Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt2vsca Structured version   Visualization version   GIF version

Theorem cnmpt2vsca 21808
 Description: Continuity of scalar multiplication; analogue of cnmpt22f 21288 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 21204 . . . . . . . . . 10 ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑌)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 691 . . . . . . . . 9 (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmpt1vsca.w . . . . . . . . . . 11 (𝜑𝑊 ∈ TopMod)
6 tlmtrg.f . . . . . . . . . . . 12 𝐹 = (Scalar‘𝑊)
76tlmscatps 21804 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
85, 7syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ TopSp)
9 eqid 2610 . . . . . . . . . . 11 (Base‘𝐹) = (Base‘𝐹)
10 cnmpt1vsca.k . . . . . . . . . . 11 𝐾 = (TopOpen‘𝐹)
119, 10istps 20551 . . . . . . . . . 10 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
128, 11sylib 207 . . . . . . . . 9 (𝜑𝐾 ∈ (TopOn‘(Base‘𝐹)))
13 cnmpt2vsca.a . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
14 cnf2 20863 . . . . . . . . 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 1318 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
16 eqid 2610 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1716fmpt2 7126 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
1815, 17sylibr 223 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹))
1918r19.21bi 2916 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐹))
2019r19.21bi 2916 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐹))
21 tlmtps 21801 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
225, 21syl 17 . . . . . . . . . 10 (𝜑𝑊 ∈ TopSp)
23 eqid 2610 . . . . . . . . . . 11 (Base‘𝑊) = (Base‘𝑊)
24 cnmpt1vsca.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝑊)
2523, 24istps 20551 . . . . . . . . . 10 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
2622, 25sylib 207 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑊)))
27 cnmpt2vsca.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
28 cnf2 20863 . . . . . . . . 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 1318 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
30 eqid 2610 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
3130fmpt2 7126 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
3229, 31sylibr 223 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3332r19.21bi 2916 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3433r19.21bi 2916 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝑊))
35 eqid 2610 . . . . . 6 ( ·sf𝑊) = ( ·sf𝑊)
36 cnmpt1vsca.t . . . . . 6 · = ( ·𝑠𝑊)
3723, 6, 9, 35, 36scafval 18705 . . . . 5 ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
3820, 34, 37syl2anc 691 . . . 4 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
39383impa 1251 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴( ·sf𝑊)𝐵) = (𝐴 · 𝐵))
4039mpt2eq3dva 6617 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)))
4135, 24, 6, 10vscacn 21799 . . . 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 21288 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
4440, 43eqeltrrd 2689 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  TopOpenctopn 15905   ·sf cscaf 18687  TopOnctopon 20518  TopSpctps 20519   Cn ccn 20838   ×t ctx 21173  TopModctlm 21771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-slot 15699  df-base 15700  df-topgen 15927  df-scaf 18689  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-tx 21175  df-tmd 21686  df-tgp 21687  df-trg 21773  df-tlm 21775 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator