Theorem va1cnlem 9684

Description: Lemma for va1cn 9685.

Hypotheses

Ref	Expression
va1cn.1	|- X = (BaseSet` U)
va1cn.2	|- G = (+v` U)
va1cn.8	|- D = (IndMet` U)
va1cn.j	|- J = (Open` D)
va1cn.f	|- F = {<.w, v>. | (w e. X /\ v = (wGA))}
va1cn.9	|- U e. NrmCVec
va1cnlem.6	|- N = (norm` U)

Assertion

Ref	Expression
va1cnlem	|- (A e. X -> F e. (J Cn J))

Distinct variable groups:   w,v,A   v,G,w   v,X,w

Proof of Theorem va1cnlem

Step	Hyp	Ref	Expression
1		va1cn.9	. . . 4 |- U e. NrmCVec
2		va1cn.8	. . . . 5 |- D = (IndMet` U)
3	2	imsmet 9656	. . . 4 |- (U e. NrmCVec -> D e. Met)
4	1, 3	ax-mp 7	. . 3 |- D e. Met
5		va1cn.1	. . . . 5 |- X = (BaseSet` U)
6	5, 2, 1	imsbai 9654	. . . 4 |- X = dom dom D
7		va1cn.j	. . . 4 |- J = (Open` D)
8	6, 7, 6, 7	metcn 9167	. . 3 |- ((D e. Met /\ D e. Met) -> (F e. (J Cn J) <-> (F:X-->X /\ A.r e. X A.s e. RR (0 < s -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s))))))
9	4, 4, 8	mp2an 761	. 2 |- (F e. (J Cn J) <-> (F:X-->X /\ A.r e. X A.s e. RR (0 < s -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s)))))
10		va1cn.2	. . . . . . 7 |- G = (+v` U)
11	5, 10	nvgcl 9571	. . . . . 6 |- ((U e. NrmCVec /\ w e. X /\ A e. X) -> (wGA) e. X)
12	1, 11	mp3an1 1178	. . . . 5 |- ((w e. X /\ A e. X) -> (wGA) e. X)
13	12	ancoms 484	. . . 4 |- ((A e. X /\ w e. X) -> (wGA) e. X)
14	13	r19.21aiva 2176	. . 3 |- (A e. X -> A.w e. X (wGA) e. X)
15		va1cn.f	. . . 4 |- F = {<.w, v>. | (w e. X /\ v = (wGA))}
16	15	fopab2 4796	. . 3 |- (A.w e. X (wGA) e. X <-> F:X-->X)
17	14, 16	sylib 215	. 2 |- (A e. X -> F:X-->X)
18		idd 75	. . . . . . . . 9 |- (u e. X -> ((rDu) < s -> (rDu) < s))
19	18	rgen 2159	. . . . . . . 8 |- A.u e. X ((rDu) < s -> (rDu) < s)
20		breq2 3342	. . . . . . . . . 10 |- (t = s -> (0 < t <-> 0 < s))
21		breq2 3342	. . . . . . . . . . . 12 |- (t = s -> ((rDu) < t <-> (rDu) < s))
22	21	imbi1d 675	. . . . . . . . . . 11 |- (t = s -> (((rDu) < t -> (rDu) < s) <-> ((rDu) < s -> (rDu) < s)))
23	22	ralbidv 2123	. . . . . . . . . 10 |- (t = s -> (A.u e. X ((rDu) < t -> (rDu) < s) <-> A.u e. X ((rDu) < s -> (rDu) < s)))
24	20, 23	anbi12d 690	. . . . . . . . 9 |- (t = s -> ((0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s)) <-> (0 < s /\ A.u e. X ((rDu) < s -> (rDu) < s))))
25	24	rcla4ev 2381	. . . . . . . 8 |- ((s e. RR /\ (0 < s /\ A.u e. X ((rDu) < s -> (rDu) < s))) -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s)))
26	19, 25	mpanr2 776	. . . . . . 7 |- ((s e. RR /\ 0 < s) -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s)))
27	26	adantll 428	. . . . . 6 |- (((r e. X /\ s e. RR) /\ 0 < s) -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s)))
28	27	adantl 424	. . . . 5 |- ((A e. X /\ ((r e. X /\ s e. RR) /\ 0 < s)) -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s)))
