Theorem tgpsubcn 21894

Description: In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)

Hypotheses

Ref	Expression
tgpsubcn.2	|- J = ( TopOpen ` G )
tgpsubcn.3	|- .- = ( -g ` G )

Assertion

Ref	Expression
tgpsubcn	|- ( G e. TopGrp -> .- e. ( ( J tX J ) Cn J ) )

Proof of Theorem tgpsubcn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
2		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
3		eqid 2622	. . 3 |- ( invg ` G ) = ( invg ` G )
4		tgpsubcn.3	. . 3 |- .- = ( -g ` G )
5	1, 2, 3, 4	grpsubfval 17464	. 2 |- .- = ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( x ( +g ` G ) ( ( invg ` G ) ` y ) ) )
6		tgpsubcn.2	. . 3 |- J = ( TopOpen ` G )
7		tgptmd 21883	. . 3 |- ( G e. TopGrp -> G e. TopMnd )
8	6, 1	tgptopon 21886	. . 3 |- ( G e. TopGrp -> J e. (TopOn ` ( Base ` G ) ) )
9	8, 8	cnmpt1st 21471	. . 3 |- ( G e. TopGrp -> ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> x ) e. ( ( J tX J ) Cn J ) )
10	8, 8	cnmpt2nd 21472	. . . 4 |- ( G e. TopGrp -> ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> y ) e. ( ( J tX J ) Cn J ) )
11	6, 3	tgpinv 21889	. . . 4 |- ( G e. TopGrp -> ( invg ` G ) e. ( J Cn J ) )
12	8, 8, 10, 11	cnmpt21f 21475	. . 3 |- ( G e. TopGrp -> ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( ( invg ` G ) ` y ) ) e. ( ( J tX J ) Cn J ) )
13	6, 2, 7, 8, 8, 9, 12	cnmpt2plusg 21892	. 2 |- ( G e. TopGrp -> ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( x ( +g ` G ) ( ( invg ` G ) ` y ) ) ) e. ( ( J tX J ) Cn J ) )
14	5, 13	syl5eqel 2705	1 |- ( G e. TopGrp -> .- e. ( ( J tX J ) Cn J ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941   TopOpenctopn 16082   invgcminusg 17423   -gcsg 17424   Cn ccn 21028   tX ctx 21363   TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-plusf 17241  df-sbg 17427  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-tx 21365  df-tmd 21876  df-tgp 21877

This theorem is referenced by:  istgp2  21895  clssubg  21912  clsnsg  21913  tgphaus  21920  tgpt0  21922  qustgplem  21924