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Theorem tgpsubcn 19666
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 2443 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2443 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2443 . . 3  |-  ( invg `  G )  =  ( invg `  G )
4 tgpsubcn.3 . . 3  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubfval 15585 . 2  |-  .-  =  ( x  e.  ( Base `  G ) ,  y  e.  ( Base `  G )  |->  ( x ( +g  `  G
) ( ( invg `  G ) `
 y ) ) )
6 tgpsubcn.2 . . 3  |-  J  =  ( TopOpen `  G )
7 tgptmd 19655 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
86, 1tgptopon 19658 . . 3  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
98, 8cnmpt1st 19246 . . 3  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  x )  e.  ( ( J  tX  J )  Cn  J
) )
108, 8cnmpt2nd 19247 . . . 4  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  y )  e.  ( ( J  tX  J )  Cn  J
) )
116, 3tgpinv 19661 . . . 4  |-  ( G  e.  TopGrp  ->  ( invg `  G )  e.  ( J  Cn  J ) )
128, 8, 10, 11cnmpt21f 19250 . . 3  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( ( invg `  G ) `
 y ) )  e.  ( ( J 
tX  J )  Cn  J ) )
136, 2, 7, 8, 8, 9, 12cnmpt2plusg 19664 . 2  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  ( x ( +g  `  G ) ( ( invg `  G ) `  y
) ) )  e.  ( ( J  tX  J )  Cn  J
) )
145, 13syl5eqel 2527 1  |-  ( G  e.  TopGrp  ->  .-  e.  (
( J  tX  J
)  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Basecbs 14179   +g cplusg 14243   TopOpenctopn 14365   invgcminusg 15416   -gcsg 15418    Cn ccn 18833    tX ctx 19138   TopGrpctgp 19647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-map 7221  df-topgen 14387  df-plusf 15421  df-sbg 15552  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cn 18836  df-tx 19140  df-tmd 19648  df-tgp 19649
This theorem is referenced by:  istgp2  19667  clssubg  19684  clsnsg  19685  tgphaus  19692  tgpt0  19694  divstgplem  19696
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