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Theorem cnmpt2vsca 23919
Description: Continuity of scalar multiplication; analogue of cnmpt22f 23399 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
tlmtrg.f 𝐹 = (Scalarβ€˜π‘Š)
cnmpt1vsca.t Β· = ( ·𝑠 β€˜π‘Š)
cnmpt1vsca.j 𝐽 = (TopOpenβ€˜π‘Š)
cnmpt1vsca.k 𝐾 = (TopOpenβ€˜πΉ)
cnmpt1vsca.w (πœ‘ β†’ π‘Š ∈ TopMod)
cnmpt1vsca.l (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘‹))
cnmpt2vsca.m (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘Œ))
cnmpt2vsca.a (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐾))
cnmpt2vsca.b (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐽))
Assertion
Ref Expression
cnmpt2vsca (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴 Β· 𝐡)) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐽))
Distinct variable groups:   π‘₯,𝑦,𝐹   π‘₯,𝐽,𝑦   π‘₯,𝐾,𝑦   π‘₯,𝐿   πœ‘,π‘₯,𝑦   π‘₯,π‘Š,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦)   Β· (π‘₯,𝑦)   𝐿(𝑦)   𝑀(π‘₯,𝑦)

Proof of Theorem cnmpt2vsca
StepHypRef Expression
1 cnmpt1vsca.l . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘‹))
2 cnmpt2vsca.m . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘Œ))
3 txtopon 23315 . . . . . . . . . 10 ((𝐿 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
41, 2, 3syl2anc 582 . . . . . . . . 9 (πœ‘ β†’ (𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
5 cnmpt1vsca.w . . . . . . . . . . 11 (πœ‘ β†’ π‘Š ∈ TopMod)
6 tlmtrg.f . . . . . . . . . . . 12 𝐹 = (Scalarβ€˜π‘Š)
76tlmscatps 23915 . . . . . . . . . . 11 (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopSp)
85, 7syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 ∈ TopSp)
9 eqid 2730 . . . . . . . . . . 11 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
10 cnmpt1vsca.k . . . . . . . . . . 11 𝐾 = (TopOpenβ€˜πΉ)
119, 10istps 22656 . . . . . . . . . 10 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOnβ€˜(Baseβ€˜πΉ)))
128, 11sylib 217 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜(Baseβ€˜πΉ)))
13 cnmpt2vsca.a . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐾))
14 cnf2 22973 . . . . . . . . 9 (((𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) ∧ 𝐾 ∈ (TopOnβ€˜(Baseβ€˜πΉ)) ∧ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜πΉ))
154, 12, 13, 14syl3anc 1369 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜πΉ))
16 eqid 2730 . . . . . . . . 9 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
1716fmpo 8056 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ (Baseβ€˜πΉ) ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜πΉ))
1815, 17sylibr 233 . . . . . . 7 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ (Baseβ€˜πΉ))
1918r19.21bi 3246 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ (Baseβ€˜πΉ))
2019r19.21bi 3246 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ (Baseβ€˜πΉ))
21 tlmtps 23912 . . . . . . . . . . 11 (π‘Š ∈ TopMod β†’ π‘Š ∈ TopSp)
225, 21syl 17 . . . . . . . . . 10 (πœ‘ β†’ π‘Š ∈ TopSp)
23 eqid 2730 . . . . . . . . . . 11 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
24 cnmpt1vsca.j . . . . . . . . . . 11 𝐽 = (TopOpenβ€˜π‘Š)
2523, 24istps 22656 . . . . . . . . . 10 (π‘Š ∈ TopSp ↔ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜π‘Š)))
2622, 25sylib 217 . . . . . . . . 9 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜π‘Š)))
27 cnmpt2vsca.b . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐽))
28 cnf2 22973 . . . . . . . . 9 (((𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)) ∧ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜π‘Š)) ∧ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐽)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜π‘Š))
294, 26, 27, 28syl3anc 1369 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜π‘Š))
30 eqid 2730 . . . . . . . . 9 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
3130fmpo 8056 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ (Baseβ€˜π‘Š) ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜π‘Š))
3229, 31sylibr 233 . . . . . . 7 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ (Baseβ€˜π‘Š))
3332r19.21bi 3246 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ (Baseβ€˜π‘Š))
3433r19.21bi 3246 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ (Baseβ€˜π‘Š))
35 eqid 2730 . . . . . 6 ( Β·sf β€˜π‘Š) = ( Β·sf β€˜π‘Š)
36 cnmpt1vsca.t . . . . . 6 Β· = ( ·𝑠 β€˜π‘Š)
3723, 6, 9, 35, 36scafval 20635 . . . . 5 ((𝐴 ∈ (Baseβ€˜πΉ) ∧ 𝐡 ∈ (Baseβ€˜π‘Š)) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
3820, 34, 37syl2anc 582 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
39383impa 1108 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
4039mpoeq3dva 7488 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴( Β·sf β€˜π‘Š)𝐡)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴 Β· 𝐡)))
4135, 24, 6, 10vscacn 23910 . . . 4 (π‘Š ∈ TopMod β†’ ( Β·sf β€˜π‘Š) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
425, 41syl 17 . . 3 (πœ‘ β†’ ( Β·sf β€˜π‘Š) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
431, 2, 13, 27, 42cnmpt22f 23399 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴( Β·sf β€˜π‘Š)𝐡)) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐽))
4440, 43eqeltrrd 2832 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴 Β· 𝐡)) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   Γ— cxp 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17148  Scalarcsca 17204   ·𝑠 cvsca 17205  TopOpenctopn 17371   Β·sf cscaf 20615  TopOnctopon 22632  TopSpctps 22654   Cn ccn 22948   Γ—t ctx 23284  TopModctlm 23882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-topgen 17393  df-scaf 20617  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cn 22951  df-tx 23286  df-tmd 23796  df-tgp 23797  df-trg 23884  df-tlm 23886
This theorem is referenced by: (None)
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