Theorem tmdlactcn 24228

Description: The left group action of element 𝐴 in a topological monoid 𝐺 is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
tgplacthmeo.1	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥))
tgplacthmeo.2	⊢ 𝑋 = (Base‘𝐺)
tgplacthmeo.3	⊢ + = (+g‘𝐺)
tgplacthmeo.4	⊢ 𝐽 = (TopOpen‘𝐺)

Assertion

Ref	Expression
tmdlactcn	⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝑥,𝐽   𝑥, +   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem tmdlactcn

Step	Hyp	Ref	Expression
1		tgplacthmeo.1	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥))
2		tgplacthmeo.4	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
3		tgplacthmeo.3	. . 3 ⊢ + = (+g‘𝐺)
4		simpl 487	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ TopMnd)
5		tgplacthmeo.2	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
6	2, 5	tmdtopon 24207	. . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
7	6	adantr 485	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8		simpr 489	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
9	7, 7, 8	cnmptc 23788	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽))
10	7	cnmptid 23787	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
11	2, 3, 4, 7, 9, 10	cnmpt1plusg 24213	. 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽))
12	1, 11	eqeltrid 2873	1 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  +_gcplusg 17310  TopOpenctopn 17474  TopOnctopon 23036   Cn ccn 23350  TopMndctmd 24196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-topgen 17496  df-plusf 18697  df-top 23020  df-topon 23037  df-topsp 23059  df-bases 23072  df-cn 23353  df-cnp 23354  df-tx 23688  df-tmd 24198

This theorem is referenced by:  tgplacthmeo  24229  ghmcnp  24241