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Theorem cnmptkk 22607
Description: The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptkk.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptkk.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptkk.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmptkk.m (𝜑𝑀 ∈ (TopOn‘𝑊))
cnmptkk.n (𝜑𝐿 ∈ 𝑛-Locally Comp)
cnmptkk.a (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
cnmptkk.b (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀ko 𝐿)))
cnmptkk.c (𝑧 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
cnmptkk (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
Distinct variable groups:   𝑧,𝐴   𝑦,𝐵   𝑥,𝐾   𝑥,𝐿   𝑥,𝑦,𝑋   𝑥,𝐽   𝑥,𝑀   𝜑,𝑥,𝑦   𝑦,𝑌   𝑦,𝑧,𝑍   𝑧,𝐶
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑧)   𝐶(𝑥,𝑦)   𝐽(𝑦,𝑧)   𝐾(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑀(𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑧)   𝑌(𝑥,𝑧)   𝑍(𝑥)

Proof of Theorem cnmptkk
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmptkk.k . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
21adantr 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3 cnmptkk.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
43adantr 484 . . . . . 6 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5 cnmptkk.j . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘𝑋))
6 topontop 21837 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
8 cnmptkk.n . . . . . . . . . 10 (𝜑𝐿 ∈ 𝑛-Locally Comp)
9 nllytop 22397 . . . . . . . . . 10 (𝐿 ∈ 𝑛-Locally Comp → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
11 eqid 2738 . . . . . . . . . 10 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1211xkotopon 22524 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
137, 10, 12syl2anc 587 . . . . . . . 8 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
14 cnmptkk.a . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
15 cnf2 22173 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
165, 13, 14, 15syl3anc 1373 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
1716fvmptelrn 6949 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
18 cnf2 22173 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
192, 4, 17, 18syl3anc 1373 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
20 eqid 2738 . . . . . 6 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2120fmpt 6946 . . . . 5 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2219, 21sylibr 237 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
23 eqidd 2739 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) = (𝑦𝑌𝐴))
24 eqidd 2739 . . . 4 ((𝜑𝑥𝑋) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
25 cnmptkk.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
2622, 23, 24, 25fmptcof 6964 . . 3 ((𝜑𝑥𝑋) → ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)) = (𝑦𝑌𝐶))
2726mpteq2dva 5165 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
28 cnmptkk.b . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀ko 𝐿)))
29 cnmptkk.m . . . . 5 (𝜑𝑀 ∈ (TopOn‘𝑊))
30 topontop 21837 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
3129, 30syl 17 . . . 4 (𝜑𝑀 ∈ Top)
32 eqid 2738 . . . . 5 (𝑀ko 𝐿) = (𝑀ko 𝐿)
3332xkotopon 22524 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
3410, 31, 33syl2anc 587 . . 3 (𝜑 → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
35 eqid 2738 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔))
3635xkococn 22584 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ 𝑛-Locally Comp ∧ 𝑀 ∈ Top) → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) ∈ (((𝑀ko 𝐿) ×t (𝐿ko 𝐾)) Cn (𝑀ko 𝐾)))
377, 8, 31, 36syl3anc 1373 . . 3 (𝜑 → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) ∈ (((𝑀ko 𝐿) ×t (𝐿ko 𝐾)) Cn (𝑀ko 𝐾)))
38 coeq1 5741 . . . 4 (𝑓 = (𝑧𝑍𝐵) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ 𝑔))
39 coeq2 5742 . . . 4 (𝑔 = (𝑦𝑌𝐴) → ((𝑧𝑍𝐵) ∘ 𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
4038, 39sylan9eq 2799 . . 3 ((𝑓 = (𝑧𝑍𝐵) ∧ 𝑔 = (𝑦𝑌𝐴)) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
415, 28, 14, 34, 13, 37, 40cnmpt12 22591 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) ∈ (𝐽 Cn (𝑀ko 𝐾)))
4227, 41eqeltrrd 2840 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  wral 3062  cmpt 5150  ccom 5570  wf 6394  cfv 6398  (class class class)co 7232  cmpo 7234  Topctop 21817  TopOnctopon 21834   Cn ccn 22148  Compccmp 22310  𝑛-Locally cnlly 22389   ×t ctx 22484  ko cxko 22485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-iun 4921  df-iin 4922  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-ord 6234  df-on 6235  df-lim 6236  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-om 7664  df-1st 7780  df-2nd 7781  df-1o 8223  df-er 8412  df-map 8531  df-en 8648  df-dom 8649  df-fin 8651  df-fi 9052  df-rest 16955  df-topgen 16976  df-top 21818  df-topon 21835  df-bases 21870  df-ntr 21944  df-nei 22022  df-cn 22151  df-cmp 22311  df-nlly 22391  df-tx 22486  df-xko 22487
This theorem is referenced by: (None)
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