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Theorem xkofvcn 23679
Description: Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 23651.) (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 23468 . . . 4 (𝑅 ∈ 𝑛-Locally Comp → 𝑅 ∈ Top)
3 eqid 2726 . . . . 5 (𝑆ko 𝑅) = (𝑆ko 𝑅)
43xkotopon 23595 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
52, 4sylan 578 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
62adantr 479 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ Top)
7 xkofvcn.1 . . . . 5 𝑋 = 𝑅
87toptopon 22910 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
96, 8sylib 217 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ (TopOn‘𝑋))
105, 9cnmpt1st 23663 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑓) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑆ko 𝑅)))
115, 9cnmpt2nd 23664 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋𝑥) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑅))
12 1on 8508 . . . . . . 7 1o ∈ On
13 distopon 22991 . . . . . . 7 (1o ∈ On → 𝒫 1o ∈ (TopOn‘1o))
1412, 13mp1i 13 . . . . . 6 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ (TopOn‘1o))
15 xkoccn 23614 . . . . . 6 ((𝒫 1o ∈ (TopOn‘1o) ∧ 𝑅 ∈ (TopOn‘𝑋)) → (𝑦𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅ko 𝒫 1o)))
1614, 9, 15syl2anc 582 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑦𝑋 ↦ (1o × {𝑦})) ∈ (𝑅 Cn (𝑅ko 𝒫 1o)))
17 sneq 4643 . . . . . 6 (𝑦 = 𝑥 → {𝑦} = {𝑥})
1817xpeq2d 5712 . . . . 5 (𝑦 = 𝑥 → (1o × {𝑦}) = (1o × {𝑥}))
195, 9, 11, 9, 16, 18cnmpt21 23666 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (1o × {𝑥})) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn (𝑅ko 𝒫 1o)))
20 distop 22989 . . . . . 6 (1o ∈ On → 𝒫 1o ∈ Top)
2112, 20mp1i 13 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝒫 1o ∈ Top)
22 eqid 2726 . . . . . 6 (𝑅ko 𝒫 1o) = (𝑅ko 𝒫 1o)
2322xkotopon 23595 . . . . 5 ((𝒫 1o ∈ Top ∧ 𝑅 ∈ Top) → (𝑅ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
2421, 6, 23syl2anc 582 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑅ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑅)))
25 simpl 481 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑅 ∈ 𝑛-Locally Comp)
26 simpr 483 . . . . 5 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝑆 ∈ Top)
27 eqid 2726 . . . . . 6 (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔)) = (𝑔 ∈ (𝑅 Cn 𝑆), ∈ (𝒫 1o Cn 𝑅) ↦ (𝑔))
2827xkococn 23655 . . . . 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 1368 . . . 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 5864 . . . . 5 (𝑔 = 𝑓 → (𝑔) = (𝑓))
31 coeq2 5865 . . . . 5 ( = (1o × {𝑥}) → (𝑓) = (𝑓 ∘ (1o × {𝑥})))
3230, 31sylan9eq 2786 . . . 4 ((𝑔 = 𝑓 = (1o × {𝑥})) → (𝑔) = (𝑓 ∘ (1o × {𝑥})))
335, 9, 10, 19, 5, 24, 29, 32cnmpt22 23669 . . 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 2726 . . . . 5 (𝑆ko 𝒫 1o) = (𝑆ko 𝒫 1o)
3534xkotopon 23595 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
3621, 26, 35syl2anc 582 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑆ko 𝒫 1o) ∈ (TopOn‘(𝒫 1o Cn 𝑆)))
37 0lt1o 8534 . . . . 5 ∅ ∈ 1o
3837a1i 11 . . . 4 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → ∅ ∈ 1o)
39 unipw 5456 . . . . . 6 𝒫 1o = 1o
4039eqcomi 2735 . . . . 5 1o = 𝒫 1o
4140xkopjcn 23651 . . . 4 ((𝒫 1o ∈ Top ∧ 𝑆 ∈ Top ∧ ∅ ∈ 1o) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
4221, 26, 38, 41syl3anc 1368 . . 3 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑔 ∈ (𝒫 1o Cn 𝑆) ↦ (𝑔‘∅)) ∈ ((𝑆ko 𝒫 1o) Cn 𝑆))
43 fveq1 6900 . . . 4 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = ((𝑓 ∘ (1o × {𝑥}))‘∅))
44 vex 3466 . . . . . . 7 𝑥 ∈ V
4544fconst 6788 . . . . . 6 (1o × {𝑥}):1o⟶{𝑥}
46 fvco3 7001 . . . . . 6 (((1o × {𝑥}):1o⟶{𝑥} ∧ ∅ ∈ 1o) → ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅)))
4745, 37, 46mp2an 690 . . . . 5 ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓‘((1o × {𝑥})‘∅))
4844fvconst2 7221 . . . . . . 7 (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
4937, 48ax-mp 5 . . . . . 6 ((1o × {𝑥})‘∅) = 𝑥
5049fveq2i 6904 . . . . 5 (𝑓‘((1o × {𝑥})‘∅)) = (𝑓𝑥)
5147, 50eqtri 2754 . . . 4 ((𝑓 ∘ (1o × {𝑥}))‘∅) = (𝑓𝑥)
5243, 51eqtrdi 2782 . . 3 (𝑔 = (𝑓 ∘ (1o × {𝑥})) → (𝑔‘∅) = (𝑓𝑥))
535, 9, 33, 36, 42, 52cnmpt21 23666 . 2 ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥)) ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑆))
541, 53eqeltrid 2830 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 394   = wceq 1534  wcel 2099  c0 4325  𝒫 cpw 4607  {csn 4633   cuni 4913  cmpt 5236   × cxp 5680  ccom 5686  Oncon0 6376  wf 6550  cfv 6554  (class class class)co 7424  cmpo 7426  1oc1o 8489  Topctop 22886  TopOnctopon 22903   Cn ccn 23219  Compccmp 23381  𝑛-Locally cnlly 23460   ×t ctx 23555  ko cxko 23556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-1o 8496  df-2o 8497  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-fin 8978  df-fi 9454  df-rest 17437  df-topgen 17458  df-pt 17459  df-top 22887  df-topon 22904  df-bases 22940  df-ntr 23015  df-nei 23093  df-cn 23222  df-cnp 23223  df-cmp 23382  df-nlly 23462  df-tx 23557  df-xko 23558
This theorem is referenced by:  cnmptk1p  23680  cnmptk2  23681
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