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Theorem xkofvcn 21397
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21369.) (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 21186 . . . 4 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3 eqid 2621 . . . . 5 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
43xkotopon 21313 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
52, 4sylan 488 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
62adantr 481 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7 xkofvcn.1 . . . . 5 𝑋 = 𝑅
87toptopon 20648 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
96, 8sylib 208 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
105, 9cnmpt1st 21381 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑓) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn (𝑆 ^ko 𝑅)))
115, 9cnmpt2nd 21382 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑥) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑅))
12 1on 7512 . . . . . . 7 1𝑜 ∈ On
13 distopon 20711 . . . . . . 7 (1𝑜 ∈ On → 𝒫 1𝑜 ∈ (TopOn‘1𝑜))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ (TopOn‘1𝑜))
15 xkoccn 21332 . . . . . 6 ((𝒫 1𝑜 ∈ (TopOn‘1𝑜) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
1614, 9, 15syl2anc 692 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
17 sneq 4158 . . . . . 6 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1817xpeq2d 5099 . . . . 5 (𝑦 = 𝑥 → (1𝑜 × {𝑦}) = (1𝑜 × {𝑥}))
195, 9, 11, 9, 16, 18cnmpt21 21384 . . . 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 20710 . . . . . 6 (1𝑜 ∈ On → 𝒫 1𝑜 ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ Top)
22 eqid 2621 . . . . . 6 (𝑅 ^ko 𝒫 1𝑜) = (𝑅 ^ko 𝒫 1𝑜)
2322xkotopon 21313 . . . . 5 ((𝒫 1𝑜 ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
2421, 6, 23syl2anc 692 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
25 simpl 473 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26 simpr 477 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27 eqid 2621 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔))
2827xkococn 21373 . . . . 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 1323 . . . 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 5239 . . . . 5 (𝑔 = 𝑓 → (𝑔) = (𝑓))
31 coeq2 5240 . . . . 5 ( = (1𝑜 × {𝑥}) → (𝑓) = (𝑓 ∘ (1𝑜 × {𝑥})))
3230, 31sylan9eq 2675 . . . 4 ((𝑔 = 𝑓 = (1𝑜 × {𝑥})) → (𝑔) = (𝑓 ∘ (1𝑜 × {𝑥})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 21387 . . 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 2621 . . . . 5 (𝑆 ^ko 𝒫 1𝑜) = (𝑆 ^ko 𝒫 1𝑜)
3534xkotopon 21313 . . . 4 ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
3621, 26, 35syl2anc 692 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
37 0lt1o 7529 . . . . 5 ∅ ∈ 1𝑜
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1𝑜)
39 unipw 4879 . . . . . 6 𝒫 1𝑜 = 1𝑜
4039eqcomi 2630 . . . . 5 1𝑜 = 𝒫 1𝑜
4140xkopjcn 21369 . . . 4 ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1𝑜) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
4221, 26, 38, 41syl3anc 1323 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
43 fveq1 6147 . . . 4 (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅))
44 vex 3189 . . . . . . 7 𝑥 ∈ V
4544fconst 6048 . . . . . 6 (1𝑜 × {𝑥}):1𝑜⟶{𝑥}
46 fvco3 6232 . . . . . 6 (((1𝑜 × {𝑥}):1𝑜⟶{𝑥} ∧ ∅ ∈ 1𝑜) → ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅)))
4745, 37, 46mp2an 707 . . . . 5 ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅))
4844fvconst2 6423 . . . . . . 7 (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
4937, 48ax-mp 5 . . . . . 6 ((1𝑜 × {𝑥})‘∅) = 𝑥
5049fveq2i 6151 . . . . 5 (𝑓‘((1𝑜 × {𝑥})‘∅)) = (𝑓𝑥)
5147, 50eqtri 2643 . . . 4 ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓𝑥)
5243, 51syl6eq 2671 . . 3 (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = (𝑓𝑥))
535, 9, 33, 36, 42, 52cnmpt21 21384 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥)) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))
541, 53syl5eqel 2702 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 384   = wceq 1480  wcel 1987  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402  cmpt 4673   × cxp 5072  ccom 5078  Oncon0 5682  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1𝑜c1o 7498  Topctop 20617  TopOnctopon 20618   Cn ccn 20938  Compccmp 21099  𝑛-Locally cnlly 21178   ×t ctx 21273   ^ko cxko 21274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-pt 16026  df-top 20621  df-bases 20622  df-topon 20623  df-ntr 20734  df-nei 20812  df-cn 20941  df-cnp 20942  df-cmp 21100  df-nlly 21180  df-tx 21275  df-xko 21276
This theorem is referenced by:  cnmptk1p  21398  cnmptk2  21399
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