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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 21968 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 2772 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 3, 4 | tmdtopon 22383 | . . . . . . 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 21551 | . . . . . 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 1351 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) |
10 | 9 | fvmptelrn 6694 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺)) |
11 | cnmpt1plusg.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) | |
12 | cnf2 21551 | . . . . . 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 1351 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) |
14 | 13 | fvmptelrn 6694 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺)) |
15 | cnmpt1plusg.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
16 | eqid 2772 | . . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
17 | 4, 15, 16 | plusfval 17706 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
18 | 10, 14, 17 | syl2anc 576 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
19 | 18 | mpteq2dva 5016 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵))) |
20 | 3, 16 | tmdcn 22385 | . . . 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 21968 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽)) |
23 | 19, 22 | eqeltrrd 2861 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ↦ cmpt 5002 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 +gcplusg 16411 TopOpenctopn 16541 +𝑓cplusf 17697 TopOnctopon 21212 Cn ccn 21526 ×t ctx 21862 TopMndctmd 22372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-map 8200 df-topgen 16563 df-plusf 17699 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cn 21529 df-tx 21864 df-tmd 22374 |
This theorem is referenced by: tmdmulg 22394 tmdgsum 22397 tmdlactcn 22404 clsnsg 22411 tgpt0 22420 cnmpt1mulr 22483 |
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