < (G` w) <-> (H` (A + 1)) < u))
29	28	notbid 614	. . . . . . . . . 10 |- ((G` w) = u -> (-. (H` (A + 1)) < (G` w) <-> -. (H` (A + 1)) < u))
30		ltnsym 5552	. . . . . . . . . . 11 |- (((G` w) e. RR /\ (H` (A + 1)) e. RR) -> ((G` w) < (H` (A + 1)) -> -. (H` (A + 1)) < (G` w)))
31		fveq2 3740	. . . . . . . . . . . . . 14 |- (w = if(w e. NN, w, 1) -> (G` w) = (G` if(w e. NN, w, 1)))
32	31	eleq1d 1547	. . . . . . . . . . . . 13 |- (w = if(w e. NN, w, 1) -> ((G` w) e. RR <-> (G` if(w e. NN, w, 1)) e. RR))
33		1nn 5948	. . . . . . . . . . . . . . 15 |- 1 e. NN
34	33	elimel 2406	. . . . . . . . . . . . . 14 |- if(w e. NN, w, 1) e. NN
35	1, 2, 3, 4, 5, 34	ruclem22 7564	. . . . . . . . . . . . 13 |- (G` if(w e. NN, w, 1)) e. RR
36	32, 35	dedth 2395	. . . . . . . . . . . 12 |- (w e. NN -> (G` w) e. RR)
37	36, 25	jctir 293	. . . . . . . . . . 11 |- (w e. NN -> ((G` w) e. RR /\ (H` (A + 1)) e. RR))
38	31	breq1d 2644	. . . . . . . . . . . 12 |- (w = if(w e. NN, w, 1) -> ((G` w) < (H` (A + 1)) <-> (G` if(w e. NN, w, 1)) < (H` (A + 1))))
39	1, 2, 3, 4, 5, 34, 12	ruclem32 7574	. . . . . . . . . . . 12 |- (G` if(w e. NN, w, 1)) < (H` (A + 1))
40	38, 39	dedth 2395	. . . . . . . . . . 11 |- (w e. NN -> (G` w) < (H` (A + 1)))
41	30, 37, 40	sylc 68	. . . . . . . . . 10 |- (w e. NN -> -. (H` (A + 1)) < (G` w))
42	29, 41	syl5cbi 209	. . . . . . . . 9 |- (w e. NN -> ((G` w) = u -> -. (H` (A + 1)) < u))
43	42	r19.23aiv 1750	. . . . . . . 8 |- (E.w e. NN (G` w) = u -> -. (H` (A + 1)) < u)
44	27, 43	sylbi 199	. . . . . . 7 |- (u e. ran G -> -. (H` (A + 1)) < u)
45	44	rgen 1705	. . . . . 6 |- A.u e. ran G -. (H` (A + 1)) < u
46	15	suprnubii 6096	. . . . . 6 |- (((H` (A + 1)) e. RR /\ A.u e. ran G -. (H` (A + 1)) < u) -> -. (H` (A + 1)) < sup(ran G, RR, < ))
47	25, 45, 46	mp2an 701	. . . . 5 |- -. (H` (A + 1)) < sup(ran G, RR, < )
48	18	breq2i 2642	. . . . 5 |- ((H` (A + 1)) < S <-> (H` (A + 1)) < sup(ran G, RR, < ))
49	47, 48	mtbir 192	. . . 4 |- -. (H` (A + 1)) < S
50	22, 25	lenlti 5598	. . . 4 |- (S <_ (H` (A + 1)) <-> -. (H` (A + 1)) < S)
51	49, 50	mpbir 190	. . 3 |- S <_ (H` (A + 1))
52	1, 2, 3, 4, 5, 6	ruclem27 7569	. . 3 |- (H` (A + 1)) < (H` A)
53	1, 2, 3, 4, 5, 6	ruclem23 7565	. . . 4 |- (H` A) e. RR
54	22, 25, 53	lelttri 5606	. . 3 |- ((S <_ (H` (A + 1)) /\ (H` (A + 1)) < (H` A)) -> S < (H` A))
55	51, 52, 54	mp2an 701	. 2 |- S < (H` A)
56	24, 55	pm3.2i 285	1 |- ((G` A) < S /\ S < (H` A))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 960   e. wcel 962  A.wral 1652  E.wrex 1653   \ cdif 2055   u. cun 2056  ifcif 2373  {csn 2421  <.cop 2423   class class class wbr 2634   X. cxp 3184  ran crn 3187   |` cres 3188   o. ccom 3190   Fn wfn 3193  -->wf 3194  ` cfv 3198  (class class class)co 3979  {copab2 3980  1stc1st 4093  2ndc2nd 4094  supcsup 4588  RRcr 5253  1c1 5255   + caddc 5257   x. cmul 5259   / cdiv 5314   <_ cle 5315  NNcn 5316   < clt 5506  2c2 5975  3c3 5976   seq1 cseq1 6508

This theorem is referenced by:  ruclem36 7578

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-rep 2708  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795  ax-un 2882  ax-inf2 4642

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 780  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-nel 1595  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-sbc 1949  df-csb 2012  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-pss 2066  df-nul 2292  df-if 2374  df-pw 2414  df-sn 2424  df-pr 2425  df-tp 2427  df-op 2428  df-uni 2518  df-int 2548  df-iun 2582  df-br 2635  df-opab 2682  df-tr 2696  df-eprel 2848  df-id 2851  df-po 2856  df-so 2866  df-fr 2933  df-we 2950  df-ord 2967  df-on 2968  df-lim 2969  df-suc 2970  df-om 3148  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-f 3210  df-f1 3211  df-fo 3212  df-f1o 3213  df-fv 3214  df-rdg 3948  df-opr 3981  df-oprab 3982  df-1st 4095  df-2nd 4096  df-1o 4149  df-oadd 4151  df-omul 4152  df-er 4277  df-ec 4279  df-qs 4282  df-en 4386  df-dom 4387  df-sdom 4388  df-sup 4589  df-ni 5020  df-pli 5021  df-mi 5022  df-lti 5023  df-plpq 5055  df-mpq 5056  df-enq 5057  df-nq 5058  df-plq 5059  df-mq 5060  df-rq 5061  df-ltq 5062  df-1q 5063  df-np 5106  df-1p 5107  df-plp 5108  df-mp 5109  df-ltp 5110  df-plpr 5184  df-mpr 5185  df-enr 5186  df-nr 5187  df-plr 5188  df-mr 5189  df-ltr 5190  df-0r 5191  df-1r 5192  df-m1r 5193  df-c 5260  df-0 5261  df-1 5262  df-i 5263  df-r 5264  df-plus 5265  df-mul 5266  df-lt 5267  df-sub 5376  df-neg 5378  df-pnf 5507  df-mnf 5508  df-xr 5509  df-ltxr 5510  df-le 5511  df-div 5723  df-n 5939  df-2 5984  df-3 5985  df-n0 6132  df-z 6168  df-seq1 6509

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