29		opreq1 4889	. . . . . . . . . . . . . . . 16 |- (w = r -> (wGA) = (rGA))
30		oprex 4907	. . . . . . . . . . . . . . . 16 |- (rGA) e. _V
31	29, 15, 30	fvopab4 4743	. . . . . . . . . . . . . . 15 |- (r e. X -> (F` r) = (rGA))
32		opreq1 4889	. . . . . . . . . . . . . . . 16 |- (w = u -> (wGA) = (uGA))
33		oprex 4907	. . . . . . . . . . . . . . . 16 |- (uGA) e. _V
34	32, 15, 33	fvopab4 4743	. . . . . . . . . . . . . . 15 |- (u e. X -> (F` u) = (uGA))
35	31, 34	opreqan12d 4902	. . . . . . . . . . . . . 14 |- ((r e. X /\ u e. X) -> ((F` r)D(F` u)) = ((rGA)D(uGA)))
36	35	adantll 428	. . . . . . . . . . . . 13 |- (((A e. X /\ r e. X) /\ u e. X) -> ((F` r)D(F` u)) = ((rGA)D(uGA)))
37	10	nvgrp 9568	. . . . . . . . . . . . . . . . . . 19 |- (U e. NrmCVec -> G e. Grp)
38	1, 37	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- G e. Grp
39	5, 10	bafval 9555	. . . . . . . . . . . . . . . . . . 19 |- X = ran G
40		eqid 1884	. . . . . . . . . . . . . . . . . . . 20 |- (-v` U) = (-v` U)
41	10, 40	vsfval 9586	. . . . . . . . . . . . . . . . . . 19 |- (-v` U) = ( /g ` G)
42	39, 41	grppnpcan2 9377	. . . . . . . . . . . . . . . . . 18 |- ((G e. Grp /\ (r e. X /\ u e. X /\ A e. X)) -> ((rGA)(-v` U)(uGA)) = (r(-v` U)u))
43	38, 42	mpan 759	. . . . . . . . . . . . . . . . 17 |- ((r e. X /\ u e. X /\ A e. X) -> ((rGA)(-v` U)(uGA)) = (r(-v` U)u))
44	43	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- ((r e. X /\ u e. X /\ A e. X) -> (N` ((rGA)(-v` U)(uGA))) = (N` (r(-v` U)u)))
45	5, 10	nvgcl 9571	. . . . . . . . . . . . . . . . . . 19 |- ((U e. NrmCVec /\ r e. X /\ A e. X) -> (rGA) e. X)
46	1, 45	mp3an1 1178	. . . . . . . . . . . . . . . . . 18 |- ((r e. X /\ A e. X) -> (rGA) e. X)
47	46	3adant2 895	. . . . . . . . . . . . . . . . 17 |- ((r e. X /\ u e. X /\ A e. X) -> (rGA) e. X)
48	5, 10	nvgcl 9571	. . . . . . . . . . . . . . . . . . 19 |- ((U e. NrmCVec /\ u e. X /\ A e. X) -> (uGA) e. X)
49	1, 48	mp3an1 1178	. . . . . . . . . . . . . . . . . 18 |- ((u e. X /\ A e. X) -> (uGA) e. X)
50	49	3adant1 894	. . . . . . . . . . . . . . . . 17 |- ((r e. X /\ u e. X /\ A e. X) -> (uGA) e. X)
51		va1cnlem.6	. . . . . . . . . . . . . . . . . . 19 |- N = (norm` U)
52	5, 40, 51, 2	imsdval 9649	. . . . . . . . . . . . . . . . . 18 |- ((U e. NrmCVec /\ (rGA) e. X /\ (uGA) e. X) -> ((rGA)D(uGA)) = (N` ((rGA)(-v` U)(uGA))))
53	1, 52	mp3an1 1178	. . . . . . . . . . . . . . . . 17 |- (((rGA) e. X /\ (uGA) e. X) -> ((rGA)D(uGA)) = (N` ((rGA)(-v` U)(uGA))))
54	47, 50, 53	syl11anc 524	. . . . . . . . . . . . . . . 16 |- ((r e. X /\ u e. X /\ A e. X) -> ((rGA)D(uGA)) = (N` ((rGA)(-v` U)(uGA))))
55	5, 40, 51, 2	imsdval 9649	. . . . . . . . . . . . . . . . . 18 |- ((U e. NrmCVec /\ r e. X /\ u e. X) -> (rDu) = (N` (r(-v` U)u)))
