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Theorem cnmpt2vsca 23628
Description: Continuity of scalar multiplication; analogue of cnmpt22f 23108 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 23024 . . . . . . . . . 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 23624 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
85, 7syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ TopSp)
9 eqid 2731 . . . . . . . . . . 11 (Base‘𝐹) = (Base‘𝐹)
10 cnmpt1vsca.k . . . . . . . . . . 11 𝐾 = (TopOpen‘𝐹)
119, 10istps 22365 . . . . . . . . . 10 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
128, 11sylib 217 . . . . . . . . 9 (𝜑𝐾 ∈ (TopOn‘(Base‘𝐹)))
13 cnmpt2vsca.a . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
14 cnf2 22682 . . . . . . . . 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 1371 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
16 eqid 2731 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1716fmpo 8036 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
1815, 17sylibr 233 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹))
1918r19.21bi 3247 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐹))
2019r19.21bi 3247 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐹))
21 tlmtps 23621 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
225, 21syl 17 . . . . . . . . . 10 (𝜑𝑊 ∈ TopSp)
23 eqid 2731 . . . . . . . . . . 11 (Base‘𝑊) = (Base‘𝑊)
24 cnmpt1vsca.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝑊)
2523, 24istps 22365 . . . . . . . . . 10 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
2622, 25sylib 217 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑊)))
27 cnmpt2vsca.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
28 cnf2 22682 . . . . . . . . 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 1371 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
30 eqid 2731 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
3130fmpo 8036 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
3229, 31sylibr 233 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3332r19.21bi 3247 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3433r19.21bi 3247 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝑊))
35 eqid 2731 . . . . . 6 ( ·sf𝑊) = ( ·sf𝑊)
36 cnmpt1vsca.t . . . . . 6 · = ( ·𝑠𝑊)
3723, 6, 9, 35, 36scafval 20440 . . . . 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 1110 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
4039mpoeq3dva 7470 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)))
4135, 24, 6, 10vscacn 23619 . . . 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 23108 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
4440, 43eqeltrrd 2833 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060   × cxp 5667  wf 6528  cfv 6532  (class class class)co 7393  cmpo 7395  Basecbs 17126  Scalarcsca 17182   ·𝑠 cvsca 17183  TopOpenctopn 17349   ·sf cscaf 20421  TopOnctopon 22341  TopSpctps 22363   Cn ccn 22657   ×t ctx 22993  TopModctlm 23591
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-map 8805  df-topgen 17371  df-scaf 20423  df-top 22325  df-topon 22342  df-topsp 22364  df-bases 22378  df-cn 22660  df-tx 22995  df-tmd 23505  df-tgp 23506  df-trg 23593  df-tlm 23595
This theorem is referenced by: (None)
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