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Theorem cnmpt1vsca 24159
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.)
Hypotheses
Ref Expression
tlmtrg.f 𝐹 = (Scalar‘𝑊)
cnmpt1vsca.t · = ( ·𝑠𝑊)
cnmpt1vsca.j 𝐽 = (TopOpen‘𝑊)
cnmpt1vsca.k 𝐾 = (TopOpen‘𝐹)
cnmpt1vsca.w (𝜑𝑊 ∈ TopMod)
cnmpt1vsca.l (𝜑𝐿 ∈ (TopOn‘𝑋))
cnmpt1vsca.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐿 Cn 𝐾))
cnmpt1vsca.b (𝜑 → (𝑥𝑋𝐵) ∈ (𝐿 Cn 𝐽))
Assertion
Ref Expression
cnmpt1vsca (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝐿   𝜑,𝑥   𝑥,𝑊   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   · (𝑥)

Proof of Theorem cnmpt1vsca
StepHypRef Expression
1 cnmpt1vsca.l . . . . . 6 (𝜑𝐿 ∈ (TopOn‘𝑋))
2 cnmpt1vsca.w . . . . . . . 8 (𝜑𝑊 ∈ TopMod)
3 tlmtrg.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
43tlmscatps 24156 . . . . . . . 8 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
52, 4syl 17 . . . . . . 7 (𝜑𝐹 ∈ TopSp)
6 eqid 2725 . . . . . . . 8 (Base‘𝐹) = (Base‘𝐹)
7 cnmpt1vsca.k . . . . . . . 8 𝐾 = (TopOpen‘𝐹)
86, 7istps 22897 . . . . . . 7 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
95, 8sylib 217 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(Base‘𝐹)))
10 cnmpt1vsca.a . . . . . 6 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐿 Cn 𝐾))
11 cnf2 23214 . . . . . 6 ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐹)) ∧ (𝑥𝑋𝐴) ∈ (𝐿 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐹))
121, 9, 10, 11syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐹))
1312fvmptelcdm 7122 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐹))
14 tlmtps 24153 . . . . . . . 8 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
152, 14syl 17 . . . . . . 7 (𝜑𝑊 ∈ TopSp)
16 eqid 2725 . . . . . . . 8 (Base‘𝑊) = (Base‘𝑊)
17 cnmpt1vsca.j . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
1816, 17istps 22897 . . . . . . 7 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
1915, 18sylib 217 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑊)))
20 cnmpt1vsca.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐿 Cn 𝐽))
21 cnf2 23214 . . . . . 6 ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥𝑋𝐵) ∈ (𝐿 Cn 𝐽)) → (𝑥𝑋𝐵):𝑋⟶(Base‘𝑊))
221, 19, 20, 21syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋⟶(Base‘𝑊))
2322fvmptelcdm 7122 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝑊))
24 eqid 2725 . . . . 5 ( ·sf𝑊) = ( ·sf𝑊)
25 cnmpt1vsca.t . . . . 5 · = ( ·𝑠𝑊)
2616, 3, 6, 24, 25scafval 20793 . . . 4 ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
2713, 23, 26syl2anc 582 . . 3 ((𝜑𝑥𝑋) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
2827mpteq2dva 5249 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴( ·sf𝑊)𝐵)) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
2924, 17, 3, 7vscacn 24151 . . . 4 (𝑊 ∈ TopMod → ( ·sf𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
302, 29syl 17 . . 3 (𝜑 → ( ·sf𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
311, 10, 20, 30cnmpt12f 23631 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴( ·sf𝑊)𝐵)) ∈ (𝐿 Cn 𝐽))
3228, 31eqeltrrd 2826 1 (𝜑 → (𝑥𝑋 ↦ (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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