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Theorem cnmptk2 23512
Description: The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptk1p.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptk1p.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptk1p.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmptk1p.n (𝜑𝐾 ∈ 𝑛-Locally Comp)
cnmptk2.a (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
Assertion
Ref Expression
cnmptk2 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Distinct variable groups:   𝑥,𝐽   𝑥,𝐾   𝑥,𝐿   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝑦,𝑍
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑦)   𝐾(𝑦)   𝐿(𝑦)   𝑍(𝑥)

Proof of Theorem cnmptk2
Dummy variables 𝑓 𝑘 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nffvmpt1 6892 . . . . 5 𝑥((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
2 nfcv 2895 . . . . 5 𝑥𝑘
31, 2nffv 6891 . . . 4 𝑥(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
4 nfcv 2895 . . . . . . 7 𝑦𝑋
5 nfmpt1 5246 . . . . . . 7 𝑦(𝑦𝑌𝐴)
64, 5nfmpt 5245 . . . . . 6 𝑦(𝑥𝑋 ↦ (𝑦𝑌𝐴))
7 nfcv 2895 . . . . . 6 𝑦𝑤
86, 7nffv 6891 . . . . 5 𝑦((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
9 nfcv 2895 . . . . 5 𝑦𝑘
108, 9nffv 6891 . . . 4 𝑦(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
11 nfcv 2895 . . . 4 𝑤(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
12 nfcv 2895 . . . 4 𝑘(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
13 fveq2 6881 . . . . . 6 (𝑤 = 𝑥 → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥))
1413fveq1d 6883 . . . . 5 (𝑤 = 𝑥 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘))
15 fveq2 6881 . . . . 5 (𝑘 = 𝑦 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
1614, 15sylan9eq 2784 . . . 4 ((𝑤 = 𝑥𝑘 = 𝑦) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
173, 10, 11, 12, 16cbvmpo 7495 . . 3 (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
18 simplr 766 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
19 cnmptk1p.j . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
20 cnmptk1p.n . . . . . . . . . . . . 13 (𝜑𝐾 ∈ 𝑛-Locally Comp)
21 nllytop 23299 . . . . . . . . . . . . 13 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
2220, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Top)
23 cnmptk1p.l . . . . . . . . . . . . 13 (𝜑𝐿 ∈ (TopOn‘𝑍))
24 topontop 22737 . . . . . . . . . . . . 13 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
2523, 24syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
26 eqid 2724 . . . . . . . . . . . . 13 (𝐿ko 𝐾) = (𝐿ko 𝐾)
2726xkotopon 23426 . . . . . . . . . . . 12 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
2822, 25, 27syl2anc 583 . . . . . . . . . . 11 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
29 cnmptk2.a . . . . . . . . . . 11 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
30 cnf2 23075 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
3119, 28, 29, 30syl3anc 1368 . . . . . . . . . 10 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
3231fvmptelcdm 7104 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
3332adantr 480 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
34 eqid 2724 . . . . . . . . 9 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
3534fvmpt2 6999 . . . . . . . 8 ((𝑥𝑋 ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3618, 33, 35syl2anc 583 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3736fveq1d 6883 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = ((𝑦𝑌𝐴)‘𝑦))
38 simpr 484 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
39 cnmptk1p.k . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝑌))
4039adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
4123adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
42 cnf2 23075 . . . . . . . . 9 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
4340, 41, 32, 42syl3anc 1368 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
4443fvmptelcdm 7104 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴𝑍)
45 eqid 2724 . . . . . . . 8 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
4645fvmpt2 6999 . . . . . . 7 ((𝑦𝑌𝐴𝑍) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
4738, 44, 46syl2anc 583 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
4837, 47eqtrd 2764 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
49483impa 1107 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
5049mpoeq3dva 7478 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)) = (𝑥𝑋, 𝑦𝑌𝐴))
5117, 50eqtrid 2776 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌𝐴))
5219, 39cnmpt1st 23494 . . . 4 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
5319, 39, 52, 29cnmpt21f 23498 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿ko 𝐾)))
5419, 39cnmpt2nd 23495 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
55 eqid 2724 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56 toponuni 22738 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
5739, 56syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
58 mpoeq12 7474 . . . . 5 (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ 𝑌 = 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
5955, 57, 58sylancr 586 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
60 eqid 2724 . . . . . 6 𝐾 = 𝐾
61 eqid 2724 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
6260, 61xkofvcn 23510 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
6320, 25, 62syl2anc 583 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
6459, 63eqeltrd 2825 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
65 fveq1 6880 . . . 4 (𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧))
66 fveq2 6881 . . . 4 (𝑧 = 𝑘 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
6765, 66sylan9eq 2784 . . 3 ((𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) ∧ 𝑧 = 𝑘) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
6819, 39, 53, 54, 28, 39, 64, 67cnmpt22 23500 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
6951, 68eqeltrrd 2826 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098   cuni 4899  cmpt 5221  wf 6529  cfv 6533  (class class class)co 7401  cmpo 7403  Topctop 22717  TopOnctopon 22734   Cn ccn 23050  Compccmp 23212  𝑛-Locally cnlly 23291   ×t ctx 23386  ko cxko 23387
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-1o 8461  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-fin 8939  df-fi 9402  df-rest 17367  df-topgen 17388  df-pt 17389  df-top 22718  df-topon 22735  df-bases 22771  df-ntr 22846  df-nei 22924  df-cn 23053  df-cnp 23054  df-cmp 23213  df-nlly 23293  df-tx 23388  df-xko 23389
This theorem is referenced by:  xkocnv  23640
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