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Theorem cnmptk2 23060
Description: The uncurrying of a curried function is 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)
cnmptk2.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
Assertion
Ref Expression
cnmptk2 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝐿   π‘₯,𝑦,𝑋   π‘₯,π‘Œ,𝑦   πœ‘,π‘₯,𝑦   𝑦,𝑍
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐽(𝑦)   𝐾(𝑦)   𝐿(𝑦)   𝑍(π‘₯)

Proof of Theorem cnmptk2
Dummy variables 𝑓 π‘˜ 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nffvmpt1 6857 . . . . 5 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)
2 nfcv 2904 . . . . 5 β„²π‘₯π‘˜
31, 2nffv 6856 . . . 4 β„²π‘₯(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜)
4 nfcv 2904 . . . . . . 7 Ⅎ𝑦𝑋
5 nfmpt1 5217 . . . . . . 7 Ⅎ𝑦(𝑦 ∈ π‘Œ ↦ 𝐴)
64, 5nfmpt 5216 . . . . . 6 Ⅎ𝑦(π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))
7 nfcv 2904 . . . . . 6 Ⅎ𝑦𝑀
86, 7nffv 6856 . . . . 5 Ⅎ𝑦((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)
9 nfcv 2904 . . . . 5 β„²π‘¦π‘˜
108, 9nffv 6856 . . . 4 Ⅎ𝑦(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜)
11 nfcv 2904 . . . 4 Ⅎ𝑀(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦)
12 nfcv 2904 . . . 4 β„²π‘˜(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦)
13 fveq2 6846 . . . . . 6 (𝑀 = π‘₯ β†’ ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€) = ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯))
1413fveq1d 6848 . . . . 5 (𝑀 = π‘₯ β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜) = (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘˜))
15 fveq2 6846 . . . . 5 (π‘˜ = 𝑦 β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘˜) = (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦))
1614, 15sylan9eq 2793 . . . 4 ((𝑀 = π‘₯ ∧ π‘˜ = 𝑦) β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜) = (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦))
173, 10, 11, 12, 16cbvmpo 7455 . . 3 (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦))
18 simplr 768 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
19 cnmptk1p.j . . . . . . . . . . 11 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
20 cnmptk1p.n . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐾 ∈ 𝑛-Locally Comp)
21 nllytop 22847 . . . . . . . . . . . . 13 (𝐾 ∈ 𝑛-Locally Comp β†’ 𝐾 ∈ Top)
2220, 21syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐾 ∈ Top)
23 cnmptk1p.l . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘))
24 topontop 22285 . . . . . . . . . . . . 13 (𝐿 ∈ (TopOnβ€˜π‘) β†’ 𝐿 ∈ Top)
2523, 24syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐿 ∈ Top)
26 eqid 2733 . . . . . . . . . . . . 13 (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
2726xkotopon 22974 . . . . . . . . . . . 12 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
2822, 25, 27syl2anc 585 . . . . . . . . . . 11 (πœ‘ β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
29 cnmptk2.a . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
30 cnf2 22623 . . . . . . . . . . 11 ((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))
3119, 28, 29, 30syl3anc 1372 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
3231fvmptelcdm 7065 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
3332adantr 482 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
34 eqid 2733 . . . . . . . . 9 (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))
3534fvmpt2 6963 . . . . . . . 8 ((π‘₯ ∈ 𝑋 ∧ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯) = (𝑦 ∈ π‘Œ ↦ 𝐴))
3618, 33, 35syl2anc 585 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯) = (𝑦 ∈ π‘Œ ↦ 𝐴))
3736fveq1d 6848 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦) = ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘¦))
38 simpr 486 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ π‘Œ)
39 cnmptk1p.k . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4039adantr 482 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4123adantr 482 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐿 ∈ (TopOnβ€˜π‘))
42 cnf2 22623 . . . . . . . . 9 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 ∈ (TopOnβ€˜π‘) ∧ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
4340, 41, 32, 42syl3anc 1372 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
4443fvmptelcdm 7065 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ 𝑍)
45 eqid 2733 . . . . . . . 8 (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴)
4645fvmpt2 6963 . . . . . . 7 ((𝑦 ∈ π‘Œ ∧ 𝐴 ∈ 𝑍) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘¦) = 𝐴)
4738, 44, 46syl2anc 585 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘¦) = 𝐴)
4837, 47eqtrd 2773 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦) = 𝐴)
49483impa 1111 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦) = 𝐴)
5049mpoeq3dva 7438 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴))
5117, 50eqtrid 2785 . 2 (πœ‘ β†’ (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴))
5219, 39cnmpt1st 23042 . . . 4 (πœ‘ β†’ (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ 𝑀) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
5319, 39, 52, 29cnmpt21f 23046 . . 3 (πœ‘ β†’ (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐿 ↑ko 𝐾)))
5419, 39cnmpt2nd 23043 . . 3 (πœ‘ β†’ (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ π‘˜) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐾))
55 eqid 2733 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56 toponuni 22286 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
5739, 56syl 17 . . . . 5 (πœ‘ β†’ π‘Œ = βˆͺ 𝐾)
58 mpoeq12 7434 . . . . 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‘§)))
5955, 57, 58sylancr 588 . . . 4 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)))
60 eqid 2733 . . . . . 6 βˆͺ 𝐾 = βˆͺ 𝐾
61 eqid 2733 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§))
6260, 61xkofvcn 23058 . . . . 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))
6320, 25, 62syl2anc 585 . . . 4 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
6459, 63eqeltrd 2834 . . 3 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
65 fveq1 6845 . . . 4 (𝑓 = ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€) β†’ (π‘“β€˜π‘§) = (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘§))
66 fveq2 6846 . . . 4 (𝑧 = π‘˜ β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘§) = (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜))
6765, 66sylan9eq 2793 . . 3 ((𝑓 = ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€) ∧ 𝑧 = π‘˜) β†’ (π‘“β€˜π‘§) = (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜))
6819, 39, 53, 54, 28, 39, 64, 67cnmpt22 23048 . 2 (πœ‘ β†’ (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
6951, 68eqeltrrd 2835 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4869   ↦ cmpt 5192  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-ixp 8842  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-pt 17334  df-top 22266  df-topon 22283  df-bases 22319  df-ntr 22394  df-nei 22472  df-cn 22601  df-cnp 22602  df-cmp 22761  df-nlly 22841  df-tx 22936  df-xko 22937
This theorem is referenced by:  xkocnv  23188
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