Theorem topgrpsubcn 14982

Description: In a topological group, the "subtraction" (or "division") is continuous. Bourbaki TG III.1 axiom GT'

Hypotheses

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

Assertion

Ref	Expression
topgrpsubcn	|- (K e. TopGrp -> D e. ((J X.t J) Cn J))

Proof of Theorem topgrpsubcn

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . 6 |- (K = if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.) -> (1st` K) = (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)))
2	1	fveq2d 4685	. . . . 5 |- (K = if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.) -> ( /g ` (1st` K)) = ( /g ` (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))))
3		fveq2 4681	. . . . . . 7 |- (K = if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.) -> (2nd` K) = (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)))
4	3, 3	opreq12d 4900	. . . . . 6 |- (K = if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.) -> ((2nd` K) X.t (2nd` K)) = ((2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)) X.t (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))))
5	4, 3	opreq12d 4900	. . . . 5 |- (K = if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.) -> (((2nd` K) X.t (2nd` K)) Cn (2nd` K)) = (((2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)) X.t (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))) Cn (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))))
6	2, 5	eleq12d 1965	. . . 4 |- (K = if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.) -> (( /g ` (1st` K)) e. (((2nd` K) X.t (2nd` K)) Cn (2nd` K)) <-> ( /g ` (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))) e. (((2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)) X.t (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))) Cn (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)))))
7		eqid 1884	. . . . 5 |- (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)) = (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))
8		eqid 1884	. . . . 5 |- (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)) = (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))
9		eqid 1884	. . . . 5 |- ( /g ` (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))) = ( /g ` (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)))
10		0ex 3446	. . . . . . 7 |- (/) e. _V
11	10	extopgrp 14980	. . . . . 6 |- <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>. e. TopGrp
12	11	elimel 3025	. . . . 5 |- if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.) e. TopGrp
13	7, 8, 9, 12	topgrpsubcnlem 14981	. . . 4 |- ( /g ` (1st` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))) e. (((2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)) X.t (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.))) Cn (2nd` if(K e. TopGrp, K, <.{<.<.(/), (/)>., (/)>.}, {(/), {(/)}}>.)))
14	6, 13	dedth 3011	. . 3 |- (K e. TopGrp -> ( /g ` (1st` K)) e. (((2nd` K) X.t (2nd` K)) Cn (2nd` K)))
15		topgrpsubcn.2	. . . . 5 |- J = (2nd` K)
16	15, 15	opreq12i 4894	. . . 4 |- (J X.t J) = ((2nd` K) X.t (2nd` K))
17	16, 15	opreq12i 4894	. . 3 |- ((J X.t J) Cn J) = (((2nd` K) X.t (2nd` K)) Cn (2nd` K))
18	14, 17	syl6eleqr 1982	. 2 |- (K e. TopGrp -> ( /g ` (1st` K)) e. ((J X.t J) Cn J))
19		topgrpsubcn.3	. . 3 |- D = ( /g ` G)
20		topgrpsubcn.1	. . . 4 |- G = (1st` K)
21	20	fveq2i 4684	. . 3 |- ( /g ` G) = ( /g ` (1st` K))
22	19, 21	eqtri 1908	. 2 |- D = ( /g ` (1st` K))
23	18, 22	syl5eqel 1975	1 |- (K e. TopGrp -> D e. ((J X.t J) Cn J))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875  ifcif 2982  {csn 3044  {cpr 3045  <.cop 3046  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019   X.t ctx 8930   Cn ccn 9028   /g cgs 9314  TopGrpctopgrp 14969

This theorem is referenced by:  trhom 14983

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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