Theorem ip1cnilem4 8376

Description: Lemma for ip1cni 8379.

Hypotheses

Ref	Expression
ip1cni.1	|- X = (Base` U)
ip1cni.2	|- G = (+v` U)
ip1cni.7	|- P = (.i` U)
ip1cni.8	|- C = (IndMet` U)
ip1cni.d	|- D = (abs o. - )
ip1cni.j	|- J = (Open` C)
ip1cni.k	|- K = (Open` D)
ip1cni.f	|- F = {<.w, v>. | (w e. X /\ v = (wPA))}
ip1cni.9	|- U e. NrmCVec
ip1cni.a	|- A e. X
ip1cnilem.4	|- S = (.s` U)
ip1cnilem.6	|- N = (norm` U)
ip1cnilem.15	|- H = {<.u, t>. | (u e. X /\ t = (((i^k) x. (1 / 4)) x. ((N` (uG((i^k)SA)))^2)))}

Assertion

Ref	Expression
ip1cnilem4	|- (k e. NN -> H e. (J Cn K))

Distinct variable groups:   t,k,u,v,w,A   u,C,w   u,D,w   k,G,t,u,v,w   v,H,w   k,J,u,w   k,K,u,w   k,N,t,u,v,w   S,k,t,u,v,w   U,k,t,u,v,w   k,X,t,u,v,w

