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Theorem tmdtopon 22690
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 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))

Proof of Theorem tmdtopon
StepHypRef Expression
1 tmdtps 22685 . 2 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
2 tgptopon.x . . 3 𝑋 = (Base‘𝐺)
3 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
42, 3istps 21543 . 2 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 221 1 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cfv 6328  Basecbs 16479  TopOpenctopn 16691  TopOnctopon 21519  TopSpctps 21541  TopMndctmd 22679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-top 21503  df-topon 21520  df-topsp 21542  df-tmd 22681
This theorem is referenced by:  cnmpt1plusg  22696  cnmpt2plusg  22697  tmdcn2  22698  tmdmulg  22701  tmdgsum  22704  tmdgsum2  22705  oppgtmd  22706  tmdlactcn  22711  submtmd  22713  ghmcnp  22724  prdstgpd  22734  tsmsxp  22764  mhmhmeotmd  31284
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