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Theorem cnmptkk 23707
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 22935 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
8 cnmptkk.n . . . . . . . . . 10 (𝜑𝐿 ∈ 𝑛-Locally Comp)
9 nllytop 23497 . . . . . . . . . 10 (𝐿 ∈ 𝑛-Locally Comp → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
11 eqid 2735 . . . . . . . . . 10 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1211xkotopon 23624 . . . . . . . . 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 23273 . . . . . . . 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 1370 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
1716fvmptelcdm 7133 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
18 cnf2 23273 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
192, 4, 17, 18syl3anc 1370 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
20 eqid 2735 . . . . . 6 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2120fmpt 7130 . . . . 5 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2219, 21sylibr 234 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
23 eqidd 2736 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) = (𝑦𝑌𝐴))
24 eqidd 2736 . . . 4 ((𝜑𝑥𝑋) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
25 cnmptkk.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
2622, 23, 24, 25fmptcof 7150 . . 3 ((𝜑𝑥𝑋) → ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)) = (𝑦𝑌𝐶))
2726mpteq2dva 5248 . 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 22935 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
3129, 30syl 17 . . . 4 (𝜑𝑀 ∈ Top)
32 eqid 2735 . . . . 5 (𝑀ko 𝐿) = (𝑀ko 𝐿)
3332xkotopon 23624 . . . 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 2735 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔))
3635xkococn 23684 . . . 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 1370 . . 3 (𝜑 → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) ∈ (((𝑀ko 𝐿) ×t (𝐿ko 𝐾)) Cn (𝑀ko 𝐾)))
38 coeq1 5871 . . . 4 (𝑓 = (𝑧𝑍𝐵) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ 𝑔))
39 coeq2 5872 . . . 4 (𝑔 = (𝑦𝑌𝐴) → ((𝑧𝑍𝐵) ∘ 𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
4038, 39sylan9eq 2795 . . 3 ((𝑓 = (𝑧𝑍𝐵) ∧ 𝑔 = (𝑦𝑌𝐴)) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
415, 28, 14, 34, 13, 37, 40cnmpt12 23691 . 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 395   = wceq 1537  wcel 2106  wral 3059  cmpt 5231  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  Compccmp 23410  𝑛-Locally cnlly 23489   ×t ctx 23584  ko cxko 23585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-cn 23251  df-cmp 23411  df-nlly 23491  df-tx 23586  df-xko 23587
This theorem is referenced by: (None)
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