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Theorem cnmptk2 21299
 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 6111 . . . . 5 𝑥((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
2 nfcv 2751 . . . . 5 𝑥𝑘
31, 2nffv 6110 . . . 4 𝑥(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
4 nfcv 2751 . . . . . . 7 𝑦𝑋
5 nfmpt1 4675 . . . . . . 7 𝑦(𝑦𝑌𝐴)
64, 5nfmpt 4674 . . . . . 6 𝑦(𝑥𝑋 ↦ (𝑦𝑌𝐴))
7 nfcv 2751 . . . . . 6 𝑦𝑤
86, 7nffv 6110 . . . . 5 𝑦((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)
9 nfcv 2751 . . . . 5 𝑦𝑘
108, 9nffv 6110 . . . 4 𝑦(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)
11 nfcv 2751 . . . 4 𝑤(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
12 nfcv 2751 . . . 4 𝑘(((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)
13 fveq2 6103 . . . . . 6 (𝑤 = 𝑥 → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥))
1413fveq1d 6105 . . . . 5 (𝑤 = 𝑥 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘))
15 fveq2 6103 . . . . 5 (𝑘 = 𝑦 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
1614, 15sylan9eq 2664 . . . 4 ((𝑤 = 𝑥𝑘 = 𝑦) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
173, 10, 11, 12, 16cbvmpt2 6632 . . 3 (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦))
18 simplr 788 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
19 cnmptk1p.j . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝑋))
20 cnmptk1p.n . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ 𝑛-Locally Comp)
21 nllytop 21086 . . . . . . . . . . . . . 14 (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
2220, 21syl 17 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Top)
23 cnmptk1p.l . . . . . . . . . . . . . 14 (𝜑𝐿 ∈ (TopOn‘𝑍))
24 topontop 20541 . . . . . . . . . . . . . 14 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
2523, 24syl 17 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ Top)
26 eqid 2610 . . . . . . . . . . . . . 14 (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
2726xkotopon 21213 . . . . . . . . . . . . 13 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
2822, 25, 27syl2anc 691 . . . . . . . . . . . 12 (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
29 cnmptk2.a . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
30 cnf2 20863 . . . . . . . . . . . 12 ((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 1318 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
32 eqid 2610 . . . . . . . . . . . 12 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
3332fmpt 6289 . . . . . . . . . . 11 (∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿) ↔ (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
3431, 33sylibr 223 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
3534r19.21bi 2916 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
3635adantr 480 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
3732fvmpt2 6200 . . . . . . . 8 ((𝑥𝑋 ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3818, 36, 37syl2anc 691 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥) = (𝑦𝑌𝐴))
3938fveq1d 6105 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = ((𝑦𝑌𝐴)‘𝑦))
40 simpr 476 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
41 cnmptk1p.k . . . . . . . . . . 11 (𝜑𝐾 ∈ (TopOn‘𝑌))
4241adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
4323adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
44 cnf2 20863 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
4542, 43, 35, 44syl3anc 1318 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
46 eqid 2610 . . . . . . . . . 10 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
4746fmpt 6289 . . . . . . . . 9 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
4845, 47sylibr 223 . . . . . . . 8 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
4948r19.21bi 2916 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴𝑍)
5046fvmpt2 6200 . . . . . . 7 ((𝑦𝑌𝐴𝑍) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
5140, 49, 50syl2anc 691 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑦𝑌𝐴)‘𝑦) = 𝐴)
5239, 51eqtrd 2644 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
53523impa 1251 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦) = 𝐴)
5453mpt2eq3dva 6617 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑥)‘𝑦)) = (𝑥𝑋, 𝑦𝑌𝐴))
5517, 54syl5eq 2656 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) = (𝑥𝑋, 𝑦𝑌𝐴))
5619, 41cnmpt1st 21281 . . . 4 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
5719, 41, 56, 29cnmpt21f 21285 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ^ko 𝐾)))
5819, 41cnmpt2nd 21282 . . 3 (𝜑 → (𝑤𝑋, 𝑘𝑌𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
59 eqid 2610 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
60 toponuni 20542 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
6141, 60syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
62 mpt2eq12 6613 . . . . 5 (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ 𝑌 = 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
6359, 61, 62sylancr 694 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)))
64 eqid 2610 . . . . . 6 𝐾 = 𝐾
65 eqid 2610 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧))
6664, 65xkofvcn 21297 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
6720, 25, 66syl2anc 691 . . . 4 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 𝐾 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
6863, 67eqeltrd 2688 . . 3 (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧𝑌 ↦ (𝑓𝑧)) ∈ (((𝐿 ^ko 𝐾) ×t 𝐾) Cn 𝐿))
69 fveq1 6102 . . . 4 (𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧))
70 fveq2 6103 . . . 4 (𝑧 = 𝑘 → (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
7169, 70sylan9eq 2664 . . 3 ((𝑓 = ((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤) ∧ 𝑧 = 𝑘) → (𝑓𝑧) = (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘))
7219, 41, 57, 58, 28, 41, 68, 71cnmpt22 21287 . 2 (𝜑 → (𝑤𝑋, 𝑘𝑌 ↦ (((𝑥𝑋 ↦ (𝑦𝑌𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
7355, 72eqeltrrd 2689 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Compccmp 20999  𝑛-Locally cnlly 21078   ×t ctx 21173   ^ko cxko 21174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-topon 20523  df-ntr 20634  df-nei 20712  df-cn 20841  df-cnp 20842  df-cmp 21000  df-nlly 21080  df-tx 21175  df-xko 21176 This theorem is referenced by:  xkocnv  21427
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