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Theorem xkofvcn 23187
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23159.) (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 22976 . . . 4 (𝑅 ∈ 𝑛-Locally Comp β†’ 𝑅 ∈ Top)
3 eqid 2732 . . . . 5 (𝑆 ↑ko 𝑅) = (𝑆 ↑ko 𝑅)
43xkotopon 23103 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑆)))
52, 4sylan 580 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝑅) ∈ (TopOnβ€˜(𝑅 Cn 𝑆)))
62adantr 481 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑅 ∈ Top)
7 xkofvcn.1 . . . . 5 𝑋 = βˆͺ 𝑅
87toptopon 22418 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOnβ€˜π‘‹))
96, 8sylib 217 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑅 ∈ (TopOnβ€˜π‘‹))
105, 9cnmpt1st 23171 . . . 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 23172 . . . . 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 8477 . . . . . . 7 1o ∈ On
13 distopon 22499 . . . . . . 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 23122 . . . . . 6 ((𝒫 1o ∈ (TopOnβ€˜1o) ∧ 𝑅 ∈ (TopOnβ€˜π‘‹)) β†’ (𝑦 ∈ 𝑋 ↦ (1o Γ— {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
1614, 9, 15syl2anc 584 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑦 ∈ 𝑋 ↦ (1o Γ— {𝑦})) ∈ (𝑅 Cn (𝑅 ↑ko 𝒫 1o)))
17 sneq 4638 . . . . . 6 (𝑦 = π‘₯ β†’ {𝑦} = {π‘₯})
1817xpeq2d 5706 . . . . 5 (𝑦 = π‘₯ β†’ (1o Γ— {𝑦}) = (1o Γ— {π‘₯}))
195, 9, 11, 9, 16, 18cnmpt21 23174 . . . 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 22497 . . . . . 6 (1o ∈ On β†’ 𝒫 1o ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝒫 1o ∈ Top)
22 eqid 2732 . . . . . 6 (𝑅 ↑ko 𝒫 1o) = (𝑅 ↑ko 𝒫 1o)
2322xkotopon 23103 . . . . 5 ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) β†’ (𝑅 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑅)))
2421, 6, 23syl2anc 584 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑅 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑅)))
25 simpl 483 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑅 ∈ 𝑛-Locally Comp)
26 simpr 485 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ 𝑆 ∈ Top)
27 eqid 2732 . . . . . 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 23163 . . . . 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 1371 . . . 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 5857 . . . . 5 (𝑔 = 𝑓 β†’ (𝑔 ∘ β„Ž) = (𝑓 ∘ β„Ž))
31 coeq2 5858 . . . . 5 (β„Ž = (1o Γ— {π‘₯}) β†’ (𝑓 ∘ β„Ž) = (𝑓 ∘ (1o Γ— {π‘₯})))
3230, 31sylan9eq 2792 . . . 4 ((𝑔 = 𝑓 ∧ β„Ž = (1o Γ— {π‘₯})) β†’ (𝑔 ∘ β„Ž) = (𝑓 ∘ (1o Γ— {π‘₯})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 23177 . . 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 2732 . . . . 5 (𝑆 ↑ko 𝒫 1o) = (𝑆 ↑ko 𝒫 1o)
3534xkotopon 23103 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑆)))
3621, 26, 35syl2anc 584 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑆 ↑ko 𝒫 1o) ∈ (TopOnβ€˜(𝒫 1o Cn 𝑆)))
37 0lt1o 8503 . . . . 5 βˆ… ∈ 1o
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ βˆ… ∈ 1o)
39 unipw 5450 . . . . . 6 βˆͺ 𝒫 1o = 1o
4039eqcomi 2741 . . . . 5 1o = βˆͺ 𝒫 1o
4140xkopjcn 23159 . . . 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 1371 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) β†’ (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (π‘”β€˜βˆ…)) ∈ ((𝑆 ↑ko 𝒫 1o) Cn 𝑆))
43 fveq1 6890 . . . 4 (𝑔 = (𝑓 ∘ (1o Γ— {π‘₯})) β†’ (π‘”β€˜βˆ…) = ((𝑓 ∘ (1o Γ— {π‘₯}))β€˜βˆ…))
44 vex 3478 . . . . . . 7 π‘₯ ∈ V
4544fconst 6777 . . . . . 6 (1o Γ— {π‘₯}):1o⟢{π‘₯}
46 fvco3 6990 . . . . . 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 690 . . . . 5 ((𝑓 ∘ (1o Γ— {π‘₯}))β€˜βˆ…) = (π‘“β€˜((1o Γ— {π‘₯})β€˜βˆ…))
4844fvconst2 7204 . . . . . . 7 (βˆ… ∈ 1o β†’ ((1o Γ— {π‘₯})β€˜βˆ…) = π‘₯)
4937, 48ax-mp 5 . . . . . 6 ((1o Γ— {π‘₯})β€˜βˆ…) = π‘₯
5049fveq2i 6894 . . . . 5 (π‘“β€˜((1o Γ— {π‘₯})β€˜βˆ…)) = (π‘“β€˜π‘₯)
5147, 50eqtri 2760 . . . 4 ((𝑓 ∘ (1o Γ— {π‘₯}))β€˜βˆ…) = (π‘“β€˜π‘₯)
5243, 51eqtrdi 2788 . . 3 (𝑔 = (𝑓 ∘ (1o Γ— {π‘₯})) β†’ (π‘”β€˜βˆ…) = (π‘“β€˜π‘₯))
535, 9, 33, 36, 42, 52cnmpt21 23174 . 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 2837 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 396   = wceq 1541   ∈ wcel 2106  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908   ↦ cmpt 5231   Γ— cxp 5674   ∘ ccom 5680  Oncon0 6364  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  1oc1o 8458  Topctop 22394  TopOnctopon 22411   Cn ccn 22727  Compccmp 22889  π‘›-Locally cnlly 22968   Γ—t ctx 23063   ↑ko cxko 23064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-1o 8465  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-pt 17389  df-top 22395  df-topon 22412  df-bases 22448  df-ntr 22523  df-nei 22601  df-cn 22730  df-cnp 22731  df-cmp 22890  df-nlly 22970  df-tx 23065  df-xko 23066
This theorem is referenced by:  cnmptk1p  23188  cnmptk2  23189
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