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Theorem xkofvcn 23058
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23030.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
xkofvcn.1 𝑋 = βˆͺ 𝑅
xkofvcn.2 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ (π‘“β€˜π‘₯))
Assertion
Ref Expression
xkofvcn ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝐹 ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn 𝑆))
Distinct variable groups:   π‘₯,𝑓,𝑅   𝑆,𝑓,π‘₯   𝑓,𝑋,π‘₯
Allowed substitution hints:   𝐹(π‘₯,𝑓)

Proof of Theorem xkofvcn
Dummy variables 𝑔 β„Ž 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkofvcn.2 . 2 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ (π‘“β€˜π‘₯))
2 nllytop 22847 . . . 4 (𝑅 ∈ 𝑛-Locally Comp β†’ 𝑅 ∈ Top)
3 eqid 2733 . . . . 5 (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
43xkotopon 22974 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑆)))
52, 4sylan 581 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑆)))
62adantr 482 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑅 ∈ Top)
7 xkofvcn.1 . . . . 5 𝑋 = βˆͺ 𝑅
87toptopon 22289 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOnβ€˜π‘‹))
96, 8sylib 217 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑅 ∈ (TopOnβ€˜π‘‹))
105, 9cnmpt1st 23042 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ 𝑓) ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn (𝑆 ↑ko 𝑅)))
115, 9cnmpt2nd 23043 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ π‘₯) ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn 𝑅))
12 1on 8428 . . . . . . 7 1o ∈ On
13 distopon 22370 . . . . . . 7 (1o ∈ On β†’ 𝒫 1o ∈ (TopOnβ€˜1o))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝒫 1o ∈ (TopOnβ€˜1o))
15 xkoccn 22993 . . . . . 6 ((𝒫 1o ∈ (TopOnβ€˜1o) ∧ 𝑅 ∈ (TopOnβ€˜π‘‹)) β†’ (𝑦 ∈ 𝑋 ↦ (1o Γ— {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
1614, 9, 15syl2anc 585 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑦 ∈ 𝑋 ↦ (1o Γ— {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
17 sneq 4600 . . . . . 6 (𝑦 = π‘₯ β†’ {𝑦} = {π‘₯})
1817xpeq2d 5667 . . . . 5 (𝑦 = π‘₯ β†’ (1o Γ— {𝑦}) = (1o Γ— {π‘₯}))
195, 9, 11, 9, 16, 18cnmpt21 23045 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ (1o Γ— {π‘₯})) ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn (𝑅 ↑ko 𝒫 1o)))
20 distop 22368 . . . . . 6 (1o ∈ On β†’ 𝒫 1o ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝒫 1o ∈ Top)
22 eqid 2733 . . . . . 6 (𝑅 ↑ko 𝒫 1o) = (𝑅 ↑ko 𝒫 1o)
2322xkotopon 22974 . . . . 5 ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) β†’ (𝑅 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑅)))
2421, 6, 23syl2anc 585 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑅 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑅)))
25 simpl 484 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑅 ∈ 𝑛-Locally Comp)
26 simpr 486 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑆 ∈ Top)
27 eqid 2733 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), β„Ž ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ β„Ž)) = (𝑔 ∈ (𝑅 Cn 𝑆), β„Ž ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ β„Ž))
2827xkococn 23034 . . . . 5 ((𝒫 1o ∈ Top ∧ 𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑔 ∈ (𝑅 Cn 𝑆), β„Ž ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ β„Ž)) ∈ (((𝑆 ↑ko 𝑅) Γ—t (𝑅 ↑ko 𝒫 1o)) Cn (𝑆 ↑ko 𝒫 1o)))
