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Theorem cnmpt2vsca 21938
Description: Continuity of scalar multiplication; analogue of cnmpt22f 21418 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))
cnmpt2vsca.m (𝜑𝑀 ∈ (TopOn‘𝑌))
cnmpt2vsca.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
cnmpt2vsca.b (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Assertion
Ref Expression
cnmpt2vsca (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿   𝜑,𝑥,𝑦   𝑥,𝑊,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   · (𝑥,𝑦)   𝐿(𝑦)   𝑀(𝑥,𝑦)

Proof of Theorem cnmpt2vsca
StepHypRef Expression
1 cnmpt1vsca.l . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘𝑋))
2 cnmpt2vsca.m . . . . . . . . . 10 (𝜑𝑀 ∈ (TopOn‘𝑌))
3 txtopon 21334 . . . . . . . . . 10 ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑌)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmpt1vsca.w . . . . . . . . . . 11 (𝜑𝑊 ∈ TopMod)
6 tlmtrg.f . . . . . . . . . . . 12 𝐹 = (Scalar‘𝑊)
76tlmscatps 21934 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
85, 7syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ TopSp)
9 eqid 2621 . . . . . . . . . . 11 (Base‘𝐹) = (Base‘𝐹)
10 cnmpt1vsca.k . . . . . . . . . . 11 𝐾 = (TopOpen‘𝐹)
119, 10istps 20678 . . . . . . . . . 10 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
128, 11sylib 208 . . . . . . . . 9 (𝜑𝐾 ∈ (TopOn‘(Base‘𝐹)))
13 cnmpt2vsca.a . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
14 cnf2 20993 . . . . . . . . 9 (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐹)) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
154, 12, 13, 14syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
16 eqid 2621 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1716fmpt2 7197 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
1815, 17sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹))
1918r19.21bi 2928 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐹))
2019r19.21bi 2928 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐹))
21 tlmtps 21931 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
225, 21syl 17 . . . . . . . . . 10 (𝜑𝑊 ∈ TopSp)
23 eqid 2621 . . . . . . . . . . 11 (Base‘𝑊) = (Base‘𝑊)
24 cnmpt1vsca.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝑊)
2523, 24istps 20678 . . . . . . . . . 10 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
2622, 25sylib 208 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑊)))
27 cnmpt2vsca.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
28 cnf2 20993 . . . . . . . . 9 (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽)) → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
294, 26, 27, 28syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
30 eqid 2621 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
3130fmpt2 7197 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
3229, 31sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3332r19.21bi 2928 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3433r19.21bi 2928 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝑊))
35 eqid 2621 . . . . . 6 ( ·sf𝑊) = ( ·sf𝑊)
36 cnmpt1vsca.t . . . . . 6 · = ( ·𝑠𝑊)
3723, 6, 9, 35, 36scafval 18822 . . . . 5 ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
3820, 34, 37syl2anc 692 . . . 4 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
39383impa 1256 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
4039mpt2eq3dva 6684 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)))
4135, 24, 6, 10vscacn 21929 . . . 4 (𝑊 ∈ TopMod → ( ·sf𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
425, 41syl 17 . . 3 (𝜑 → ( ·sf𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
431, 2, 13, 27, 42cnmpt22f 21418 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
4440, 43eqeltrrd 2699 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908   × cxp 5082  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617  Basecbs 15800  Scalarcsca 15884   ·𝑠 cvsca 15885  TopOpenctopn 16022   ·sf cscaf 18804  TopOnctopon 20655  TopSpctps 20676   Cn ccn 20968   ×t ctx 21303  TopModctlm 21901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819  df-slot 15804  df-base 15805  df-topgen 16044  df-scaf 18806  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cn 20971  df-tx 21305  df-tmd 21816  df-tgp 21817  df-trg 21903  df-tlm 21905
This theorem is referenced by: (None)
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