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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 476 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ TopMnd) | |
5 | tgplacthmeo.2 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
6 | 2, 5 | tmdtopon 22304 | . . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋)) |
7 | 6 | adantr 474 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
8 | simpr 479 | . . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
9 | 7, 7, 8 | cnmptc 21885 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽)) |
10 | 7 | cnmptid 21884 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
11 | 2, 3, 4, 7, 9, 10 | cnmpt1plusg 22310 | . 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽)) |
12 | 1, 11 | syl5eqel 2863 | 1 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 +gcplusg 16349 TopOpenctopn 16479 TopOnctopon 21133 Cn ccn 21447 TopMndctmd 22293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-map 8144 df-topgen 16501 df-plusf 17638 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cn 21450 df-cnp 21451 df-tx 21785 df-tmd 22295 |
This theorem is referenced by: tgplacthmeo 22326 ghmcnp 22337 |
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