56	1, 55	mp3an1 1178	. . . . . . . . . . . . . . . . 17 |- ((r e. X /\ u e. X) -> (rDu) = (N` (r(-v` U)u)))
57	56	3adant3 896	. . . . . . . . . . . . . . . 16 |- ((r e. X /\ u e. X /\ A e. X) -> (rDu) = (N` (r(-v` U)u)))
58	44, 54, 57	3eqtr4d 1937	. . . . . . . . . . . . . . 15 |- ((r e. X /\ u e. X /\ A e. X) -> ((rGA)D(uGA)) = (rDu))
59	58	3comr 1076	. . . . . . . . . . . . . 14 |- ((A e. X /\ r e. X /\ u e. X) -> ((rGA)D(uGA)) = (rDu))
60	59	3expa 1067	. . . . . . . . . . . . 13 |- (((A e. X /\ r e. X) /\ u e. X) -> ((rGA)D(uGA)) = (rDu))
61	36, 60	eqtrd 1925	. . . . . . . . . . . 12 |- (((A e. X /\ r e. X) /\ u e. X) -> ((F` r)D(F` u)) = (rDu))
62	61	breq1d 3348	. . . . . . . . . . 11 |- (((A e. X /\ r e. X) /\ u e. X) -> (((F` r)D(F` u)) < s <-> (rDu) < s))
63	62	imbi2d 674	. . . . . . . . . 10 |- (((A e. X /\ r e. X) /\ u e. X) -> (((rDu) < t -> ((F` r)D(F` u)) < s) <-> ((rDu) < t -> (rDu) < s)))
64	63	ralbidva 2119	. . . . . . . . 9 |- ((A e. X /\ r e. X) -> (A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s) <-> A.u e. X ((rDu) < t -> (rDu) < s)))
65	64	anbi2d 678	. . . . . . . 8 |- ((A e. X /\ r e. X) -> ((0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s)) <-> (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s))))
66	65	rexbidv 2124	. . . . . . 7 |- ((A e. X /\ r e. X) -> (E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s)) <-> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s))))
67	66	adantrr 431	. . . . . 6 |- ((A e. X /\ (r e. X /\ s e. RR)) -> (E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s)) <-> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s))))
68	67	adantrr 431	. . . . 5 |- ((A e. X /\ ((r e. X /\ s e. RR) /\ 0 < s)) -> (E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s)) <-> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> (rDu) < s))))
69	28, 68	mpbird 213	. . . 4 |- ((A e. X /\ ((r e. X /\ s e. RR) /\ 0 < s)) -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s)))
70	69	exp32 408	. . 3 |- (A e. X -> ((r e. X /\ s e. RR) -> (0 < s -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s)))))
71	70	r19.21aivv 2183	. 2 |- (A e. X -> A.r e. X A.s e. RR (0 < s -> E.t e. RR (0 < t /\ A.u e. X ((rDu) < t -> ((F` r)D(F` u)) < s))))
72	9, 17, 71	sylanbrc 527	1 |- (A e. X -> F e. (J Cn J))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   < clt 6653   Cn ccn 9028  Metcme 9066  Opencopn 9069  Grpcgr 9311  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  -vcnsb 9540  normcnm 9541  IndMetcims 9542

This theorem is referenced by:  va1cn 9685

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552

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