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Theorem cnmptkk 21296
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 (𝐿 ^ko 𝐾)))
cnmptkk.b (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀 ^ko 𝐿)))
cnmptkk.c (𝑧 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
cnmptkk (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀 ^ko 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 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝑋))
6 topontop 20541 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
8 cnmptkk.n . . . . . . . . . . 11 (𝜑𝐿 ∈ 𝑛-Locally Comp)
9 nllytop 21086 . . . . . . . . . . 11 (𝐿 ∈ 𝑛-Locally Comp → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ Top)
11 eqid 2610 . . . . . . . . . . 11 (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
1211xkotopon 21213 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
137, 10, 12syl2anc 691 . . . . . . . . 9 (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
14 cnmptkk.a . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
15 cnf2 20863 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
165, 13, 14, 15syl3anc 1318 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
17 eqid 2610 . . . . . . . . 9 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
1817fmpt 6289 . . . . . . . 8 (∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿) ↔ (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
1916, 18sylibr 223 . . . . . . 7 (𝜑 → ∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
2019r19.21bi 2916 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
21 cnf2 20863 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
222, 4, 20, 21syl3anc 1318 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
23 eqid 2610 . . . . . 6 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2423fmpt 6289 . . . . 5 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2522, 24sylibr 223 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
26 eqidd 2611 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) = (𝑦𝑌𝐴))
27 eqidd 2611 . . . 4 ((𝜑𝑥𝑋) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
28 cnmptkk.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
2925, 26, 27, 28fmptcof 6304 . . 3 ((𝜑𝑥𝑋) → ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)) = (𝑦𝑌𝐶))
3029mpteq2dva 4672 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
31 cnmptkk.b . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀 ^ko 𝐿)))
32 cnmptkk.m . . . . 5 (𝜑𝑀 ∈ (TopOn‘𝑊))
33 topontop 20541 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
3432, 33syl 17 . . . 4 (𝜑𝑀 ∈ Top)
35 eqid 2610 . . . . 5 (𝑀 ^ko 𝐿) = (𝑀 ^ko 𝐿)
3635xkotopon 21213 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
3710, 34, 36syl2anc 691 . . 3 (𝜑 → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
38 eqid 2610 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔))
3938xkococn 21273 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ 𝑛-Locally Comp ∧ 𝑀 ∈ Top) → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) ∈ (((𝑀 ^ko 𝐿) ×t (𝐿 ^ko 𝐾)) Cn (𝑀 ^ko 𝐾)))
407, 8, 34, 39syl3anc 1318 . . 3 (𝜑 → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) ∈ (((𝑀 ^ko 𝐿) ×t (𝐿 ^ko 𝐾)) Cn (𝑀 ^ko 𝐾)))
41 coeq1 5201 . . . 4 (𝑓 = (𝑧𝑍𝐵) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ 𝑔))
42 coeq2 5202 . . . 4 (𝑔 = (𝑦𝑌𝐴) → ((𝑧𝑍𝐵) ∘ 𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
4341, 42sylan9eq 2664 . . 3 ((𝑓 = (𝑧𝑍𝐵) ∧ 𝑔 = (𝑦𝑌𝐴)) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
445, 31, 14, 37, 13, 40, 43cnmpt12 21280 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))
4530, 44eqeltrrd 2689 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  cmpt 4643  ccom 5042  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-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-ntr 20634  df-nei 20712  df-cn 20841  df-cmp 21000  df-nlly 21080  df-tx 21175  df-xko 21176
This theorem is referenced by: (None)
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