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Theorem cnmpt1vsca 24114
Description: Continuity of scalar multiplication; analogue of cnmpt12f 23586 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 24111 . . . . . . . 8 (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopSp)
52, 4syl 17 . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ TopSp)
6 eqid 2725 . . . . . . . 8 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
7 cnmpt1vsca.k . . . . . . . 8 𝐾 = (TopOpenβ€˜πΉ)
86, 7istps 22852 . . . . . . 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 23169 . . . . . 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 7117 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ (Baseβ€˜πΉ))
14 tlmtps 24108 . . . . . . . 8 (π‘Š ∈ TopMod β†’ π‘Š ∈ TopSp)
152, 14syl 17 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ TopSp)
16 eqid 2725 . . . . . . . 8 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
17 cnmpt1vsca.j . . . . . . . 8 𝐽 = (TopOpenβ€˜π‘Š)
1816, 17istps 22852 . . . . . . 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 23169 . . . . . 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 7117 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ (Baseβ€˜π‘Š))
24 eqid 2725 . . . . 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 20766 . . . 4 ((𝐴 ∈ (Baseβ€˜πΉ) ∧ 𝐡 ∈ (Baseβ€˜π‘Š)) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
2713, 23, 26syl2anc 582 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
2827mpteq2dva 5243 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴( Β·sf β€˜π‘Š)𝐡)) = (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)))
2924, 17, 3, 7vscacn 24106 . . . 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 23586 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴( Β·sf β€˜π‘Š)𝐡)) ∈ (𝐿 Cn 𝐽))
3228, 31eqeltrrd 2826 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)) ∈ (𝐿 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5226  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415  Basecbs 17177  Scalarcsca 17233   ·𝑠 cvsca 17234  TopOpenctopn 17400   Β·sf cscaf 20746  TopOnctopon 22828  TopSpctps 22850   Cn ccn 23144   Γ—t ctx 23480  TopModctlm 24078
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-map 8843  df-topgen 17422  df-scaf 20748  df-top 22812  df-topon 22829  df-topsp 22851  df-bases 22865  df-cn 23147  df-tx 23482  df-tmd 23992  df-tgp 23993  df-trg 24080  df-tlm 24082
This theorem is referenced by:  tlmtgp  24116
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