Theorem tmdtopon 24005

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

Step	Hyp	Ref	Expression
1		tmdtps 24000	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
2		tgptopon.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
3		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
4	2, 3	istps 22856	. 2 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
5	1, 4	sylib 217	1 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ‘cfv 6553  Basecbs 17187  TopOpenctopn 17410  TopOnctopon 22832  TopSpctps 22854  TopMndctmd 23994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-top 22816  df-topon 22833  df-topsp 22855  df-tmd 23996

This theorem is referenced by:  cnmpt1plusg  24011  cnmpt2plusg  24012  tmdcn2  24013  tmdmulg  24016  tmdgsum  24019  tmdgsum2  24020  oppgtmd  24021  tmdlactcn  24026  submtmd  24028  ghmcnp  24039  prdstgpd  24049  tsmsxp  24079  mhmhmeotmd  33561