Theorem tmdlactcn 22707

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 486	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ TopMnd)
5		tgplacthmeo.2	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
6	2, 5	tmdtopon 22686	. . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
7	6	adantr 484	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8		simpr 488	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
9	7, 7, 8	cnmptc 22267	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽))
10	7	cnmptid 22266	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
11	2, 3, 4, 7, 9, 10	cnmpt1plusg 22692	. 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽))
12	1, 11	eqeltrid 2894	1 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  TopOpenctopn 16687  TopOnctopon 21515   Cn ccn 21829  TopMndctmd 22675

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-topgen 16709  df-plusf 17843  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cn 21832  df-cnp 21833  df-tx 22167  df-tmd 22677

This theorem is referenced by:  tgplacthmeo  22708  ghmcnp  22720