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Theorem cnmptkk 23691
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 480 . . . . . 6 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3 cnmptkk.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
43adantr 480 . . . . . 6 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5 cnmptkk.j . . . . . . . 8 (𝜑𝐽 ∈ (TopOn‘𝑋))
6 topontop 22919 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
8 cnmptkk.n . . . . . . . . . 10 (𝜑𝐿 ∈ 𝑛-Locally Comp)
9 nllytop 23481 . . . . . . . . . 10 (𝐿 ∈ 𝑛-Locally Comp → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
11 eqid 2737 . . . . . . . . . 10 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1211xkotopon 23608 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
137, 10, 12syl2anc 584 . . . . . . . 8 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
14 cnmptkk.a . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
15 cnf2 23257 . . . . . . . 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))
1716fvmptelcdm 7133 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
18 cnf2 23257 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
192, 4, 17, 18syl3anc 1373 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
20 eqid 2737 . . . . . 6 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2120fmpt 7130 . . . . 5 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2219, 21sylibr 234 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
23 eqidd 2738 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) = (𝑦𝑌𝐴))
24 eqidd 2738 . . . 4 ((𝜑𝑥𝑋) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
25 cnmptkk.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
2622, 23, 24, 25fmptcof 7150 . . 3 ((𝜑𝑥𝑋) → ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)) = (𝑦𝑌𝐶))
2726mpteq2dva 5242 . 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 22919 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
3129, 30syl 17 . . . 4 (𝜑𝑀 ∈ Top)
32 eqid 2737 . . . . 5 (𝑀ko 𝐿) = (𝑀ko 𝐿)
3332xkotopon 23608 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
3410, 31, 33syl2anc 584 . . 3 (𝜑 → (𝑀ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
35 eqid 2737 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔))
3635xkococn 23668 . . . 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 5868 . . . 4 (𝑓 = (𝑧𝑍𝐵) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ 𝑔))
39 coeq2 5869 . . . 4 (𝑔 = (𝑦𝑌𝐴) → ((𝑧𝑍𝐵) ∘ 𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
4038, 39sylan9eq 2797 . . 3 ((𝑓 = (𝑧𝑍𝐵) ∧ 𝑔 = (𝑦𝑌𝐴)) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
415, 28, 14, 34, 13, 37, 40cnmpt12 23675 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) ∈ (𝐽 Cn (𝑀ko 𝐾)))
4227, 41eqeltrrd 2842 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  cmpt 5225  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  Topctop 22899  TopOnctopon 22916   Cn ccn 23232  Compccmp 23394  𝑛-Locally cnlly 23473   ×t ctx 23568  ko cxko 23569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-map 8868  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-ntr 23028  df-nei 23106  df-cn 23235  df-cmp 23395  df-nlly 23475  df-tx 23570  df-xko 23571
This theorem is referenced by: (None)
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