2921, 25, 26, 28syl3anc 1372 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑔 ∈ (𝑅 Cn 𝑆), β„Ž ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔 ∘ β„Ž)) ∈ (((𝑆 ↑ko 𝑅) Γ—t (𝑅 ↑ko 𝒫 1o)) Cn (𝑆 ↑ko 𝒫 1o)))
30 coeq1 5817 . . . . 5 (𝑔 = 𝑓 β†’ (𝑔 ∘ β„Ž) = (𝑓 ∘ β„Ž))
31 coeq2 5818 . . . . 5 (β„Ž = (1o Γ— {π‘₯}) β†’ (𝑓 ∘ β„Ž) = (𝑓 ∘ (1o Γ— {π‘₯})))
3230, 31sylan9eq 2793 . . . 4 ((𝑔 = 𝑓 ∧ β„Ž = (1o Γ— {π‘₯})) β†’ (𝑔 ∘ β„Ž) = (𝑓 ∘ (1o Γ— {π‘₯})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 23048 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ (𝑓 ∘ (1o Γ— {π‘₯}))) ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn (𝑆 ↑ko 𝒫 1o)))
34 eqid 2733 . . . . 5 (𝑆 ↑ko 𝒫 1o) = (𝑆 ↑ko 𝒫 1o)
3534xkotopon 22974 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑆)))
3621, 26, 35syl2anc 585 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑆)))
37 0lt1o 8454 . . . . 5 βˆ… ∈ 1o
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ βˆ… ∈ 1o)
39 unipw 5411 . . . . . 6 βˆͺ 𝒫 1o = 1o
4039eqcomi 2742 . . . . 5 1o = βˆͺ 𝒫 1o
4140xkopjcn 23030 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top ∧ βˆ… ∈ 1o) β†’ (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (π‘”β€˜βˆ…)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
4221, 26, 38, 41syl3anc 1372 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (π‘”β€˜βˆ…)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
43 fveq1 6845 . . . 4 (𝑔 = (𝑓 ∘ (1o Γ— {π‘₯})) β†’ (π‘”β€˜βˆ…) = ((𝑓 ∘ (1o Γ— {π‘₯}))β€˜βˆ…))
44 vex 3451 . . . . . . 7 π‘₯ ∈ V
4544fconst 6732 . . . . . 6 (1o Γ— {π‘₯}):1o⟢{π‘₯}
46 fvco3 6944 . . . . . 6 (((1o Γ— {π‘₯}):1o⟢{π‘₯} ∧ βˆ… ∈ 1o) β†’ ((𝑓 ∘ (1o Γ— {π‘₯}))β€˜βˆ…) = (π‘“β€˜((1o Γ— {π‘₯})β€˜βˆ…)))
4745, 37, 46mp2an 691 . . . . 5 ((𝑓 ∘ (1o Γ— {π‘₯}))β€˜βˆ…) = (π‘“β€˜((1o Γ— {π‘₯})β€˜βˆ…))
4844fvconst2 7157 . . . . . . 7 (βˆ… ∈ 1o β†’ ((1o Γ— {π‘₯})β€˜βˆ…) = π‘₯)
4937, 48ax-mp 5 . . . . . 6 ((1o Γ— {π‘₯})β€˜βˆ…) = π‘₯
5049fveq2i 6849 . . . . 5 (π‘“β€˜((1o Γ— {π‘₯})β€˜βˆ…)) = (π‘“β€˜π‘₯)
5147, 50eqtri 2761 . . . 4 ((𝑓 ∘ (1o Γ— {π‘₯}))β€˜βˆ…) = (π‘“β€˜π‘₯)
5243, 51eqtrdi 2789 . . 3 (𝑔 = (𝑓 ∘ (1o Γ— {π‘₯})) β†’ (π‘”β€˜βˆ…) = (π‘“β€˜π‘₯))
535, 9, 33, 36, 42, 52cnmpt21 23045 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑓 ∈ (𝑅 Cn 𝑆), π‘₯ ∈ 𝑋 ↦ (π‘“β€˜π‘₯)) ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn 𝑆))
541, 53eqeltrid 2838 1 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝐹 ∈ (((𝑆 ↑ko 𝑅) Γ—t 𝑅) Cn 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ…c0 4286  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869   ↦ cmpt 5192   Γ— cxp 5635   ∘ ccom 5641  Oncon0 6321  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  1oc1o 8409  Topctop 22265  TopOnctopon 22282   Cn ccn 22598  Compccmp 22760  π‘›-Locally cnlly 22839   Γ—t ctx 22934   ↑ko cxko 22935
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-ixp 8842  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-pt 17334  df-top 22266  df-topon 22283  df-bases 22319  df-ntr 22394  df-nei 22472  df-cn 22601  df-cnp 22602  df-cmp 22761  df-nlly 22841  df-tx 22936  df-xko 22937
This theorem is referenced by:  cnmptk1p  23059  cnmptk2  23060
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