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Theorem tmdtopon 22106
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 22101 . 2 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
2 tgptopon.x . . 3 𝑋 = (Base‘𝐺)
3 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
42, 3istps 20960 . 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 1632  wcel 2139  cfv 6049  Basecbs 16079  TopOpenctopn 16304  TopOnctopon 20937  TopSpctps 20958  TopMndctmd 22095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-top 20921  df-topon 20938  df-topsp 20959  df-tmd 22097
This theorem is referenced by:  cnmpt1plusg  22112  cnmpt2plusg  22113  tmdcn2  22114  tmdmulg  22117  tmdgsum  22120  tmdgsum2  22121  oppgtmd  22122  tmdlactcn  22127  submtmd  22129  ghmcnp  22139  prdstgpd  22149  tsmsxp  22179  mhmhmeotmd  30303
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