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Theorem tmdtopon 21866
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 21861 . 2 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
2 tgptopon.x . . 3 𝑋 = (Base‘𝐺)
3 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
42, 3istps 20719 . 2 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 208 1 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cfv 5876  Basecbs 15838  TopOpenctopn 16063  TopOnctopon 20696  TopSpctps 20717  TopMndctmd 21855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-top 20680  df-topon 20697  df-topsp 20718  df-tmd 21857
This theorem is referenced by:  cnmpt1plusg  21872  cnmpt2plusg  21873  tmdcn2  21874  tmdmulg  21877  tmdgsum  21880  tmdgsum2  21881  oppgtmd  21882  tmdlactcn  21887  submtmd  21889  ghmcnp  21899  prdstgpd  21909  tsmsxp  21939  mhmhmeotmd  29947
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