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Theorem tgpsubcn 17773
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 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2283 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
4 tgpsubcn.3 . . 3  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubfval 14524 . 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 17762 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
86, 1tgptopon 17765 . . 3  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
98, 8cnmpt1st 17362 . . 3  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  x )  e.  ( ( J  tX  J )  Cn  J
) )
108, 8cnmpt2nd 17363 . . . 4  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  y )  e.  ( ( J  tX  J )  Cn  J
) )
116, 3tgpinv 17768 . . . 4  |-  ( G  e.  TopGrp  ->  ( inv g `  G )  e.  ( J  Cn  J ) )
128, 8, 10, 11cnmpt21f 17366 . . 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 17771 . 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 2367 1  |-  ( G  e.  TopGrp  ->  .-  e.  (
( J  tX  J
)  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   TopOpenctopn 13326   inv gcminusg 14363   -gcsg 14365    Cn ccn 16954    tX ctx 17255   TopGrpctgp 17754
This theorem is referenced by:  istgp2  17774  clssubg  17791  clsnsg  17792  tgphaus  17799  tgpt0  17801  divstgplem  17803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-plusf 14368  df-sbg 14491  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-tx 17257  df-tmd 17755  df-tgp 17756
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