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Theorem xkofvcn 23708
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23680.) (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 ∈ (((𝑆ko 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 23497 . . . 4 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3 eqid 2735 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
43xkotopon 23624 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
52, 4sylan 580 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
62adantr 480 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7 xkofvcn.1 . . . . 5 𝑋 = 𝑅
87toptopon 22939 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
96, 8sylib 218 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
105, 9cnmpt1st 23692 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑓) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑆ko 𝑅)))
115, 9cnmpt2nd 23693 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑥) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑅))
12 1on 8517 . . . . . . 7 1o ∈ On
13 distopon 23020 . . . . . . 7 (1o ∈ On → 𝒫 1o ∈ (TopOn‘1o))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ (TopOn‘1o))
15 xkoccn 23643 . . . . . 6 ((𝒫 1o ∈ (TopOn‘1o) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅ko 𝒫 1o)))
1614, 9, 15syl2anc 584 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅ko 𝒫 1o)))
17 sneq 4641 . . . . . 6 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1817xpeq2d 5719 . . . . 5 (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
195, 9, 11, 9, 16, 18cnmpt21 23695 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (1o × {𝑥})) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑅ko 𝒫 1o)))
20 distop 23018 . . . . . 6 (1o ∈ On → 𝒫 1o ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22 eqid 2735 . . . . . 6 (𝑅ko 𝒫 1o) = (𝑅ko 𝒫 1o)
2322xkotopon 23624 . . . . 5 ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) → (𝑅ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
2421, 6, 23syl2anc 584 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
25 simpl 482 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26 simpr 484 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27 eqid 2735 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔))
2827xkococn 23684 . . . . 5 ((𝒫 1o ∈ Top ∧ 𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔)) ∈ (((𝑆ko 𝑅) ×t (𝑅ko 𝒫 1o)) Cn (𝑆ko 𝒫 1o)))
2921, 25, 26, 28syl3anc 1370 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔)) ∈ (((𝑆ko 𝑅) ×t (𝑅ko 𝒫 1o)) Cn (𝑆ko 𝒫 1o)))
30 coeq1 5871 . . . . 5 (𝑔 = 𝑓 → (𝑔) = (𝑓))
31 coeq2 5872 . . . . 5 ( = (1o × {𝑥}) → (𝑓) = (𝑓 ∘ (1o × {𝑥})))
3230, 31sylan9eq 2795 . . . 4 ((𝑔 = 𝑓 = (1o × {𝑥})) → (𝑔) = (𝑓 ∘ (1o × {𝑥})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 23698 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓 ∘ (1o × {𝑥}))) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑆ko 𝒫 1o)))
34 eqid 2735 . . . . 5 (𝑆ko 𝒫 1o) = (𝑆ko 𝒫 1o)
3534xkotopon 23624 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
3621, 26, 35syl2anc 584 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
37 0lt1o 8541 . . . . 5 ∅ ∈ 1o
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39 unipw 5461 . . . . . 6 𝒫 1o = 1o
4039eqcomi 2744 . . . . 5 1o = 𝒫 1o
4140xkopjcn 23680 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1o) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
4221, 26, 38, 41syl3anc 1370 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
43 fveq1 6906 . . . 4 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44 vex 3482 . . . . . . 7 𝑥 ∈ V
4544fconst 6795 . . . . . 6 (1o × {𝑥}):1o⟶{𝑥}
46 fvco3 7008 . . . . . 6 (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
4745, 37, 46mp2an 692 . . . . 5 ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
4844fvconst2 7224 . . . . . . 7 (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
4937, 48ax-mp 5 . . . . . 6 ((1o × {𝑥})‘∅) = 𝑥
5049fveq2i 6910 . . . . 5 (𝑓‘((1o × {𝑥})‘∅)) = (𝑓𝑥)
5147, 50eqtri 2763 . . . 4 ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓𝑥)
5243, 51eqtrdi 2791 . . 3 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓𝑥))
535, 9, 33, 36, 42, 52cnmpt21 23695 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥)) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑆))
541, 53eqeltrid 2843 1 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231   × cxp 5687  ccom 5693  Oncon0 6386  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1oc1o 8498  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  Compccmp 23410  𝑛-Locally cnlly 23489   ×t ctx 23584  ko cxko 23585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-pt 17491  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-cn 23251  df-cnp 23252  df-cmp 23411  df-nlly 23491  df-tx 23586  df-xko 23587
This theorem is referenced by:  cnmptk1p  23709  cnmptk2  23710
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