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Theorem cnmpt2vsca 24219
Description: Continuity of scalar multiplication; analogue of cnmpt22f 23699 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 23615 . . . . . . . . . 10 ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑌)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmpt1vsca.w . . . . . . . . . . 11 (𝜑𝑊 ∈ TopMod)
6 tlmtrg.f . . . . . . . . . . . 12 𝐹 = (Scalar‘𝑊)
76tlmscatps 24215 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
85, 7syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ TopSp)
9 eqid 2735 . . . . . . . . . . 11 (Base‘𝐹) = (Base‘𝐹)
10 cnmpt1vsca.k . . . . . . . . . . 11 𝐾 = (TopOpen‘𝐹)
119, 10istps 22956 . . . . . . . . . 10 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
128, 11sylib 218 . . . . . . . . 9 (𝜑𝐾 ∈ (TopOn‘(Base‘𝐹)))
13 cnmpt2vsca.a . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
14 cnf2 23273 . . . . . . . . 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 1370 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
16 eqid 2735 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1716fmpo 8092 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
1815, 17sylibr 234 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹))
1918r19.21bi 3249 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐹))
2019r19.21bi 3249 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐹))
21 tlmtps 24212 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
225, 21syl 17 . . . . . . . . . 10 (𝜑𝑊 ∈ TopSp)
23 eqid 2735 . . . . . . . . . . 11 (Base‘𝑊) = (Base‘𝑊)
24 cnmpt1vsca.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝑊)
2523, 24istps 22956 . . . . . . . . . 10 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
2622, 25sylib 218 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑊)))
27 cnmpt2vsca.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
28 cnf2 23273 . . . . . . . . 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 1370 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
30 eqid 2735 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
3130fmpo 8092 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
3229, 31sylibr 234 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3332r19.21bi 3249 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3433r19.21bi 3249 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝑊))
35 eqid 2735 . . . . . 6 ( ·sf𝑊) = ( ·sf𝑊)
36 cnmpt1vsca.t . . . . . 6 · = ( ·𝑠𝑊)
3723, 6, 9, 35, 36scafval 20896 . . . . 5 ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
3820, 34, 37syl2anc 584 . . . 4 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
39383impa 1109 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
4039mpoeq3dva 7510 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)))
4135, 24, 6, 10vscacn 24210 . . . 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 23699 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
4440, 43eqeltrrd 2840 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  TopOpenctopn 17468   ·sf cscaf 20876  TopOnctopon 22932  TopSpctps 22954   Cn ccn 23248   ×t ctx 23584  TopModctlm 24182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-scaf 20878  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-tx 23586  df-tmd 24096  df-tgp 24097  df-trg 24184  df-tlm 24186
This theorem is referenced by: (None)
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