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Mirrors > Home > MPE Home > Th. List > cnmpt1vsca | Structured version Visualization version GIF version |
Description: Continuity of scalar multiplication; analogue of cnmpt12f 21963 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tlmtrg.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cnmpt1vsca.t | ⊢ · = ( ·𝑠 ‘𝑊) |
cnmpt1vsca.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
cnmpt1vsca.k | ⊢ 𝐾 = (TopOpen‘𝐹) |
cnmpt1vsca.w | ⊢ (𝜑 → 𝑊 ∈ TopMod) |
cnmpt1vsca.l | ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋)) |
cnmpt1vsca.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) |
cnmpt1vsca.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) |
Ref | Expression |
---|---|
cnmpt1vsca | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1vsca.l | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋)) | |
2 | cnmpt1vsca.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ TopMod) | |
3 | tlmtrg.f | . . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 3 | tlmscatps 22487 | . . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ TopSp) |
6 | eqid 2795 | . . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | cnmpt1vsca.k | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹) | |
8 | 6, 7 | istps 21231 | . . . . . . 7 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹))) |
9 | 5, 8 | sylib 219 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹))) |
10 | cnmpt1vsca.a | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) | |
11 | cnf2 21546 | . . . . . 6 ⊢ ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐹)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹)) | |
12 | 1, 9, 10, 11 | syl3anc 1364 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹)) |
13 | 12 | fvmptelrn 6745 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹)) |
14 | tlmtps 22484 | . . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp) | |
15 | 2, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp) |
16 | eqid 2795 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
17 | cnmpt1vsca.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊) | |
18 | 16, 17 | istps 21231 | . . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
19 | 15, 18 | sylib 219 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
20 | cnmpt1vsca.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) | |
21 | cnf2 21546 | . . . . . 6 ⊢ ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊)) | |
22 | 1, 19, 20, 21 | syl3anc 1364 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊)) |
23 | 22 | fvmptelrn 6745 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊)) |
24 | eqid 2795 | . . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
25 | cnmpt1vsca.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
26 | 16, 3, 6, 24, 25 | scafval 19348 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵)) |
27 | 13, 23, 26 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵)) |
28 | 27 | mpteq2dva 5060 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
29 | 24, 17, 3, 7 | vscacn 22482 | . . . 4 ⊢ (𝑊 ∈ TopMod → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
30 | 2, 29 | syl 17 | . . 3 ⊢ (𝜑 → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
31 | 1, 10, 20, 30 | cnmpt12f 21963 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽)) |
32 | 28, 31 | eqeltrrd 2884 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ↦ cmpt 5045 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 Basecbs 16317 Scalarcsca 16402 ·𝑠 cvsca 16403 TopOpenctopn 16529 ·sf cscaf 19330 TopOnctopon 21207 TopSpctps 21229 Cn ccn 21521 ×t ctx 21857 TopModctlm 22454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-fv 6238 df-ov 7024 df-oprab 7025 df-mpo 7026 df-1st 7550 df-2nd 7551 df-map 8263 df-slot 16321 df-base 16323 df-topgen 16551 df-scaf 19332 df-top 21191 df-topon 21208 df-topsp 21230 df-bases 21243 df-cn 21524 df-tx 21859 df-tmd 22369 df-tgp 22370 df-trg 22456 df-tlm 22458 |
This theorem is referenced by: tlmtgp 22492 |
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