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Theorem cnmptk1p 23188
Description: The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptk1p.j (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmptk1p.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmptk1p.l (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
cnmptk1p.n (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally Comp)
cnmptk1p.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
cnmptk1p.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))
cnmptk1p.c (𝑦 = 𝐡 β†’ 𝐴 = 𝐢)
Assertion
Ref Expression
cnmptk1p (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝐿   𝑦,𝐡   𝑦,𝐢   π‘₯,𝑦,𝑋   π‘₯,π‘Œ,𝑦   πœ‘,π‘₯,𝑦   𝑦,𝑍
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯)   𝐢(π‘₯)   𝐽(𝑦)   𝐾(𝑦)   𝐿(𝑦)   𝑍(π‘₯)

Proof of Theorem cnmptk1p
Dummy variables 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . 4 (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴)
2 cnmptk1p.c . . . 4 (𝑦 = 𝐡 β†’ 𝐴 = 𝐢)
3 cnmptk1p.j . . . . . 6 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 cnmptk1p.k . . . . . 6 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
5 cnmptk1p.b . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))
6 cnf2 22752 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆπ‘Œ)
73, 4, 5, 6syl3anc 1371 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆπ‘Œ)
87fvmptelcdm 7112 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ π‘Œ)
92eleq1d 2818 . . . . 5 (𝑦 = 𝐡 β†’ (𝐴 ∈ 𝑍 ↔ 𝐢 ∈ 𝑍))
104adantr 481 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
11 cnmptk1p.l . . . . . . . 8 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
1211adantr 481 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐿 ∈ (TopOnβ€˜π‘))
13 cnmptk1p.n . . . . . . . . . . 11 (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally Comp)
14 nllytop 22976 . . . . . . . . . . 11 (𝐾 ∈ 𝑛-Locally Comp β†’ 𝐾 ∈ Top)
1513, 14syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ Top)
16 topontop 22414 . . . . . . . . . . 11 (𝐿 ∈ (TopOnβ€˜π‘) β†’ 𝐿 ∈ Top)
1711, 16syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐿 ∈ Top)
18 eqid 2732 . . . . . . . . . . 11 (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
1918xkotopon 23103 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
2015, 17, 19syl2anc 584 . . . . . . . . 9 (πœ‘ β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
21 cnmptk1p.a . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
22 cnf2 22752 . . . . . . . . 9 ((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))
233, 20, 21, 22syl3anc 1371 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
2423fvmptelcdm 7112 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
25 cnf2 22752 . . . . . . 7 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 ∈ (TopOnβ€˜π‘) ∧ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
2610, 12, 24, 25syl3anc 1371 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
271fmpt 7109 . . . . . 6 (βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ 𝑍 ↔ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
2826, 27sylibr 233 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐴 ∈ 𝑍)
299, 28, 8rspcdva 3613 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ 𝑍)
301, 2, 8, 29fvmptd3 7021 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅) = 𝐢)
3130mpteq2dva 5248 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅)) = (π‘₯ ∈ 𝑋 ↦ 𝐢))
32 eqid 2732 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
33 toponuni 22415 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
344, 33syl 17 . . . . 5 (πœ‘ β†’ π‘Œ = βˆͺ 𝐾)
35 mpoeq12 7481 . . . . 5 (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ π‘Œ = βˆͺ 𝐾) β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)))
3632, 34, 35sylancr 587 . . . 4 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)))
37 eqid 2732 . . . . . 6 βˆͺ 𝐾 = βˆͺ 𝐾
38 eqid 2732 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§))
3937, 38xkofvcn 23187 . . . . 5 ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
4013, 17, 39syl2anc 584 . . . 4 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
4136, 40eqeltrd 2833 . . 3 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
42 fveq1 6890 . . . 4 (𝑓 = (𝑦 ∈ π‘Œ ↦ 𝐴) β†’ (π‘“β€˜π‘§) = ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§))
43 fveq2 6891 . . . 4 (𝑧 = 𝐡 β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘§) = ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅))
4442, 43sylan9eq 2792 . . 3 ((𝑓 = (𝑦 ∈ π‘Œ ↦ 𝐴) ∧ 𝑧 = 𝐡) β†’ (π‘“β€˜π‘§) = ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅))
453, 21, 5, 20, 4, 41, 44cnmpt12 23170 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π΅)) ∈ (𝐽 Cn 𝐿))
4631, 45eqeltrrd 2834 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆͺ cuni 4908   ↦ cmpt 5231  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-ixp 8891  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-pt 17389  df-top 22395  df-topon 22412  df-bases 22448  df-ntr 22523  df-nei 22601  df-cn 22730  df-cnp 22731  df-cmp 22890  df-nlly 22970  df-tx 23065  df-xko 23066
This theorem is referenced by:  xkohmeo  23318
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