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

Theorem cnmpt2vsca 24204
Description: Continuity of scalar multiplication; analogue of cnmpt22f 23684 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 23600 . . . . . . . . . 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 24200 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
85, 7syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ TopSp)
9 eqid 2736 . . . . . . . . . . 11 (Base‘𝐹) = (Base‘𝐹)
10 cnmpt1vsca.k . . . . . . . . . . 11 𝐾 = (TopOpen‘𝐹)
119, 10istps 22941 . . . . . . . . . 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 23258 . . . . . . . . 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 1372 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
16 eqid 2736 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1716fmpo 8094 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
1815, 17sylibr 234 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐹))
1918r19.21bi 3250 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐹))
2019r19.21bi 3250 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐹))
21 tlmtps 24197 . . . . . . . . . . 11 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
225, 21syl 17 . . . . . . . . . 10 (𝜑𝑊 ∈ TopSp)
23 eqid 2736 . . . . . . . . . . 11 (Base‘𝑊) = (Base‘𝑊)
24 cnmpt1vsca.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝑊)
2523, 24istps 22941 . . . . . . . . . 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 23258 . . . . . . . . 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 1372 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
30 eqid 2736 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
3130fmpo 8094 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
3229, 31sylibr 234 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3332r19.21bi 3250 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝑊))
3433r19.21bi 3250 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝑊))
35 eqid 2736 . . . . . 6 ( ·sf𝑊) = ( ·sf𝑊)
36 cnmpt1vsca.t . . . . . 6 · = ( ·𝑠𝑊)
3723, 6, 9, 35, 36scafval 20880 . . . . 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 7511 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)))
4135, 24, 6, 10vscacn 24195 . . . 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 23684 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴( ·sf𝑊)𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
4440, 43eqeltrrd 2841 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060   × cxp 5682  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  Basecbs 17248  Scalarcsca 17301   ·𝑠 cvsca 17302  TopOpenctopn 17467   ·sf cscaf 20860  TopOnctopon 22917  TopSpctps 22939   Cn ccn 23233   ×t ctx 23569  TopModctlm 24167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-topgen 17489  df-scaf 20862  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-cn 23236  df-tx 23571  df-tmd 24081  df-tgp 24082  df-trg 24169  df-tlm 24171
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator