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Theorem cnmptkk 23186
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 481 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3 cnmptkk.l . . . . . . 7 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
43adantr 481 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐿 ∈ (TopOnβ€˜π‘))
5 cnmptkk.j . . . . . . . 8 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
6 topontop 22414 . . . . . . . . . 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 22976 . . . . . . . . . 10 (𝐿 ∈ 𝑛-Locally Comp β†’ 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
11 eqid 2732 . . . . . . . . . 10 (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
1211xkotopon 23103 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
137, 10, 12syl2anc 584 . . . . . . . 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 22752 . . . . . . . 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 1371 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
1716fvmptelcdm 7112 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
18 cnf2 22752 . . . . . 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 1371 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
20 eqid 2732 . . . . . 6 (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴)
2120fmpt 7109 . . . . 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 2733 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴))
24 eqidd 2733 . . . 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 7127 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴)) = (𝑦 ∈ π‘Œ ↦ 𝐢))
2726mpteq2dva 5248 . 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 22414 . . . . 5 (𝑀 ∈ (TopOnβ€˜π‘Š) β†’ 𝑀 ∈ Top)
3129, 30syl 17 . . . 4 (πœ‘ β†’ 𝑀 ∈ Top)
32 eqid 2732 . . . . 5 (𝑀 ↑ko 𝐿) = (𝑀 ↑ko 𝐿)
3332xkotopon 23103 . . . 4 ((𝐿 ∈ Top ∧ 𝑀 ∈ Top) β†’ (𝑀 ↑ko 𝐿) ∈ (TopOnβ€˜(𝐿 Cn 𝑀)))
3410, 31, 33syl2anc 584 . . 3 (πœ‘ β†’ (𝑀 ↑ko 𝐿) ∈ (TopOnβ€˜(𝐿 Cn 𝑀)))
35 eqid 2732 . . . . 5 (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ (𝐿 Cn 𝑀), 𝑔 ∈ (𝐾 Cn 𝐿) ↦ (𝑓 ∘ 𝑔))
3635xkococn 23163 . . . 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 1371 . . 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 5857 . . . 4 (𝑓 = (𝑧 ∈ 𝑍 ↦ 𝐡) β†’ (𝑓 ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ 𝑔))
39 coeq2 5858 . . . 4 (𝑔 = (𝑦 ∈ π‘Œ ↦ 𝐴) β†’ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ 𝑔) = ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴)))
4038, 39sylan9eq 2792 . . 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 23170 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑧 ∈ 𝑍 ↦ 𝐡) ∘ (𝑦 ∈ π‘Œ ↦ 𝐴))) ∈ (𝐽 Cn (𝑀 ↑ko 𝐾)))
4227, 41eqeltrrd 2834 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   ↦ cmpt 5231   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Topctop 22394  TopOnctopon 22411   Cn ccn 22727  Compccmp 22889  π‘›-Locally cnlly 22968   Γ—t ctx 23063   ↑ko cxko 23064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-ntr 22523  df-nei 22601  df-cn 22730  df-cmp 22890  df-nlly 22970  df-tx 23065  df-xko 23066
This theorem is referenced by: (None)
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