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Theorem xkofvcn 22284
 Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 22256.) (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 22073 . . . 4 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3 eqid 2819 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
43xkotopon 22200 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
52, 4sylan 582 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
62adantr 483 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7 xkofvcn.1 . . . . 5 𝑋 = 𝑅
87toptopon 21517 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
96, 8sylib 220 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
105, 9cnmpt1st 22268 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑓) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑆ko 𝑅)))
115, 9cnmpt2nd 22269 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑥) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑅))
12 1on 8101 . . . . . . 7 1o ∈ On
13 distopon 21597 . . . . . . 7 (1o ∈ On → 𝒫 1o ∈ (TopOn‘1o))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ (TopOn‘1o))
15 xkoccn 22219 . . . . . 6 ((𝒫 1o ∈ (TopOn‘1o) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅ko 𝒫 1o)))
1614, 9, 15syl2anc 586 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅ko 𝒫 1o)))
17 sneq 4569 . . . . . 6 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1817xpeq2d 5578 . . . . 5 (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
195, 9, 11, 9, 16, 18cnmpt21 22271 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (1o × {𝑥})) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑅ko 𝒫 1o)))
20 distop 21595 . . . . . 6 (1o ∈ On → 𝒫 1o ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22 eqid 2819 . . . . . 6 (𝑅ko 𝒫 1o) = (𝑅ko 𝒫 1o)
2322xkotopon 22200 . . . . 5 ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) → (𝑅ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
2421, 6, 23syl2anc 586 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
25 simpl 485 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26 simpr 487 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27 eqid 2819 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔))
2827xkococn 22260 . . . . 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 1365 . . . 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 5721 . . . . 5 (𝑔 = 𝑓 → (𝑔) = (𝑓))
31 coeq2 5722 . . . . 5 ( = (1o × {𝑥}) → (𝑓) = (𝑓 ∘ (1o × {𝑥})))
3230, 31sylan9eq 2874 . . . 4 ((𝑔 = 𝑓 = (1o × {𝑥})) → (𝑔) = (𝑓 ∘ (1o × {𝑥})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 22274 . . 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 2819 . . . . 5 (𝑆ko 𝒫 1o) = (𝑆ko 𝒫 1o)
3534xkotopon 22200 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
3621, 26, 35syl2anc 586 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
37 0lt1o 8121 . . . . 5 ∅ ∈ 1o
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39 unipw 5333 . . . . . 6 𝒫 1o = 1o
4039eqcomi 2828 . . . . 5 1o = 𝒫 1o
4140xkopjcn 22256 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1o) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
4221, 26, 38, 41syl3anc 1365 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
43 fveq1 6662 . . . 4 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44 vex 3496 . . . . . . 7 𝑥 ∈ V
4544fconst 6558 . . . . . 6 (1o × {𝑥}):1o⟶{𝑥}
46 fvco3 6753 . . . . . 6 (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
4745, 37, 46mp2an 690 . . . . 5 ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
4844fvconst2 6959 . . . . . . 7 (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
4937, 48ax-mp 5 . . . . . 6 ((1o × {𝑥})‘∅) = 𝑥
5049fveq2i 6666 . . . . 5 (𝑓‘((1o × {𝑥})‘∅)) = (𝑓𝑥)
5147, 50eqtri 2842 . . . 4 ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓𝑥)
5243, 51syl6eq 2870 . . 3 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓𝑥))
535, 9, 33, 36, 42, 52cnmpt21 22271 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥)) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑆))
541, 53eqeltrid 2915 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 398   = wceq 1530   ∈ wcel 2107  ∅c0 4289  𝒫 cpw 4537  {csn 4559  ∪ cuni 4830   ↦ cmpt 5137   × cxp 5546   ∘ ccom 5552  Oncon0 6184  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148   ∈ cmpo 7150  1oc1o 8087  Topctop 21493  TopOnctopon 21510   Cn ccn 21824  Compccmp 21986  𝑛-Locally cnlly 22065   ×t ctx 22160   ↑ko cxko 22161 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-rest 16688  df-topgen 16709  df-pt 16710  df-top 21494  df-topon 21511  df-bases 21546  df-ntr 21620  df-nei 21698  df-cn 21827  df-cnp 21828  df-cmp 21987  df-nlly 22067  df-tx 22162  df-xko 22163 This theorem is referenced by:  cnmptk1p  22285  cnmptk2  22286
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