![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tmdlactcn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
tgplacthmeo.1 | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) |
tgplacthmeo.2 | ⊢ 𝑋 = (Base‘𝐺) |
tgplacthmeo.3 | ⊢ + = (+g‘𝐺) |
tgplacthmeo.4 | ⊢ 𝐽 = (TopOpen‘𝐺) |
Ref | Expression |
---|---|
tmdlactcn | ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽)) |
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 482 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ TopMnd) | |
5 | tgplacthmeo.2 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
6 | 2, 5 | tmdtopon 24105 | . . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
8 | simpr 484 | . . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
9 | 7, 7, 8 | cnmptc 23686 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽)) |
10 | 7 | cnmptid 23685 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
11 | 2, 3, 4, 7, 9, 10 | cnmpt1plusg 24111 | . 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽)) |
12 | 1, 11 | eqeltrid 2843 | 1 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 TopOpenctopn 17468 TopOnctopon 22932 Cn ccn 23248 TopMndctmd 24094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-topgen 17490 df-plusf 18665 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-cnp 23252 df-tx 23586 df-tmd 24096 |
This theorem is referenced by: tgplacthmeo 24127 ghmcnp 24139 |
Copyright terms: Public domain | W3C validator |