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Theorem cnmptk2 22010
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 (𝐿 ^ko 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 6507 . . . . 5 𝑥((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
2 nfcv 2926 . . . . 5 𝑥𝑘
31, 2nffv 6506 . . . 4 𝑥(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
4 nfcv 2926 . . . . . . 7 𝑦𝑋
5 nfmpt1 5021 . . . . . . 7 𝑦(𝑦𝑌𝐴)
64, 5nfmpt 5020 . . . . . 6 𝑦(𝑥𝑋 ↦ (𝑦𝑌𝐴))
7 nfcv 2926 . . . . . 6 𝑦𝑤
86, 7nffv 6506 . . . . 5 𝑦((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
9 nfcv 2926 . . . . 5 𝑦𝑘
108, 9nffv 6506 . . . 4 𝑦(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
11 nfcv 2926 . . . 4 𝑤(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
12 nfcv 2926 . . . 4 𝑘(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
13 fveq2 6496 . . . . . 6 (𝑤 = 𝑥 → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥))
1413fveq1d 6498 . . . . 5 (𝑤 = 𝑥 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘))
15 fveq2 6496 . . . . 5 (𝑘 = 𝑦 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
1614, 15sylan9eq 2828 . . . 4 ((𝑤 = 𝑥𝑘 = 𝑦) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
173, 10, 11, 12, 16cbvmpo 7062 . . 3 (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
18 simplr 756 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
19 cnmptk1p.j . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
20 cnmptk1p.n . . . . . . . . . . . . 13 (𝜑𝐾 ∈ 𝑛-Locally Comp)
21 nllytop 21797 . . . . . . . . . . . . 13 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
2220, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Top)
23 cnmptk1p.l . . . . . . . . . . . . 13 (𝜑𝐿 ∈ (TopOn‘𝑍))
24 topontop 21237 . . . . . . . . . . . . 13 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
2523, 24syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
26 eqid 2772 . . . . . . . . . . . . 13 (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
2726xkotopon 21924 . . . . . . . . . . . 12 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
2822, 25, 27syl2anc 576 . . . . . . . . . . 11 (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
29 cnmptk2.a . . . . . . . . . . 11 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
30 cnf2 21573 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
3119, 28, 29, 30syl3anc 1351 . . . . . . . . . 10 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
3231fvmptelrn 6698 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
3332adantr 473 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
34 eqid 2772 . . . . . . . . 9 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
3534fvmpt2 6603 . . . . . . . 8 ((𝑥𝑋 ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3618, 33, 35syl2anc 576 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3736fveq1d 6498 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = ((𝑦𝑌𝐴)‘𝑦))
38 simpr 477 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
39 cnmptk1p.k . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝑌))
4039adantr 473 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
4123adantr 473 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
42 cnf2 21573 . . . . . . . . 9 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
4340, 41, 32, 42syl3anc 1351 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
4443fvmptelrn 6698 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴𝑍)
45 eqid 2772 . . . . . . . 8 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
4645fvmpt2 6603 . . . . . . 7 ((𝑦𝑌𝐴𝑍) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
4738, 44, 46syl2anc 576 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
4837, 47eqtrd 2808 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
49483impa 1090 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
5049mpoeq3dva 7047 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)) = (𝑥𝑋, 𝑦𝑌𝐴))
5117, 50syl5eq 2820 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌𝐴))
5219, 39cnmpt1st 21992 . . . 4 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
5319, 39, 52, 29cnmpt21f 21996 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ^ko 𝐾)))
5419, 39cnmpt2nd 21993 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
55 eqid 2772 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56 toponuni 21238 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
5739, 56syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
58 mpoeq12 7043 . . . . 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 578 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
60 eqid 2772 . . . . . 6 𝐾 = 𝐾
61 eqid 2772 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
6260, 61xkofvcn 22008 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
6320, 25, 62syl2anc 576 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
6459, 63eqeltrd 2860 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
65 fveq1 6495 . . . 4 (𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧))
66 fveq2 6496 . . . 4 (𝑧 = 𝑘 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
6765, 66sylan9eq 2828 . . 3 ((𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) ∧ 𝑧 = 𝑘) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
6819, 39, 53, 54, 28, 39, 64, 67cnmpt22 21998 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
6951, 68eqeltrrd 2861 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050   cuni 4708  cmpt 5004  wf 6181  cfv 6185  (class class class)co 6974  cmpo 6976  Topctop 21217  TopOnctopon 21234   Cn ccn 21548  Compccmp 21710  𝑛-Locally cnlly 21789   ×t ctx 21884   ^ko cxko 21885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-2o 7904  df-oadd 7907  df-er 8087  df-map 8206  df-ixp 8258  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-fi 8668  df-rest 16550  df-topgen 16571  df-pt 16572  df-top 21218  df-topon 21235  df-bases 21270  df-ntr 21344  df-nei 21422  df-cn 21551  df-cnp 21552  df-cmp 21711  df-nlly 21791  df-tx 21886  df-xko 21887
This theorem is referenced by:  xkocnv  22138
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