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Theorem cnmptk2 23541
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 6895 . . . . 5 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)
2 nfcv 2897 . . . . 5 β„²π‘₯π‘˜
31, 2nffv 6894 . . . 4 β„²π‘₯(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜)
4 nfcv 2897 . . . . . . 7 Ⅎ𝑦𝑋
5 nfmpt1 5249 . . . . . . 7 Ⅎ𝑦(𝑦 ∈ π‘Œ ↦ 𝐴)
64, 5nfmpt 5248 . . . . . 6 Ⅎ𝑦(π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))
7 nfcv 2897 . . . . . 6 Ⅎ𝑦𝑀
86, 7nffv 6894 . . . . 5 Ⅎ𝑦((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)
9 nfcv 2897 . . . . 5 β„²π‘¦π‘˜
108, 9nffv 6894 . . . 4 Ⅎ𝑦(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€)β€˜π‘˜)
11 nfcv 2897 . . . 4 Ⅎ𝑀(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦)
12 nfcv 2897 . . . 4 β„²π‘˜(((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦)
13 fveq2 6884 . . . . . 6 (𝑀 = π‘₯ β†’ ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘€) = ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯))
1413fveq1d 6886 . . . . 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 6884 . . . . 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 2786 . . . 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 7498 . . 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 766 . . . . . . . 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 23328 . . . . . . . . . . . . 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 22766 . . . . . . . . . . . . 13 (𝐿 ∈ (TopOnβ€˜π‘) β†’ 𝐿 ∈ Top)
2523, 24syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐿 ∈ Top)
26 eqid 2726 . . . . . . . . . . . . 13 (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
2726xkotopon 23455 . . . . . . . . . . . 12 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) β†’ (𝐿 ↑ko 𝐾) ∈ (TopOnβ€˜(𝐾 Cn 𝐿)))
2822, 25, 27syl2anc 583 . . . . . . . . . . 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 23104 . . . . . . . . . . 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 1368 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)):π‘‹βŸΆ(𝐾 Cn 𝐿))
3231fvmptelcdm 7107 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
3332adantr 480 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
34 eqid 2726 . . . . . . . . 9 (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))
3534fvmpt2 7002 . . . . . . . 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 583 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ ((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯) = (𝑦 ∈ π‘Œ ↦ 𝐴))
3736fveq1d 6886 . . . . . 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 484 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ π‘Œ)
39 cnmptk1p.k . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4039adantr 480 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4123adantr 480 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐿 ∈ (TopOnβ€˜π‘))
42 cnf2 23104 . . . . . . . . 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 1368 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ 𝐴):π‘ŒβŸΆπ‘)
4443fvmptelcdm 7107 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ 𝑍)
45 eqid 2726 . . . . . . . 8 (𝑦 ∈ π‘Œ ↦ 𝐴) = (𝑦 ∈ π‘Œ ↦ 𝐴)
4645fvmpt2 7002 . . . . . . 7 ((𝑦 ∈ π‘Œ ∧ 𝐴 ∈ 𝑍) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘¦) = 𝐴)
4738, 44, 46syl2anc 583 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐴)β€˜π‘¦) = 𝐴)
4837, 47eqtrd 2766 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦) = 𝐴)
49483impa 1107 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ (((π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴))β€˜π‘₯)β€˜π‘¦) = 𝐴)
5049mpoeq3dva 7481 . . 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 2778 . 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 23523 . . . 4 (πœ‘ β†’ (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ 𝑀) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
5319, 39, 52, 29cnmpt21f 23527 . . 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 23524 . . 3 (πœ‘ β†’ (𝑀 ∈ 𝑋, π‘˜ ∈ π‘Œ ↦ π‘˜) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐾))
55 eqid 2726 . . . . 5 (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56 toponuni 22767 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
5739, 56syl 17 . . . . 5 (πœ‘ β†’ π‘Œ = βˆͺ 𝐾)
58 mpoeq12 7477 . . . . 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 586 . . . 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 2726 . . . . . 6 βˆͺ 𝐾 = βˆͺ 𝐾
61 eqid 2726 . . . . . 6 (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§))
6260, 61xkofvcn 23539 . . . . 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 583 . . . 4 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ βˆͺ 𝐾 ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
6459, 63eqeltrd 2827 . . 3 (πœ‘ β†’ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ π‘Œ ↦ (π‘“β€˜π‘§)) ∈ (((𝐿 ↑ko 𝐾) Γ—t 𝐾) Cn 𝐿))
65 fveq1 6883 . . . 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 6884 . . . 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 2786 . . 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 23529 . 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 2828 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 395   = wceq 1533   ∈ wcel 2098  βˆͺ cuni 4902   ↦ cmpt 5224  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  Topctop 22746  TopOnctopon 22763   Cn ccn 23079  Compccmp 23241  π‘›-Locally cnlly 23320   Γ—t ctx 23415   ↑ko cxko 23416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8464  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-rest 17375  df-topgen 17396  df-pt 17397  df-top 22747  df-topon 22764  df-bases 22800  df-ntr 22875  df-nei 22953  df-cn 23082  df-cnp 23083  df-cmp 23242  df-nlly 23322  df-tx 23417  df-xko 23418
This theorem is referenced by:  xkocnv  23669
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