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Theorem tgpsubcn 17789
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 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2296 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
4 tgpsubcn.3 . . 3  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubfval 14540 . 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 17778 . . 3  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
86, 1tgptopon 17781 . . 3  |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  ( Base `  G
) ) )
98, 8cnmpt1st 17378 . . 3  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  x )  e.  ( ( J  tX  J )  Cn  J
) )
108, 8cnmpt2nd 17379 . . . 4  |-  ( G  e.  TopGrp  ->  ( x  e.  ( Base `  G
) ,  y  e.  ( Base `  G
)  |->  y )  e.  ( ( J  tX  J )  Cn  J
) )
116, 3tgpinv 17784 . . . 4  |-  ( G  e.  TopGrp  ->  ( inv g `  G )  e.  ( J  Cn  J ) )
128, 8, 10, 11cnmpt21f 17382 . . 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 17787 . 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 2380 1  |-  ( G  e.  TopGrp  ->  .-  e.  (
( J  tX  J
)  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   +g cplusg 13224   TopOpenctopn 13342   inv gcminusg 14379   -gcsg 14381    Cn ccn 16970    tX ctx 17271   TopGrpctgp 17770
This theorem is referenced by:  istgp2  17790  clssubg  17807  clsnsg  17808  tgphaus  17815  tgpt0  17817  divstgplem  17819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-topgen 13360  df-plusf 14384  df-sbg 14507  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-tx 17273  df-tmd 17771  df-tgp 17772
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