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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 23674 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 2737 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 5 | 3, 4 | tmdtopon 24089 | . . . . . . 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 23257 | . . . . . 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 1373 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) | 
| 10 | 9 | fvmptelcdm 7133 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺)) | 
| 11 | cnmpt1plusg.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) | |
| 12 | cnf2 23257 | . . . . . 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 1373 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) | 
| 14 | 13 | fvmptelcdm 7133 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺)) | 
| 15 | cnmpt1plusg.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
| 16 | eqid 2737 | . . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
| 17 | 4, 15, 16 | plusfval 18660 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) | 
| 18 | 10, 14, 17 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) | 
| 19 | 18 | mpteq2dva 5242 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵))) | 
| 20 | 3, 16 | tmdcn 24091 | . . . 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 23674 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽)) | 
| 23 | 19, 22 | eqeltrrd 2842 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 TopOpenctopn 17466 +𝑓cplusf 18650 TopOnctopon 22916 Cn ccn 23232 ×t ctx 23568 TopMndctmd 24078 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-topgen 17488 df-plusf 18652 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-tx 23570 df-tmd 24080 | 
| This theorem is referenced by: tmdmulg 24100 tmdgsum 24103 tmdlactcn 24110 clsnsg 24118 tgpt0 24127 cnmpt1mulr 24190 | 
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