Proof of Theorem ip1cnilem4

Step	Hyp	Ref	Expression
1		ip1cni.1	. . . 4 |- X = (Base` U)
2		ip1cni.8	. . . 4 |- C = (IndMet` U)
3		ip1cni.9	. . . 4 |- U e. NrmCVec
4	1, 2, 3	imsbai 8322	. . 3 |- X = dom dom C
5		ip1cni.d	. . . 4 |- D = (abs o. - )
6	5	cnmetba 7903	. . 3 |- CC = dom dom D
7	2	imsmet 8324	. . . 4 |- (U e. NrmCVec -> C e. Met)
8	3, 7	ax-mp 7	. . 3 |- C e. Met
9	5	cnmet 7904	. . 3 |- D e. Met
10		ip1cni.j	. . 3 |- J = (Open` C)
11		ip1cni.k	. . 3 |- K = (Open` D)
12		eqid 1475	. . 3 |- (Open` {<.<.c, d>., f>. | ((c e. (CC X. CC) /\ d e. (CC X. CC)) /\ f = sup({((1st` c)D(1st` d)), ((2nd` c)D(2nd` d))}, RR, < ))}) = (Open` {<.<.c, d>., f>. | ((c e. (CC X. CC) /\ d e. (CC X. CC)) /\ f = sup({((1st` c)D(1st` d)), ((2nd` c)D(2nd` d))}, RR, < ))})
13		eqid 1475	. . 3 |- {<.<.c, d>., f>. | ((c e. (CC X. CC) /\ d e. (CC X. CC)) /\ f = sup({((1st` c)D(1st` d)), ((2nd` c)D(2nd` d))}, RR, < ))} = {<.<.c, d>., f>. | ((c e. (CC X. CC) /\ d e. (CC X. CC)) /\ f = sup({((1st` c)D(1st` d)), ((2nd` c)D(2nd` d))}, RR, < ))}
14	5, 13, 11, 12	mulcn 7988	. . 3 |- x. e. ((Open` {<.<.c, d>., f>. | ((c e. (CC X. CC) /\ d e. (CC X. CC)) /\ f = sup({((1st` c)D(1st` d)), ((2nd` c)D(2nd` d))}, RR, < ))}) Cn K)
15		ip1cnilem.15	. . . 4 |- H = {<.u, t>. | (u e. X /\ t = (((i^k) x. (1 / 4)) x. ((N` (uG((i^k)SA)))^2)))}
16		opreq1 3968	. . . . . . . . . . 11 |- (a = u -> (aG((i^k)SA)) = (uG((i^k)SA)))
17	16	fveq2d 3728	. . . . . . . . . 10 |- (a = u -> (N` (aG((i^k)SA))) = (N` (uG((i^k)SA))))
18	17	opreq1d 3975	. . . . . . . . 9 |- (a = u -> ((N` (aG((i^k)SA)))^2) = ((N` (uG((i^k)SA)))^2))
19		eqid 1475	. . . . . . . . 9 |- {<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))} = {<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))}
20		oprex 3983	. . . . . . . . 9 |- ((N` (uG((i^k)SA)))^2) e. V
21	18, 19, 20	fvopab4 3780	. . . . . . . 8 |- (u e. X -> ({<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))}` u) = ((N` (uG((i^k)SA)))^2))
22	21	opreq2d 3976	. . . . . . 7 |- (u e. X -> (((i^k) x. (1 / 4)) x. ({<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))}` u)) = (((i^k) x. (1 / 4)) x. ((N` (uG((i^k)SA)))^2)))
23	22	eqeq2d 1486	. . . . . 6 |- (u e. X -> (t = (((i^k) x. (1 / 4)) x. ({<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))}` u)) <-> t = (((i^k) x. (1 / 4)) x. ((N` (uG((i^k)SA)))^2))))
24	23	pm5.32i 645	. . . . 5 |- ((u e. X /\ t = (((i^k) x. (1 / 4)) x. ({<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))}` u))) <-> (u e. X /\ t = (((i^k) x. (1 / 4)) x. ((N` (uG((i^k)SA)))^2))))
25	24	opabbii 2671	. . . 4 |- {<.u, t>. | (u e. X /\ t = (((i^k) x. (1 / 4)) x. ({<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))}` u)))} = {<.u, t>. | (u e. X /\ t = (((i^k) x. (1 / 4)) x. ((N` (uG((i^k)SA)))^2)))}
26	15, 25	eqtr4 1498	. . 3 |- H = {<.u, t>. | (u e. X /\ t = (((i^k) x. (1 / 4)) x. ({<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))}` u)))}
27	4, 6, 6, 8, 9, 9, 9, 10, 11, 11, 12, 11, 13, 14, 26	opr2cn 7979	. 2 |- ((((i^k) x. (1 / 4)) e. CC /\ {<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))} e. (J Cn K)) -> H e. (J Cn K))
28		nnnn0t 6106	. . 3 |- (k e. NN -> k e. NN0)
29		axicn 5270	. . . 4 |- i e. CC
30		expclt 6581	. . . 4 |- ((i e. CC /\ k e. NN0) -> (i^k) e. CC)
31	29, 30	mpan 695	. . 3 |- (k e. NN0 -> (i^k) e. CC)
32		4re 5982	. . . . . 6 |- 4 e. RR
33	32	recn 5314	. . . . 5 |- 4 e. CC
34		4pos 5992	. . . . . 6 |- 0 < 4
35	32, 34	gt0ne0i 5617	. . . . 5 |- 4 =/= 0
36	33, 35	reccl 5713	. . . 4 |- (1 / 4) e. CC
37		axmulcl 5273	. . . 4 |- (((i^k) e. CC /\ (1 / 4) e. CC) -> ((i^k) x. (1 / 4)) e. CC)
38	36, 37	mpan2 696	. . 3 |- ((i^k) e. CC -> ((i^k) x. (1 / 4)) e. CC)
39	28, 31, 38	3syl 20	. 2 |- (k e. NN -> ((i^k) x. (1 / 4)) e. CC)
40		ip1cni.2	. . 3 |- G = (+v` U)
41		ip1cni.7	. . 3 |- P = (.i` U)
42		ip1cni.f	. . 3 |- F = {<.w, v>. | (w e. X /\ v = (wPA))}
43		ip1cni.a	. . 3 |- A e. X
44		ip1cnilem.4	. . 3 |- S = (.s` U)
45		ip1cnilem.6	. . 3 |- N = (norm` U)
46	1, 40, 41, 2, 5, 10, 11, 42, 3, 43, 44, 45, 19	ip1cnilem3 8375	. 2 |- (k e. NN -> {<.a, b>. | (a e. X /\ b = ((N` (aG((i^k)SA)))^2))} e. (J Cn K))
47	27, 39, 46	sylanc 471	1 |- (k e. NN -> H e. (J Cn K))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cpr 2410  {copab 2666   X. cxp 3168   o. ccom 3174  ` cfv 3182  (class class class)co 3963  {copab2 3964  1stc1st 4077  2ndc2nd 4078  supcsup 4573  CCcc 5232  RRcr 5233  1c1 5235  ici 5236   x. cmul 5239   - cmin 5292   / cdiv 5294  NNcn 5296  NN0cn0 5297   < clt 5486  2c2 5961  4c4 5963  ^cexp 6568  abscabs 6750   Cn ccn 7752  Metcme 7789  Opencopn 7792  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  normcnm 8209  IndMetcims 8210  .icip 8349

This theorem is referenced by:  ip1cnilem5 8377

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-reg 4593  ax-inf2 4625  ax-ac 4744

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-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  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-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  d