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Theorem tgptmd 17778
Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tgptmd  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )

Proof of Theorem tgptmd
StepHypRef Expression
1 eqid 2296 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2296 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
31, 2istgp 17776 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  ( inv g `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) ) )
43simp2bi 971 1  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   ` cfv 5271  (class class class)co 5874   TopOpenctopn 13342   Grpcgrp 14378   inv gcminusg 14379    Cn ccn 16970  TopMndctmd 17769   TopGrpctgp 17770
This theorem is referenced by:  tgptps  17779  tgpcn  17783  tgpsubcn  17789  tgpmulg  17792  oppgtgp  17797  tgplacthmeo  17802  subgtgp  17804  clsnsg  17808  tgpt0  17817  prdstgpd  17823  tsmssub  17847  tsmsxp  17853  trgtmd2  17867  nlmtlm  18220
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-tgp 17772
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