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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 472 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐺 ∈ TopMnd) | |
5 | tgplacthmeo.2 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
6 | 2, 5 | tmdtopon 21695 | . . . 4 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
8 | simpr 476 | . . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
9 | 7, 7, 8 | cnmptc 21275 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐽)) |
10 | 7 | cnmptid 21274 | . . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
11 | 2, 3, 4, 7, 9, 10 | cnmpt1plusg 21701 | . 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) ∈ (𝐽 Cn 𝐽)) |
12 | 1, 11 | syl5eqel 2692 | 1 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 TopOpenctopn 15905 TopOnctopon 20518 Cn ccn 20838 TopMndctmd 21684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-topgen 15927 df-plusf 17064 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-tmd 21686 |
This theorem is referenced by: tgplacthmeo 21717 ghmcnp 21728 |
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