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Theorem cnmptk2 22291
 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 6656 . . . . 5 𝑥((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
2 nfcv 2955 . . . . 5 𝑥𝑘
31, 2nffv 6655 . . . 4 𝑥(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
4 nfcv 2955 . . . . . . 7 𝑦𝑋
5 nfmpt1 5128 . . . . . . 7 𝑦(𝑦𝑌𝐴)
64, 5nfmpt 5127 . . . . . 6 𝑦(𝑥𝑋 ↦ (𝑦𝑌𝐴))
7 nfcv 2955 . . . . . 6 𝑦𝑤
86, 7nffv 6655 . . . . 5 𝑦((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
9 nfcv 2955 . . . . 5 𝑦𝑘
108, 9nffv 6655 . . . 4 𝑦(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
11 nfcv 2955 . . . 4 𝑤(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
12 nfcv 2955 . . . 4 𝑘(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
13 fveq2 6645 . . . . . 6 (𝑤 = 𝑥 → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥))
1413fveq1d 6647 . . . . 5 (𝑤 = 𝑥 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘))
15 fveq2 6645 . . . . 5 (𝑘 = 𝑦 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
1614, 15sylan9eq 2853 . . . 4 ((𝑤 = 𝑥𝑘 = 𝑦) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
173, 10, 11, 12, 16cbvmpo 7227 . . 3 (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
18 simplr 768 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
19 cnmptk1p.j . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
20 cnmptk1p.n . . . . . . . . . . . . 13 (𝜑𝐾 ∈ 𝑛-Locally Comp)
21 nllytop 22078 . . . . . . . . . . . . 13 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
2220, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Top)
23 cnmptk1p.l . . . . . . . . . . . . 13 (𝜑𝐿 ∈ (TopOn‘𝑍))
24 topontop 21518 . . . . . . . . . . . . 13 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
2523, 24syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
26 eqid 2798 . . . . . . . . . . . . 13 (𝐿ko 𝐾) = (𝐿ko 𝐾)
2726xkotopon 22205 . . . . . . . . . . . 12 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
2822, 25, 27syl2anc 587 . . . . . . . . . . 11 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
29 cnmptk2.a . . . . . . . . . . 11 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
30 cnf2 21854 . . . . . . . . . . 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))
3231fvmptelrn 6854 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
3332adantr 484 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
34 eqid 2798 . . . . . . . . 9 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
3534fvmpt2 6756 . . . . . . . 8 ((𝑥𝑋 ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3618, 33, 35syl2anc 587 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3736fveq1d 6647 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = ((𝑦𝑌𝐴)‘𝑦))
38 simpr 488 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
39 cnmptk1p.k . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝑌))
4039adantr 484 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
4123adantr 484 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
42 cnf2 21854 . . . . . . . . 9 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
4340, 41, 32, 42syl3anc 1368 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
4443fvmptelrn 6854 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴𝑍)
45 eqid 2798 . . . . . . . 8 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
4645fvmpt2 6756 . . . . . . 7 ((𝑦𝑌𝐴𝑍) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
4738, 44, 46syl2anc 587 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
4837, 47eqtrd 2833 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
49483impa 1107 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
5049mpoeq3dva 7210 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)) = (𝑥𝑋, 𝑦𝑌𝐴))
5117, 50syl5eq 2845 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌𝐴))
5219, 39cnmpt1st 22273 . . . 4 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
5319, 39, 52, 29cnmpt21f 22277 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿ko 𝐾)))
5419, 39cnmpt2nd 22274 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
55 eqid 2798 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56 toponuni 21519 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
5739, 56syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
58 mpoeq12 7206 . . . . 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 590 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
60 eqid 2798 . . . . . 6 𝐾 = 𝐾
61 eqid 2798 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
6260, 61xkofvcn 22289 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
6320, 25, 62syl2anc 587 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
6459, 63eqeltrd 2890 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿ko 𝐾) ×t 𝐾) Cn 𝐿))
65 fveq1 6644 . . . 4 (𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧))
66 fveq2 6645 . . . 4 (𝑧 = 𝑘 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
6765, 66sylan9eq 2853 . . 3 ((𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) ∧ 𝑧 = 𝑘) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
6819, 39, 53, 54, 28, 39, 64, 67cnmpt22 22279 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
6951, 68eqeltrrd 2891 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∪ cuni 4800   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  Compccmp 21991  𝑛-Locally cnlly 22070   ×t ctx 22165   ↑ko cxko 22166 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-pt 16710  df-top 21499  df-topon 21516  df-bases 21551  df-ntr 21625  df-nei 21703  df-cn 21832  df-cnp 21833  df-cmp 21992  df-nlly 22072  df-tx 22167  df-xko 22168 This theorem is referenced by:  xkocnv  22419
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