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Theorem cnmpt2vsca 23562
Description: Continuity of scalar multiplication; analogue of cnmpt22f 23042 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 22958 . . . . . . . . . 10 ((𝐿 ∈ (TopOnβ€˜π‘‹) ∧ 𝑀 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐿 Γ—t 𝑀) ∈ (TopOnβ€˜(𝑋 Γ— π‘Œ)))
41, 2, 3syl2anc 585 . . . . . . . . 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 23558 . . . . . . . . . . 11 (π‘Š ∈ TopMod β†’ 𝐹 ∈ TopSp)
85, 7syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 ∈ TopSp)
9 eqid 2733 . . . . . . . . . . 11 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
10 cnmpt1vsca.k . . . . . . . . . . 11 𝐾 = (TopOpenβ€˜πΉ)
119, 10istps 22299 . . . . . . . . . 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 22616 . . . . . . . . 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 1372 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜πΉ))
16 eqid 2733 . . . . . . . . 9 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴)
1716fmpo 8001 . . . . . . . 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 3233 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ (Baseβ€˜πΉ))
2019r19.21bi 3233 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ (Baseβ€˜πΉ))
21 tlmtps 23555 . . . . . . . . . . 11 (π‘Š ∈ TopMod β†’ π‘Š ∈ TopSp)
225, 21syl 17 . . . . . . . . . 10 (πœ‘ β†’ π‘Š ∈ TopSp)
23 eqid 2733 . . . . . . . . . . 11 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
24 cnmpt1vsca.j . . . . . . . . . . 11 𝐽 = (TopOpenβ€˜π‘Š)
2523, 24istps 22299 . . . . . . . . . 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 22616 . . . . . . . . 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 1372 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡):(𝑋 Γ— π‘Œ)⟢(Baseβ€˜π‘Š))
30 eqid 2733 . . . . . . . . 9 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡)
3130fmpo 8001 . . . . . . . 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 3233 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ (Baseβ€˜π‘Š))
3433r19.21bi 3233 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐡 ∈ (Baseβ€˜π‘Š))
35 eqid 2733 . . . . . 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 20356 . . . . 5 ((𝐴 ∈ (Baseβ€˜πΉ) ∧ 𝐡 ∈ (Baseβ€˜π‘Š)) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
3820, 34, 37syl2anc 585 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
39383impa 1111 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (𝐴( Β·sf β€˜π‘Š)𝐡) = (𝐴 Β· 𝐡))
4039mpoeq3dva 7435 . 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 23553 . . . 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 23042 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴( Β·sf β€˜π‘Š)𝐡)) ∈ ((𝐿 Γ—t 𝑀) Cn 𝐽))
4440, 43eqeltrrd 2835 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   Γ— cxp 5632  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  TopOpenctopn 17308   Β·sf cscaf 20337  TopOnctopon 22275  TopSpctps 22297   Cn ccn 22591   Γ—t ctx 22927  TopModctlm 23525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-topgen 17330  df-scaf 20339  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cn 22594  df-tx 22929  df-tmd 23439  df-tgp 23440  df-trg 23527  df-tlm 23529
This theorem is referenced by: (None)
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