Theorem topgrpsubcnlem 14981

Description: Lemma for topgrpsubcn 14982.

Hypotheses

Ref	Expression
topgrpsubcn.1	|- G = (1st` K)
topgrpsubcn.2	|- J = (2nd` K)
topgrpsubcn.3	|- D = ( /g ` G)
topgrpsubcn.4	|- K e. TopGrp

Assertion

Ref	Expression
topgrpsubcnlem	|- D e. ((J X.t J) Cn J)

Proof of Theorem topgrpsubcnlem

Step	Hyp	Ref	Expression
1		topgrpsubcn.4	. . 3 |- K e. TopGrp
2		topgrpsubcn.1	. . . . 5 |- G = (1st` K)
3	2	topgrpgrp 14976	. . . 4 |- (K e. TopGrp -> G e. Grp)
4		eqid 1884	. . . . 5 |- ran G = ran G
5		eqid 1884	. . . . 5 |- (inv` G) = (inv` G)
6		topgrpsubcn.3	. . . . 5 |- D = ( /g ` G)
7	4, 5, 6	grpdivfval 9366	. . . 4 |- (G e. Grp -> D = {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))})
8	3, 7	syl 12	. . 3 |- (K e. TopGrp -> D = {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))})
9	1, 8	ax-mp 7	. 2 |- D = {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))}
10	4	grpfo 9323	. . . . . 6 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
11		fofn 4619	. . . . . 6 |- (G:(ran G X. ran G)-onto->ran G -> G Fn (ran G X. ran G))
12	3, 10, 11	3syl 24	. . . . 5 |- (K e. TopGrp -> G Fn (ran G X. ran G))
13		simpl 346	. . . . . 6 |- ((x e. ran G /\ y e. ran G) -> x e. ran G)
14	4, 5	grpinvcl 9352	. . . . . . . . . 10 |- ((G e. Grp /\ y e. ran G) -> ((inv` G)` y) e. ran G)
15	14	ex 402	. . . . . . . . 9 |- (G e. Grp -> (y e. ran G -> ((inv` G)` y) e. ran G))
16	3, 15	syl 12	. . . . . . . 8 |- (K e. TopGrp -> (y e. ran G -> ((inv` G)` y) e. ran G))
17	1, 16	ax-mp 7	. . . . . . 7 |- (y e. ran G -> ((inv` G)` y) e. ran G)
18	17	adantl 424	. . . . . 6 |- ((x e. ran G /\ y e. ran G) -> ((inv` G)` y) e. ran G)
19		eqid 1884	. . . . . 6 |- {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)} = {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)}
20		eqid 1884	. . . . . 6 |- {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))} = {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))}
21	13, 18, 19, 20	oprab2co 10160	. . . . 5 |- (G Fn (ran G X. ran G) -> {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))} = (G o. {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)}))
22	12, 21	syl 12	. . . 4 |- (K e. TopGrp -> {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))} = (G o. {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)}))
23	1, 22	ax-mp 7	. . 3 |- {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))} = (G o. {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)})
24		topgrpsubcn.2	. . . . . . . . 9 |- J = (2nd` K)
25	24	topgrptop 14977	. . . . . . . 8 |- (K e. TopGrp -> J e. Top)
26		eqid 1884	. . . . . . . . 9 |- (J X.t J) = (J X.t J)
27	26	txtop 8934	. . . . . . . 8 |- ((J e. Top /\ J e. Top) -> (J X.t J) e. Top)
28	25, 25, 27	syl11anc 524	. . . . . . 7 |- (K e. TopGrp -> (J X.t J) e. Top)
29	28, 28, 25	3jca 1050	. . . . . 6 |- (K e. TopGrp -> ((J X.t J) e. Top /\ (J X.t J) e. Top /\ J e. Top))
30	2, 24	topgrpbs 14974	. . . . . . . . . . 11 |- (K e. TopGrp -> ran G = U.J)
31	1, 30	ax-mp 7	. . . . . . . . . 10 |- ran G = U.J
32	31	idcn 9042	. . . . . . . . 9 |- (J e. Top -> ( _I |` ran G) e. (J Cn J))
33	25, 32	syl 12	. . . . . . . 8 |- (K e. TopGrp -> ( _I |` ran G) e. (J Cn J))
34	2, 24, 5	topgrpinv 14973	. . . . . . . 8 |- (K e. TopGrp -> (inv` G) e. (J Cn J))
35		eqid 1884	. . . . . . . . 9 |- {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.(( _I |` ran G)` x), ((inv` G)` y)>.)} = {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.(( _I |` ran G)` x), ((inv` G)` y)>.)}
