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Theorem cnmpt1vsca 24053
Description: Continuity of scalar multiplication; analogue of cnmpt12f 23525 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‘‹))
cnmpt1vsca.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))
cnmpt1vsca.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐿 Cn 𝐽))
Assertion
Ref Expression
cnmpt1vsca (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)) ∈ (𝐿 Cn 𝐽))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝐿   πœ‘,π‘₯   π‘₯,π‘Š   π‘₯,𝑋
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘₯)   Β· (π‘₯)

Proof of Theorem cnmpt1vsca
StepHypRef Expression
1 cnmpt1vsca.l . . . . . 6 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘‹))
2 cnmpt1vsca.w . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ TopMod)
3 tlmtrg.f . . . . . . . . 9 𝐹 = (Scalarβ€˜π‘Š)
43tlmscatps 24050 . . . . . . . 8 (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopSp)
52, 4syl 17 . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ TopSp)
6 eqid 2726 . . . . . . . 8 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
7 cnmpt1vsca.k . . . . . . . 8 𝐾 = (TopOpenβ€˜πΉ)
86, 7istps 22791 . . . . . . 7 (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOnβ€˜(Baseβ€˜πΉ)))
95, 8sylib 217 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜(Baseβ€˜πΉ)))
10 cnmpt1vsca.a . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))
11 cnf2 23108 . . . . . 6 ((𝐿 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜(Baseβ€˜πΉ)) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆ(Baseβ€˜πΉ))
121, 9, 10, 11syl3anc 1368 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆ(Baseβ€˜πΉ))
1312fvmptelcdm 7108 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ (Baseβ€˜πΉ))
14 tlmtps 24047 . . . . . . . 8 (π‘Š ∈ TopMod β†’ π‘Š ∈ TopSp)
152, 14syl 17 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ TopSp)
16 eqid 2726 . . . . . . . 8 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
17 cnmpt1vsca.j . . . . . . . 8 𝐽 = (TopOpenβ€˜π‘Š)
1816, 17istps 22791 . . . . . . 7 (π‘Š ∈ TopSp ↔ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜π‘Š)))
1915, 18sylib 217 . . . . . 6 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜π‘Š)))
20 cnmpt1vsca.b . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐿 Cn 𝐽))
21 cnf2 23108 . . . . . 6 ((𝐿 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜π‘Š)) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐿 Cn 𝐽)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆ(Baseβ€˜π‘Š))
221, 19, 20, 21syl3anc 1368 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆ(Baseβ€˜π‘Š))
2322fvmptelcdm 7108 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ (Baseβ€˜π‘Š))
24 eqid 2726 . . . . 5 ( Β·sf β€˜π‘Š) = ( Β·sf β€˜π‘Š)
25 cnmpt1vsca.t . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
2616, 3, 6, 24, 25scafval 20727 . . . 4 ((𝐴 ∈ (Baseβ€˜πΉ) ∧ 𝐡 ∈ (Baseβ€˜π‘Š)) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
2713, 23, 26syl2anc 583 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
2827mpteq2dva 5241 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴( Β·sf β€˜π‘Š)𝐡)) = (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)))
2924, 17, 3, 7vscacn 24045 . . . 4 (π‘Š ∈ TopMod β†’ ( Β·sf β€˜π‘Š) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
302, 29syl 17 . . 3 (πœ‘ β†’ ( Β·sf β€˜π‘Š) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
311, 10, 20, 30cnmpt12f 23525 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴( Β·sf β€˜π‘Š)𝐡)) ∈ (𝐿 Cn 𝐽))
3228, 31eqeltrrd 2828 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)) ∈ (𝐿 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5224  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  Scalarcsca 17209   ·𝑠 cvsca 17210  TopOpenctopn 17376   Β·sf cscaf 20707  TopOnctopon 22767  TopSpctps 22789   Cn ccn 23083   Γ—t ctx 23419  TopModctlm 24017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824  df-topgen 17398  df-scaf 20709  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cn 23086  df-tx 23421  df-tmd 23931  df-tgp 23932  df-trg 24019  df-tlm 24021
This theorem is referenced by:  tlmtgp  24055
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