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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 482 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ TopMnd) | |
| 5 | tgplacthmeo.2 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
| 6 | 2, 5 | tmdtopon 24090 | . . . 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 23671 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽)) | 
| 10 | 7 | cnmptid 23670 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) | 
| 11 | 2, 3, 4, 7, 9, 10 | cnmpt1plusg 24096 | . 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽)) | 
| 12 | 1, 11 | eqeltrid 2844 | 1 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 TopOpenctopn 17467 TopOnctopon 22917 Cn ccn 23233 TopMndctmd 24079 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 df-topgen 17489 df-plusf 18653 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cn 23236 df-cnp 23237 df-tx 23571 df-tmd 24081 | 
| This theorem is referenced by: tgplacthmeo 24112 ghmcnp 24124 | 
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