36	26, 26, 31, 31, 35	2txcn 10229	. . . . . . . 8 |- (((J e. Top /\ J e. Top) /\ (J e. Top /\ J e. Top) /\ (( _I |` ran G) e. (J Cn J) /\ (inv` G) e. (J Cn J))) -> {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.(( _I |` ran G)` x), ((inv` G)` y)>.)} e. ((J X.t J) Cn (J X.t J)))
37	25, 25, 25, 25, 33, 34, 36	syl222anc 1116	. . . . . . 7 |- (K e. TopGrp -> {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.(( _I |` ran G)` x), ((inv` G)` y)>.)} e. ((J X.t J) Cn (J X.t J)))
38		fvresi 4819	. . . . . . . . . . . . 13 |- (x e. ran G -> (( _I |` ran G)` x) = x)
39	38	adantr 425	. . . . . . . . . . . 12 |- ((x e. ran G /\ y e. ran G) -> (( _I |` ran G)` x) = x)
40	39	eqcomd 1889	. . . . . . . . . . 11 |- ((x e. ran G /\ y e. ran G) -> x = (( _I |` ran G)` x))
41	40	opeq1d 3164	. . . . . . . . . 10 |- ((x e. ran G /\ y e. ran G) -> <.x, ((inv` G)` y)>. = <.(( _I |` ran G)` x), ((inv` G)` y)>.)
42	41	eqeq2d 1895	. . . . . . . . 9 |- ((x e. ran G /\ y e. ran G) -> (u = <.x, ((inv` G)` y)>. <-> u = <.(( _I |` ran G)` x), ((inv` G)` y)>.))
43	42	pm5.32i 707	. . . . . . . 8 |- (((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.) <-> ((x e. ran G /\ y e. ran G) /\ u = <.(( _I |` ran G)` x), ((inv` G)` y)>.))
44	43	oprabbii 4923	. . . . . . 7 |- {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)} = {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.(( _I |` ran G)` x), ((inv` G)` y)>.)}
45	37, 44	syl5eqel 1975	. . . . . 6 |- (K e. TopGrp -> {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)} e. ((J X.t J) Cn (J X.t J)))
46	2, 24	topgrpcn 14975	. . . . . 6 |- (K e. TopGrp -> G e. ((J X.t J) Cn J))
47	29, 45, 46	jca32 312	. . . . 5 |- (K e. TopGrp -> (((J X.t J) e. Top /\ (J X.t J) e. Top /\ J e. Top) /\ ({<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)} e. ((J X.t J) Cn (J X.t J)) /\ G e. ((J X.t J) Cn J))))
48	1, 47	ax-mp 7	. . . 4 |- (((J X.t J) e. Top /\ (J X.t J) e. Top /\ J e. Top) /\ ({<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)} e. ((J X.t J) Cn (J X.t J)) /\ G e. ((J X.t J) Cn J)))
49		cnco 9045	. . . 4 |- ((((J X.t J) e. Top /\ (J X.t J) e. Top /\ J e. Top) /\ ({<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)} e. ((J X.t J) Cn (J X.t J)) /\ G e. ((J X.t J) Cn J))) -> (G o. {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)}) e. ((J X.t J) Cn J))
50	48, 49	ax-mp 7	. . 3 |- (G o. {<.<.x, y>., u>. | ((x e. ran G /\ y e. ran G) /\ u = <.x, ((inv` G)` y)>.)}) e. ((J X.t J) Cn J)
51	23, 50	eqeltri 1967	. 2 |- {<.<.x, y>., z>. | ((x e. ran G /\ y e. ran G) /\ z = (xG((inv` G)` y)))} e. ((J X.t J) Cn J)
52	9, 51	eqeltri 1967	1 |- D e. ((J X.t J) Cn J)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  <.cop 3046  U.cuni 3177   _I cid 3582   X. cxp 3984  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  Topctop 8857   X.t ctx 8930   Cn ccn 9028  Grpcgr 9311  invcgn 9313   /g cgs 9314  TopGrpctopgrp 14969

This theorem is referenced by:  topgrpsubcn 14982

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-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-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  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-1st 5020  df-2nd 5021  df-map 5383  df-top 8861  df-bases 8863  df-topgen 8864  df-tx 8931  df-cn 9030  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-topgrp 14970

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