MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmptkk Structured version   Visualization version   GIF version

Theorem cnmptkk 23581
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmptkk.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmptkk.l (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
cnmptkk.m (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘Š))
cnmptkk.n (πœ‘ β†’ 𝐿 ∈ 𝑛-Locally Comp)
cnmptkk.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
cnmptkk.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑧 ∈ 𝑍 ↦ 𝐡)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐿)))
cnmptkk.c (𝑧 = 𝐴 β†’ 𝐡 = 𝐢)
Assertion
Ref Expression
cnmptkk (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐢)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
Distinct variable groups:   𝑧,𝐴   𝑦,𝐡   π‘₯,𝐾   π‘₯,𝐿   π‘₯,𝑦,𝑋   π‘₯,𝐽   π‘₯,𝑀   πœ‘,π‘₯,𝑦   𝑦,π‘Œ   𝑦,𝑧,𝑍   𝑧,𝐢
Allowed substitution hints:   πœ‘(𝑧)   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑧)   𝐢(π‘₯,𝑦)   𝐽(𝑦,𝑧)   𝐾(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑀(𝑦,𝑧)   π‘Š(π‘₯,𝑦,𝑧)   𝑋(𝑧)   π‘Œ(π‘₯,𝑧)   𝑍(π‘₯)

Proof of Theorem cnmptkk
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmptkk.k . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
21adantr 480 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3 cnmptkk.l . . . . . . 7 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
43adantr 480 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐿 ∈ (TopOnβ€˜π‘))
5 cnmptkk.j . . . . . . . 8 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
6 topontop 22809 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
71, 6syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ Top)
8 cnmptkk.n . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ 𝑛-Locally Comp)
9 nllytop 23371 . . . . . . . . . 10 (𝐿 ∈ 𝑛-Locally Comp β†’ 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
11 eqid 2728 . . . . . . . . . 10 (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
1211xkotopon 23498 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
137, 10, 12syl2anc 583 . . . . . . . 8 (πœ‘ β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
14 cnmptkk.a . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
15 cnf2 23147 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)) ∧ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
165, 13, 14, 15syl3anc 1369 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
1716fvmptelcdm 7118 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
18 cnf2 23147 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 ∈ (TopOnβ€˜π‘) ∧ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
192, 4, 17, 18syl3anc 1369 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
20 eqid 2728 . . . . . 6 (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴)
2120fmpt 7115 . . . . 5 (βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
2219, 21sylibr 233 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ 𝑍)
23 eqidd 2729 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴))
24 eqidd 2729 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑧 ∈ 𝑍 ↦ 𝐡) = (𝑧 ∈ 𝑍 ↦ 𝐡))
25 cnmptkk.c . . . 4 (𝑧 = 𝐴 β†’ 𝐡 = 𝐢)
2622, 23, 24, 25fmptcof 7134 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴)) = (𝑦 ∈ π‘Œ ↦ 𝐢))
2726mpteq2dva 5243 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴))) = (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐢)))
28 cnmptkk.b . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑧 ∈ 𝑍 ↦ 𝐡)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐿)))
29 cnmptkk.m . . . . 5 (πœ‘ β†’ 𝑀 ∈ (TopOnβ€˜π‘Š))
30 topontop 22809 . . . . 5 (𝑀 ∈ (TopOnβ€˜π‘Š) β†’ 𝑀 ∈ Top)
3129, 30syl 17 . . . 4 (πœ‘ β†’ 𝑀 ∈ Top)
32 eqid 2728 . . . . 5 (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿)
3332xkotopon 23498 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) β†’ (𝑀 ↑ko 𝐿) ∈ (TopOnβ€˜(𝐿 Cn 𝑀)))
3410, 31, 33syl2anc 583 . . 3 (πœ‘ β†’ (𝑀 ↑ko 𝐿) ∈ (TopOnβ€˜(𝐿 Cn 𝑀)))
35 eqid 2728 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔))
3635xkococn 23558 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ 𝑛-Locally Comp ∧ 𝑀 ∈ Top) β†’ (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) ∈ (((𝑀 ↑ko 𝐿) Γ—t (𝐿 ↑ko 𝐾)) Cn (𝑀 ↑ko 𝐾)))
377, 8, 31, 36syl3anc 1369 . . 3 (πœ‘ β†’ (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) ∈ (((𝑀 ↑ko 𝐿) Γ—t (𝐿 ↑ko 𝐾)) Cn (𝑀 ↑ko 𝐾)))
38 coeq1 5855 . . . 4 (𝑓 = (𝑧 ∈ 𝑍 ↦ 𝐡) β†’ (𝑓 ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ 𝑔))
39 coeq2 5856 . . . 4 (𝑔 = (𝑦 ∈ π‘Œ ↦ 𝐴) β†’ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴)))
4038, 39sylan9eq 2788 . . 3 ((𝑓 = (𝑧 ∈ 𝑍 ↦ 𝐡) ∧ 𝑔 = (𝑦 ∈ π‘Œ ↦ 𝐴)) β†’ (𝑓 ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴)))
415, 28, 14, 34, 13, 37, 40cnmpt12 23565 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴))) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
4227, 41eqeltrrd 2830 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐢)) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3057   ↦ cmpt 5226   ∘ ccom 5677  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7415   ∈ cmpo 7417  Topctop 22789  TopOnctopon 22806   Cn ccn 23122  Compccmp 23284  π‘›-Locally cnlly 23363   Γ—t ctx 23458   ↑ko cxko 23459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-1o 8481  df-er 8719  df-map 8841  df-en 8959  df-dom 8960  df-fin 8962  df-fi 9429  df-rest 17398  df-topgen 17419  df-top 22790  df-topon 22807  df-bases 22843  df-ntr 22918  df-nei 22996  df-cn 23125  df-cmp 23285  df-nlly 23365  df-tx 23460  df-xko 23461
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator