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Theorem xkofvcn 21535
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21507.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
xkofvcn.1 𝑋 = 𝑅
xkofvcn.2 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥))
Assertion
Ref Expression
xkofvcn ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))
Distinct variable groups:   𝑥,𝑓,𝑅   𝑆,𝑓,𝑥   𝑓,𝑋,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑓)

Proof of Theorem xkofvcn
Dummy variables 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xkofvcn.2 . 2 𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥))
2 nllytop 21324 . . . 4 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3 eqid 2651 . . . . 5 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
43xkotopon 21451 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
52, 4sylan 487 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
62adantr 480 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7 xkofvcn.1 . . . . 5 𝑋 = 𝑅
87toptopon 20770 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
96, 8sylib 208 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
105, 9cnmpt1st 21519 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑓) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn (𝑆 ^ko 𝑅)))
115, 9cnmpt2nd 21520 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑥) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑅))
12 1on 7612 . . . . . . 7 1𝑜 ∈ On
13 distopon 20849 . . . . . . 7 (1𝑜 ∈ On → 𝒫 1𝑜 ∈ (TopOn‘1𝑜))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ (TopOn‘1𝑜))
15 xkoccn 21470 . . . . . 6 ((𝒫 1𝑜 ∈ (TopOn‘1𝑜) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
1614, 9, 15syl2anc 694 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
17 sneq 4220 . . . . . 6 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1817xpeq2d 5173 . . . . 5 (𝑦 = 𝑥 → (1𝑜 × {𝑦}) = (1𝑜 × {𝑥}))
195, 9, 11, 9, 16, 18cnmpt21 21522 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (1𝑜 × {𝑥})) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn (𝑅 ^ko 𝒫 1𝑜)))
20 distop 20847 . . . . . 6 (1𝑜 ∈ On → 𝒫 1𝑜 ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ Top)
22 eqid 2651 . . . . . 6 (𝑅 ^ko 𝒫 1𝑜) = (𝑅 ^ko 𝒫 1𝑜)
2322xkotopon 21451 . . . . 5 ((𝒫 1𝑜 ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
2421, 6, 23syl2anc 694 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
25 simpl 472 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26 simpr 476 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27 eqid 2651 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔))
2827xkococn 21511 . . . . 5 ((𝒫 1𝑜 ∈ Top ∧ 𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔)) ∈ (((𝑆 ^ko 𝑅) ×t (𝑅 ^ko 𝒫 1𝑜)) Cn (𝑆 ^ko 𝒫 1𝑜)))
2921, 25, 26, 28syl3anc 1366 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔)) ∈ (((𝑆 ^ko 𝑅) ×t (𝑅 ^ko 𝒫 1𝑜)) Cn (𝑆 ^ko 𝒫 1𝑜)))
30 coeq1 5312 . . . . 5 (𝑔 = 𝑓 → (𝑔) = (𝑓))
31 coeq2 5313 . . . . 5 ( = (1𝑜 × {𝑥}) → (𝑓) = (𝑓 ∘ (1𝑜 × {𝑥})))
3230, 31sylan9eq 2705 . . . 4 ((𝑔 = 𝑓 = (1𝑜 × {𝑥})) → (𝑔) = (𝑓 ∘ (1𝑜 × {𝑥})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 21525 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓 ∘ (1𝑜 × {𝑥}))) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn (𝑆 ^ko 𝒫 1𝑜)))
34 eqid 2651 . . . . 5 (𝑆 ^ko 𝒫 1𝑜) = (𝑆 ^ko 𝒫 1𝑜)
3534xkotopon 21451 . . . 4 ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
3621, 26, 35syl2anc 694 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
37 0lt1o 7629 . . . . 5 ∅ ∈ 1𝑜
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1𝑜)
39 unipw 4948 . . . . . 6 𝒫 1𝑜 = 1𝑜
4039eqcomi 2660 . . . . 5 1𝑜 = 𝒫 1𝑜
4140xkopjcn 21507 . . . 4 ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1𝑜) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
4221, 26, 38, 41syl3anc 1366 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
43 fveq1 6228 . . . 4 (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅))
44 vex 3234 . . . . . . 7 𝑥 ∈ V
4544fconst 6129 . . . . . 6 (1𝑜 × {𝑥}):1𝑜⟶{𝑥}
46 fvco3 6314 . . . . . 6 (((1𝑜 × {𝑥}):1𝑜⟶{𝑥} ∧ ∅ ∈ 1𝑜) → ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅)))
4745, 37, 46mp2an 708 . . . . 5 ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅))
4844fvconst2 6510 . . . . . . 7 (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
4937, 48ax-mp 5 . . . . . 6 ((1𝑜 × {𝑥})‘∅) = 𝑥
5049fveq2i 6232 . . . . 5 (𝑓‘((1𝑜 × {𝑥})‘∅)) = (𝑓𝑥)
5147, 50eqtri 2673 . . . 4 ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓𝑥)
5243, 51syl6eq 2701 . . 3 (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = (𝑓𝑥))
535, 9, 33, 36, 42, 52cnmpt21 21522 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥)) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))
541, 53syl5eqel 2734 1 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762   × cxp 5141  ccom 5147  Oncon0 5761  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1𝑜c1o 7598  Topctop 20746  TopOnctopon 20763   Cn ccn 21076  Compccmp 21237  𝑛-Locally cnlly 21316   ×t ctx 21411   ^ko cxko 21412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-pt 16152  df-top 20747  df-topon 20764  df-bases 20798  df-ntr 20872  df-nei 20950  df-cn 21079  df-cnp 21080  df-cmp 21238  df-nlly 21318  df-tx 21413  df-xko 21414
This theorem is referenced by:  cnmptk1p  21536  cnmptk2  21537
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