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Mirrors > Home > MPE Home > Th. List > cnmpt1plusg | Structured version Visualization version GIF version |
Description: Continuity of the group sum; analogue of cnmpt12f 22274 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 2821 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 3, 4 | tmdtopon 22689 | . . . . . . 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 21857 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) |
10 | 9 | fvmptelrn 6877 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺)) |
11 | cnmpt1plusg.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) | |
12 | cnf2 21857 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) |
14 | 13 | fvmptelrn 6877 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺)) |
15 | cnmpt1plusg.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
16 | eqid 2821 | . . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
17 | 4, 15, 16 | plusfval 17859 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
18 | 10, 14, 17 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
19 | 18 | mpteq2dva 5161 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵))) |
20 | 3, 16 | tmdcn 22691 | . . . 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 22274 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽)) |
23 | 19, 22 | eqeltrrd 2914 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 TopOpenctopn 16695 +𝑓cplusf 17849 TopOnctopon 21518 Cn ccn 21832 ×t ctx 22168 TopMndctmd 22678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-topgen 16717 df-plusf 17851 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-tx 22170 df-tmd 22680 |
This theorem is referenced by: tmdmulg 22700 tmdgsum 22703 tmdlactcn 22710 clsnsg 22718 tgpt0 22727 cnmpt1mulr 22790 |
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