![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnmpt1plusg | Structured version Visualization version GIF version |
Description: Continuity of the group sum; analogue of cnmpt12f 22271 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
cnmpt1plusg.p | ⊢ + = (+g‘𝐺) |
cnmpt1plusg.g | ⊢ (𝜑 → 𝐺 ∈ TopMnd) |
cnmpt1plusg.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) |
cnmpt1plusg.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) |
cnmpt1plusg.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) |
Ref | Expression |
---|---|
cnmpt1plusg | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1plusg.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) | |
2 | cnmpt1plusg.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TopMnd) | |
3 | tgpcn.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺) | |
4 | eqid 2798 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 3, 4 | tmdtopon 22686 | . . . . . . 7 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
6 | 2, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
7 | cnmpt1plusg.a | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) | |
8 | cnf2 21854 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) | |
9 | 1, 6, 7, 8 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) |
10 | 9 | fvmptelrn 6854 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺)) |
11 | cnmpt1plusg.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) | |
12 | cnf2 21854 | . . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) | |
13 | 1, 6, 11, 12 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) |
14 | 13 | fvmptelrn 6854 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺)) |
15 | cnmpt1plusg.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
16 | eqid 2798 | . . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
17 | 4, 15, 16 | plusfval 17851 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
18 | 10, 14, 17 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
19 | 18 | mpteq2dva 5125 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵))) |
20 | 3, 16 | tmdcn 22688 | . . . 4 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
21 | 2, 20 | syl 17 | . . 3 ⊢ (𝜑 → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
22 | 1, 7, 11, 21 | cnmpt12f 22271 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽)) |
23 | 19, 22 | eqeltrrd 2891 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 TopOpenctopn 16687 +𝑓cplusf 17841 TopOnctopon 21515 Cn ccn 21829 ×t ctx 22165 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-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-tx 22167 df-tmd 22677 |
This theorem is referenced by: tmdmulg 22697 tmdgsum 22700 tmdlactcn 22707 clsnsg 22715 tgpt0 22724 cnmpt1mulr 22787 |
Copyright terms: Public domain | W3C validator |