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Theorem tmdtopon 20308
Description: The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tgpcn.j  |-  J  =  ( TopOpen `  G )
tgptopon.x  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
tmdtopon  |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  X ) )

Proof of Theorem tmdtopon
StepHypRef Expression
1 tmdtps 20303 . 2  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
2 tgptopon.x . . 3  |-  X  =  ( Base `  G
)
3 tgpcn.j . . 3  |-  J  =  ( TopOpen `  G )
42, 3istps 19197 . 2  |-  ( G  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
51, 4sylib 196 1  |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5579   Basecbs 14479   TopOpenctopn 14666  TopOnctopon 19155   TopSpctps 19157  TopMndctmd 20297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-top 19159  df-topon 19162  df-topsp 19163  df-tmd 20299
This theorem is referenced by:  cnmpt1plusg  20314  cnmpt2plusg  20315  tmdcn2  20316  tmdmulg  20319  tmdgsum  20322  tmdgsum2  20323  oppgtmd  20324  tmdlactcn  20329  submtmd  20331  ghmcnp  20341  prdstgpd  20351  tsmsxp  20385  mhmhmeotmd  27531
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