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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 22535 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 23060 | . . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ TopSp) |
6 | eqid 2734 | . . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | cnmpt1vsca.k | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹) | |
8 | 6, 7 | istps 21803 | . . . . . . 7 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹))) |
9 | 5, 8 | sylib 221 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹))) |
10 | cnmpt1vsca.a | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) | |
11 | cnf2 22118 | . . . . . 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 1373 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹)) |
13 | 12 | fvmptelrn 6919 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹)) |
14 | tlmtps 23057 | . . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp) | |
15 | 2, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp) |
16 | eqid 2734 | . . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
17 | cnmpt1vsca.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊) | |
18 | 16, 17 | istps 21803 | . . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
19 | 15, 18 | sylib 221 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊))) |
20 | cnmpt1vsca.b | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) | |
21 | cnf2 22118 | . . . . . 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 1373 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊)) |
23 | 22 | fvmptelrn 6919 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊)) |
24 | eqid 2734 | . . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊) | |
25 | cnmpt1vsca.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
26 | 16, 3, 6, 24, 25 | scafval 19890 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵)) |
27 | 13, 23, 26 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵)) |
28 | 27 | mpteq2dva 5139 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
29 | 24, 17, 3, 7 | vscacn 23055 | . . . 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 22535 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽)) |
32 | 28, 31 | eqeltrrd 2835 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5124 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 Scalarcsca 16770 ·𝑠 cvsca 16771 TopOpenctopn 16898 ·sf cscaf 19872 TopOnctopon 21779 TopSpctps 21801 Cn ccn 22093 ×t ctx 22429 TopModctlm 23027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-map 8499 df-topgen 16920 df-scaf 19874 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cn 22096 df-tx 22431 df-tmd 22941 df-tgp 22942 df-trg 23029 df-tlm 23031 |
This theorem is referenced by: tlmtgp 23065 |
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