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Theorem xkofvcn 21297
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21269.) (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 21086 . . . 4 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3 eqid 2610 . . . . 5 (𝑆 ^ko 𝑅) = (𝑆 ^ko 𝑅)
43xkotopon 21213 . . . 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 20548 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
96, 8sylib 207 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
105, 9cnmpt1st 21281 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑓) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn (𝑆 ^ko 𝑅)))
115, 9cnmpt2nd 21282 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑥) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑅))
12 1on 7454 . . . . . . 7 1𝑜 ∈ On
13 distopon 20611 . . . . . . 7 (1𝑜 ∈ On → 𝒫 1𝑜 ∈ (TopOn‘1𝑜))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ (TopOn‘1𝑜))
15 xkoccn 21232 . . . . . 6 ((𝒫 1𝑜 ∈ (TopOn‘1𝑜) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
1614, 9, 15syl2anc 691 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦𝑋 ↦ (1𝑜 × {𝑦})) ∈ (𝑅 Cn (𝑅 ^ko 𝒫 1𝑜)))
17 sneq 4135 . . . . . 6 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1817xpeq2d 5063 . . . . 5 (𝑦 = 𝑥 → (1𝑜 × {𝑦}) = (1𝑜 × {𝑥}))
195, 9, 11, 9, 16, 18cnmpt21 21284 . . . 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 20610 . . . . . 6 (1𝑜 ∈ On → 𝒫 1𝑜 ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1𝑜 ∈ Top)
22 eqid 2610 . . . . . 6 (𝑅 ^ko 𝒫 1𝑜) = (𝑅 ^ko 𝒫 1𝑜)
2322xkotopon 21213 . . . . 5 ((𝒫 1𝑜 ∈ Top ∧ 𝑅 ∈ Top) → (𝑅 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑅)))
2421, 6, 23syl2anc 691 . . . 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 2610 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1𝑜 Cn 𝑅) ↦ (𝑔))
2827xkococn 21273 . . . . 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 1318 . . . 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 5201 . . . . 5 (𝑔 = 𝑓 → (𝑔) = (𝑓))
31 coeq2 5202 . . . . 5 ( = (1𝑜 × {𝑥}) → (𝑓) = (𝑓 ∘ (1𝑜 × {𝑥})))
3230, 31sylan9eq 2664 . . . 4 ((𝑔 = 𝑓 = (1𝑜 × {𝑥})) → (𝑔) = (𝑓 ∘ (1𝑜 × {𝑥})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 21287 . . 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 2610 . . . . 5 (𝑆 ^ko 𝒫 1𝑜) = (𝑆 ^ko 𝒫 1𝑜)
3534xkotopon 21213 . . . 4 ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
3621, 26, 35syl2anc 691 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝒫 1𝑜) ∈ (TopOn‘(𝒫 1𝑜 Cn 𝑆)))
37 0lt1o 7471 . . . . 5 ∅ ∈ 1𝑜
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1𝑜)
39 unipw 4845 . . . . . 6 𝒫 1𝑜 = 1𝑜
4039eqcomi 2619 . . . . 5 1𝑜 = 𝒫 1𝑜
4140xkopjcn 21269 . . . 4 ((𝒫 1𝑜 ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1𝑜) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
4221, 26, 38, 41syl3anc 1318 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1𝑜 Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆 ^ko 𝒫 1𝑜) Cn 𝑆))
43 fveq1 6102 . . . 4 (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅))
44 vex 3176 . . . . . . 7 𝑥 ∈ V
4544fconst 6004 . . . . . 6 (1𝑜 × {𝑥}):1𝑜⟶{𝑥}
46 fvco3 6185 . . . . . 6 (((1𝑜 × {𝑥}):1𝑜⟶{𝑥} ∧ ∅ ∈ 1𝑜) → ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅)))
4745, 37, 46mp2an 704 . . . . 5 ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓‘((1𝑜 × {𝑥})‘∅))
4844fvconst2 6374 . . . . . . 7 (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
4937, 48ax-mp 5 . . . . . 6 ((1𝑜 × {𝑥})‘∅) = 𝑥
5049fveq2i 6106 . . . . 5 (𝑓‘((1𝑜 × {𝑥})‘∅)) = (𝑓𝑥)
5147, 50eqtri 2632 . . . 4 ((𝑓 ∘ (1𝑜 × {𝑥}))‘∅) = (𝑓𝑥)
5243, 51syl6eq 2660 . . 3 (𝑔 = (𝑓 ∘ (1𝑜 × {𝑥})) → (𝑔‘∅) = (𝑓𝑥))
535, 9, 33, 36, 42, 52cnmpt21 21284 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥)) ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))
541, 53syl5eqel 2692 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 1475  wcel 1977  c0 3874  𝒫 cpw 4108  {csn 4125   cuni 4372  cmpt 4643   × cxp 5036  ccom 5042  Oncon0 5640  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  1𝑜c1o 7440  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Compccmp 20999  𝑛-Locally cnlly 21078   ×t ctx 21173   ^ko cxko 21174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-topon 20523  df-ntr 20634  df-nei 20712  df-cn 20841  df-cnp 20842  df-cmp 21000  df-nlly 21080  df-tx 21175  df-xko 21176
This theorem is referenced by:  cnmptk1p  21298  cnmptk2  21299
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