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Theorem cnmpt1vsca 22802
 Description: Continuity of scalar multiplication; analogue of cnmpt12f 22274 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 22799 . . . . . . . 8 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
52, 4syl 17 . . . . . . 7 (𝜑𝐹 ∈ TopSp)
6 eqid 2801 . . . . . . . 8 (Base‘𝐹) = (Base‘𝐹)
7 cnmpt1vsca.k . . . . . . . 8 𝐾 = (TopOpen‘𝐹)
86, 7istps 21542 . . . . . . 7 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
95, 8sylib 221 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(Base‘𝐹)))
10 cnmpt1vsca.a . . . . . 6 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐿 Cn 𝐾))
11 cnf2 21857 . . . . . 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))
1312fvmptelrn 6858 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐹))
14 tlmtps 22796 . . . . . . . 8 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
152, 14syl 17 . . . . . . 7 (𝜑𝑊 ∈ TopSp)
16 eqid 2801 . . . . . . . 8 (Base‘𝑊) = (Base‘𝑊)
17 cnmpt1vsca.j . . . . . . . 8 𝐽 = (TopOpen‘𝑊)
1816, 17istps 21542 . . . . . . 7 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
1915, 18sylib 221 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑊)))
20 cnmpt1vsca.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐿 Cn 𝐽))
21 cnf2 21857 . . . . . 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))
2322fvmptelrn 6858 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝑊))
24 eqid 2801 . . . . 5 ( ·sf𝑊) = ( ·sf𝑊)
25 cnmpt1vsca.t . . . . 5 · = ( ·𝑠𝑊)
2616, 3, 6, 24, 25scafval 19649 . . . 4 ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
2713, 23, 26syl2anc 587 . . 3 ((𝜑𝑥𝑋) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
2827mpteq2dva 5128 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴( ·sf𝑊)𝐵)) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
2924, 17, 3, 7vscacn 22794 . . . 4 (𝑊 ∈ TopMod → ( ·sf𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
302, 29syl 17 . . 3 (𝜑 → ( ·sf𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
311, 10, 20, 30cnmpt12f 22274 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴( ·sf𝑊)𝐵)) ∈ (𝐿 Cn 𝐽))
3228, 31eqeltrrd 2894 1 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ↦ cmpt 5113  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  Scalarcsca 16563   ·𝑠 cvsca 16564  TopOpenctopn 16690   ·sf cscaf 19631  TopOnctopon 21518  TopSpctps 21540   Cn ccn 21832   ×t ctx 22168  TopModctlm 22766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-topgen 16712  df-scaf 19633  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cn 21835  df-tx 22170  df-tmd 22680  df-tgp 22681  df-trg 22768  df-tlm 22770 This theorem is referenced by:  tlmtgp  22804
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