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Theorem tgpsubcn 18108
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.
StepHypRef Expression
1 eqid 2435 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2435 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2435 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
4 tgpsubcn.3 . . 3  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubfval 14835 . 2  |-  .-  =  ( x  e.  ( Base `  G ) ,  y  e.  ( Base `  G )  |->  ( x ( +g  `  G
) ( ( inv g `  G ) `
 y ) ) )
6 tgpsubcn.2 . . 3  |-  J  =  ( TopOpen `  G )
7 tgptmd 18097 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
86, 1tgptopon 18100 . . 3  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
98, 8cnmpt1st 17688 . . 3  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  x )  e.  ( ( J  tX  J )  Cn  J
) )
108, 8cnmpt2nd 17689 . . . 4  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  y )  e.  ( ( J  tX  J )  Cn  J
) )
116, 3tgpinv 18103 . . . 4  |-  ( G  e.  TopGrp  ->  ( inv g `  G )  e.  ( J  Cn  J ) )
128, 8, 10, 11cnmpt21f 17692 . . 3  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( ( inv g `  G ) `
 y ) )  e.  ( ( J 
tX  J )  Cn  J ) )
136, 2, 7, 8, 8, 9, 12cnmpt2plusg 18106 . 2  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) ( ( inv g `  G ) `  y
) ) )  e.  ( ( J  tX  J )  Cn  J
) )
145, 13syl5eqel 2519 1  |-  ( G  e.  TopGrp  ->  .-  e.  (
( J  tX  J
)  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   Basecbs 13457   +g cplusg 13517   TopOpenctopn 13637   inv gcminusg 14674   -gcsg 14676    Cn ccn 17276    tX ctx 17580   TopGrpctgp 18089
This theorem is referenced by:  istgp2  18109  clssubg  18126  clsnsg  18127  tgphaus  18134  tgpt0  18136  divstgplem  18138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-map 7011  df-topgen 13655  df-plusf 14679  df-sbg 14802  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cn 17279  df-tx 17582  df-tmd 18090  df-tgp 18091
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