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Theorem cnmptkk 21706
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 466 . . . . . 6 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
3 cnmptkk.l . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
43adantr 466 . . . . . 6 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘𝑍))
5 cnmptkk.j . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝑋))
6 topontop 20937 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
8 cnmptkk.n . . . . . . . . . . 11 (𝜑𝐿 ∈ 𝑛-Locally Comp)
9 nllytop 21496 . . . . . . . . . . 11 (𝐿 ∈ 𝑛-Locally Comp → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ Top)
11 eqid 2770 . . . . . . . . . . 11 (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
1211xkotopon 21623 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
137, 10, 12syl2anc 565 . . . . . . . . 9 (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
14 cnmptkk.a . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
15 cnf2 21273 . . . . . . . . 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 1475 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
17 eqid 2770 . . . . . . . . 9 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
1817fmpt 6523 . . . . . . . 8 (∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿) ↔ (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
1916, 18sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
2019r19.21bi 3080 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
21 cnf2 21273 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌𝑍)
222, 4, 20, 21syl3anc 1475 . . . . 5 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌𝑍)
23 eqid 2770 . . . . . 6 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2423fmpt 6523 . . . . 5 (∀𝑦𝑌 𝐴𝑍 ↔ (𝑦𝑌𝐴):𝑌𝑍)
2522, 24sylibr 224 . . . 4 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴𝑍)
26 eqidd 2771 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) = (𝑦𝑌𝐴))
27 eqidd 2771 . . . 4 ((𝜑𝑥𝑋) → (𝑧𝑍𝐵) = (𝑧𝑍𝐵))
28 cnmptkk.c . . . 4 (𝑧 = 𝐴𝐵 = 𝐶)
2925, 26, 27, 28fmptcof 6539 . . 3 ((𝜑𝑥𝑋) → ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)) = (𝑦𝑌𝐶))
3029mpteq2dva 4876 . 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 20937 . . . . 5 (𝑀 ∈ (TopOn‘𝑊) → 𝑀 ∈ Top)
3432, 33syl 17 . . . 4 (𝜑𝑀 ∈ Top)
35 eqid 2770 . . . . 5 (𝑀 ^ko 𝐿) = (𝑀 ^ko 𝐿)
3635xkotopon 21623 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
3710, 34, 36syl2anc 565 . . 3 (𝜑 → (𝑀 ^ko 𝐿) ∈ (TopOn‘(𝐿 Cn 𝑀)))
38 eqid 2770 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔))
3938xkococn 21683 . . . 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 1475 . . 3 (𝜑 → (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓𝑔)) ∈ (((𝑀 ^ko 𝐿) ×t (𝐿 ^ko 𝐾)) Cn (𝑀 ^ko 𝐾)))
41 coeq1 5418 . . . 4 (𝑓 = (𝑧𝑍𝐵) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ 𝑔))
42 coeq2 5419 . . . 4 (𝑔 = (𝑦𝑌𝐴) → ((𝑧𝑍𝐵) ∘ 𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
4341, 42sylan9eq 2824 . . 3 ((𝑓 = (𝑧𝑍𝐵) ∧ 𝑔 = (𝑦𝑌𝐴)) → (𝑓𝑔) = ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴)))
445, 31, 14, 37, 13, 40, 43cnmpt12 21690 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑧𝑍𝐵) ∘ (𝑦𝑌𝐴))) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))
4530, 44eqeltrrd 2850 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  cmpt 4861  ccom 5253  wf 6027  cfv 6031  (class class class)co 6792  cmpt2 6794  Topctop 20917  TopOnctopon 20934   Cn ccn 21248  Compccmp 21409  𝑛-Locally cnlly 21488   ×t ctx 21583   ^ko cxko 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-fin 8112  df-fi 8472  df-rest 16290  df-topgen 16311  df-top 20918  df-topon 20935  df-bases 20970  df-ntr 21044  df-nei 21122  df-cn 21251  df-cmp 21410  df-nlly 21490  df-tx 21585  df-xko 21586
This theorem is referenced by: (None)
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