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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 23631 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 24156 | . . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ TopSp) |
6 | eqid 2725 | . . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | cnmpt1vsca.k | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹) | |
8 | 6, 7 | istps 22897 | . . . . . . 7 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹))) |
9 | 5, 8 | sylib 217 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹))) |
10 | cnmpt1vsca.a | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) | |
11 | cnf2 23214 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹)) |
13 | 12 | fvmptelcdm 7122 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹)) |
14 | tlmtps 24153 | . . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp) | |
15 | 2, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp) |
16 | eqid 2725 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
17 | cnmpt1vsca.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊) | |
18 | 16, 17 | istps 22897 | . . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
19 | 15, 18 | sylib 217 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
20 | cnmpt1vsca.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) | |
21 | cnf2 23214 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊)) |
23 | 22 | fvmptelcdm 7122 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊)) |
24 | eqid 2725 | . . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
25 | cnmpt1vsca.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
26 | 16, 3, 6, 24, 25 | scafval 20793 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵)) |
27 | 13, 23, 26 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵)) |
28 | 27 | mpteq2dva 5249 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
29 | 24, 17, 3, 7 | vscacn 24151 | . . . 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 23631 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽)) |
32 | 28, 31 | eqeltrrd 2826 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5232 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Basecbs 17199 Scalarcsca 17255 ·𝑠 cvsca 17256 TopOpenctopn 17422 ·sf cscaf 20773 TopOnctopon 22873 TopSpctps 22895 Cn ccn 23189 ×t ctx 23525 TopModctlm 24123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 df-topgen 17444 df-scaf 20775 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-cn 23192 df-tx 23527 df-tmd 24037 df-tgp 24038 df-trg 24125 df-tlm 24127 |
This theorem is referenced by: tlmtgp 24161 |
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