Theorem topgrpi 14972

Description: "Axioms" of a topological group.

Hypotheses

Ref	Expression
topgrpi.1	|- G = (1st` K)
topgrpi.2	|- J = (2nd` K)
topgrpi.3	|- N = (inv` G)

Assertion

Ref	Expression
topgrpi	|- (K e. TopGrp -> ((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ N e. (J Cn J)))

Proof of Theorem topgrpi

Step	Hyp	Ref	Expression
1		df-topgrp 14970	. . 3 |- TopGrp = {<.g, j>. | ((g e. Grp /\ j e. Top) /\ g e. ((j X.t j) Cn j) /\ (inv` g) e. (j Cn j))}
2	1	eleq2i 1961	. 2 |- (K e. TopGrp <-> K e. {<.g, j>. | ((g e. Grp /\ j e. Top) /\ g e. ((j X.t j) Cn j) /\ (inv` g) e. (j Cn j))})
3		topgrpi.1	. . . 4 |- G = (1st` K)
4		eqtr3 1907	. . . . . 6 |- ((G = (1st` K) /\ g = (1st` K)) -> G = g)
5	4	eqcomd 1889	. . . . 5 |- ((G = (1st` K) /\ g = (1st` K)) -> g = G)
6		eleq1 1957	. . . . . . 7 |- (g = G -> (g e. Grp <-> G e. Grp))
7	6	anbi1d 679	. . . . . 6 |- (g = G -> ((g e. Grp /\ j e. Top) <-> (G e. Grp /\ j e. Top)))
8		eleq1 1957	. . . . . 6 |- (g = G -> (g e. ((j X.t j) Cn j) <-> G e. ((j X.t j) Cn j)))
9		fveq2 4681	. . . . . . . 8 |- (g = G -> (inv` g) = (inv` G))
10		topgrpi.3	. . . . . . . 8 |- N = (inv` G)
11	9, 10	syl6eqr 1946	. . . . . . 7 |- (g = G -> (inv` g) = N)
12	11	eleq1d 1963	. . . . . 6 |- (g = G -> ((inv` g) e. (j Cn j) <-> N e. (j Cn j)))
13	7, 8, 12	3anbi123d 1168	. . . . 5 |- (g = G -> (((g e. Grp /\ j e. Top) /\ g e. ((j X.t j) Cn j) /\ (inv` g) e. (j Cn j)) <-> ((G e. Grp /\ j e. Top) /\ G e. ((j X.t j) Cn j) /\ N e. (j Cn j))))
14	5, 13	syl 12	. . . 4 |- ((G = (1st` K) /\ g = (1st` K)) -> (((g e. Grp /\ j e. Top) /\ g e. ((j X.t j) Cn j) /\ (inv` g) e. (j Cn j)) <-> ((G e. Grp /\ j e. Top) /\ G e. ((j X.t j) Cn j) /\ N e. (j Cn j))))
15	3, 14	mpan 759	. . 3 |- (g = (1st` K) -> (((g e. Grp /\ j e. Top) /\ g e. ((j X.t j) Cn j) /\ (inv` g) e. (j Cn j)) <-> ((G e. Grp /\ j e. Top) /\ G e. ((j X.t j) Cn j) /\ N e. (j Cn j))))
16		topgrpi.2	. . . 4 |- J = (2nd` K)
17		eqtr3 1907	. . . . . 6 |- ((J = (2nd` K) /\ j = (2nd` K)) -> J = j)
18	17	eqcomd 1889	. . . . 5 |- ((J = (2nd` K) /\ j = (2nd` K)) -> j = J)
19		eleq1 1957	. . . . . . 7 |- (j = J -> (j e. Top <-> J e. Top))
20	19	anbi2d 678	. . . . . 6 |- (j = J -> ((G e. Grp /\ j e. Top) <-> (G e. Grp /\ J e. Top)))
21		opreq12 4891	. . . . . . . . 9 |- ((j = J /\ j = J) -> (j X.t j) = (J X.t J))
22	21	anidms 480	. . . . . . . 8 |- (j = J -> (j X.t j) = (J X.t J))
23		opreq12 4891	. . . . . . . 8 |- (((j X.t j) = (J X.t J) /\ j = J) -> ((j X.t j) Cn j) = ((J X.t J) Cn J))
24	22, 23	mpancom 769	. . . . . . 7 |- (j = J -> ((j X.t j) Cn j) = ((J X.t J) Cn J))
25	24	eleq2d 1964	. . . . . 6 |- (j = J -> (G e. ((j X.t j) Cn j) <-> G e. ((J X.t J) Cn J)))
26		opreq12 4891	. . . . . . . 8 |- ((j = J /\ j = J) -> (j Cn j) = (J Cn J))
27	26	anidms 480	. . . . . . 7 |- (j = J -> (j Cn j) = (J Cn J))
28	27	eleq2d 1964	. . . . . 6 |- (j = J -> (N e. (j Cn j) <-> N e. (J Cn J)))
29	20, 25, 28	3anbi123d 1168	. . . . 5 |- (j = J -> (((G e. Grp /\ j e. Top) /\ G e. ((j X.t j) Cn j) /\ N e. (j Cn j)) <-> ((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ N e. (J Cn J))))
30	18, 29	syl 12	. . . 4 |- ((J = (2nd` K) /\ j = (2nd` K)) -> (((G e. Grp /\ j e. Top) /\ G e. ((j X.t j) Cn j) /\ N e. (j Cn j)) <-> ((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ N e. (J Cn J))))
31	16, 30	mpan 759	. . 3 |- (j = (2nd` K) -> (((G e. Grp /\ j e. Top) /\ G e. ((j X.t j) Cn j) /\ N e. (j Cn j)) <-> ((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ N e. (J Cn J))))
32	15, 31	elopabi 5059	. 2 |- (K e. {<.g, j>. | ((g e. Grp /\ j e. Top) /\ g e. ((j X.t j) Cn j) /\ (inv` g) e. (j Cn j))} -> ((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ N e. (J Cn J)))
33	2, 32	sylbi 216	1 |- (K e. TopGrp -> ((G e. Grp /\ J e. Top) /\ G e. ((J X.t J) Cn J) /\ N e. (J Cn J)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {copab 3395  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Topctop 8857   X.t ctx 8930   Cn ccn 9028  Grpcgr 9311  invcgn 9313  TopGrpctopgrp 14969

This theorem is referenced by:  topgrpinv 14973  topgrpbs 14974  topgrpcn 14975  topgrpgrp 14976  topgrptop 14977

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-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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  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-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-topgrp 14970

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