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Theorem cnmptkk 23057
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 482 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3 cnmptkk.l . . . . . . 7 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
43adantr 482 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐿 ∈ (TopOnβ€˜π‘))
5 cnmptkk.j . . . . . . . 8 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
6 topontop 22285 . . . . . . . . . 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 22847 . . . . . . . . . 10 (𝐿 ∈ 𝑛-Locally Comp β†’ 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
11 eqid 2733 . . . . . . . . . 10 (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
1211xkotopon 22974 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
137, 10, 12syl2anc 585 . . . . . . . 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 22623 . . . . . . . 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 1372 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
1716fvmptelcdm 7065 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
18 cnf2 22623 . . . . . 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 1372 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
20 eqid 2733 . . . . . 6 (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴)
2120fmpt 7062 . . . . 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 2734 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴))
24 eqidd 2734 . . . 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 7080 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴)) = (𝑦 ∈ π‘Œ ↦ 𝐢))
2726mpteq2dva 5209 . 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 22285 . . . . 5 (𝑀 ∈ (TopOnβ€˜π‘Š) β†’ 𝑀 ∈ Top)
3129, 30syl 17 . . . 4 (πœ‘ β†’ 𝑀 ∈ Top)
32 eqid 2733 . . . . 5 (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿)
3332xkotopon 22974 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) β†’ (𝑀 ↑ko 𝐿) ∈ (TopOnβ€˜(𝐿 Cn 𝑀)))
3410, 31, 33syl2anc 585 . . 3 (πœ‘ β†’ (𝑀 ↑ko 𝐿) ∈ (TopOnβ€˜(𝐿 Cn 𝑀)))
35 eqid 2733 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔))
3635xkococn 23034 . . . 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 1372 . . 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 5817 . . . 4 (𝑓 = (𝑧 ∈ 𝑍 ↦ 𝐡) β†’ (𝑓 ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ 𝑔))
39 coeq2 5818 . . . 4 (𝑔 = (𝑦 ∈ π‘Œ ↦ 𝐴) β†’ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴)))
4038, 39sylan9eq 2793 . . 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 23041 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴))) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
4227, 41eqeltrrd 2835 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   ↦ cmpt 5192   ∘ ccom 5641  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Topctop 22265  TopOnctopon 22282   Cn ccn 22598  Compccmp 22760  π‘›-Locally cnlly 22839   Γ—t ctx 22934   ↑ko cxko 22935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-ntr 22394  df-nei 22472  df-cn 22601  df-cmp 22761  df-nlly 22841  df-tx 22936  df-xko 22937
This theorem is referenced by: (None)
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