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Theorem xkofvcn 23692
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23664.) (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 23481 . . . 4 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3 eqid 2737 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
43xkotopon 23608 . . . 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 22923 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
96, 8sylib 218 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
105, 9cnmpt1st 23676 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑓) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑆ko 𝑅)))
115, 9cnmpt2nd 23677 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑥) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑅))
12 1on 8518 . . . . . . 7 1o ∈ On
13 distopon 23004 . . . . . . 7 (1o ∈ On → 𝒫 1o ∈ (TopOn‘1o))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ (TopOn‘1o))
15 xkoccn 23627 . . . . . 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 4636 . . . . . 6 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1817xpeq2d 5715 . . . . 5 (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
195, 9, 11, 9, 16, 18cnmpt21 23679 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (1o × {𝑥})) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑅ko 𝒫 1o)))
20 distop 23002 . . . . . 6 (1o ∈ On → 𝒫 1o ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22 eqid 2737 . . . . . 6 (𝑅ko 𝒫 1o) = (𝑅ko 𝒫 1o)
2322xkotopon 23608 . . . . 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 2737 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔))
2827xkococn 23668 . . . . 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 1373 . . . 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 5868 . . . . 5 (𝑔 = 𝑓 → (𝑔) = (𝑓))
31 coeq2 5869 . . . . 5 ( = (1o × {𝑥}) → (𝑓) = (𝑓 ∘ (1o × {𝑥})))
3230, 31sylan9eq 2797 . . . 4 ((𝑔 = 𝑓 = (1o × {𝑥})) → (𝑔) = (𝑓 ∘ (1o × {𝑥})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 23682 . . 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 2737 . . . . 5 (𝑆ko 𝒫 1o) = (𝑆ko 𝒫 1o)
3534xkotopon 23608 . . . 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 8542 . . . . 5 ∅ ∈ 1o
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39 unipw 5455 . . . . . 6 𝒫 1o = 1o
4039eqcomi 2746 . . . . 5 1o = 𝒫 1o
4140xkopjcn 23664 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1o) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
4221, 26, 38, 41syl3anc 1373 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
43 fveq1 6905 . . . 4 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44 vex 3484 . . . . . . 7 𝑥 ∈ V
4544fconst 6794 . . . . . 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 6909 . . . . 5 (𝑓‘((1o × {𝑥})‘∅)) = (𝑓𝑥)
5147, 50eqtri 2765 . . . 4 ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓𝑥)
5243, 51eqtrdi 2793 . . 3 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓𝑥))
535, 9, 33, 36, 42, 52cnmpt21 23679 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥)) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑆))
541, 53eqeltrid 2845 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 1540  wcel 2108  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  ccom 5689  Oncon0 6384  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1oc1o 8499  Topctop 22899  TopOnctopon 22916   Cn ccn 23232  Compccmp 23394  𝑛-Locally cnlly 23473   ×t ctx 23568  ko cxko 23569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-pt 17489  df-top 22900  df-topon 22917  df-bases 22953  df-ntr 23028  df-nei 23106  df-cn 23235  df-cnp 23236  df-cmp 23395  df-nlly 23475  df-tx 23570  df-xko 23571
This theorem is referenced by:  cnmptk1p  23693  cnmptk2  23694
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