Theorem cnmptk2 23037

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	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmptk1p.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmptk1p.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmptk1p.n	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
cnmptk2.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))

Assertion

Ref	Expression
cnmptk2	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Distinct variable groups:   𝑥,𝐽   𝑥,𝐾   𝑥,𝐿   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝑦,𝑍

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑦)   𝐾(𝑦)   𝐿(𝑦)   𝑍(𝑥)

Proof of Theorem cnmptk2

Dummy variables 𝑓 𝑘 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nffvmpt1 6853	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
2		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝑘
3	1, 2	nffv 6852	. . . 4 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)
4		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
5		nfmpt1 5213	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌 ↦ 𝐴)
6	4, 5	nfmpt 5212	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
7		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦𝑤
8	6, 7	nffv 6852	. . . . 5 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
9		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦𝑘
10	8, 9	nffv 6852	. . . 4 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)
11		nfcv 2907	. . . 4 ⊢ Ⅎ𝑤(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)
12		nfcv 2907	. . . 4 ⊢ Ⅎ𝑘(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)
13		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥))
14	13	fveq1d 6844	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘))
15		fveq2 6842	. . . . 5 ⊢ (𝑘 = 𝑦 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦))
16	14, 15	sylan9eq 2796	. . . 4 ⊢ ((𝑤 = 𝑥 ∧ 𝑘 = 𝑦) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦))
17	3, 10, 11, 12, 16	cbvmpo 7451	. . 3 ⊢ (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦))
18		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑥 ∈ 𝑋)
19		cnmptk1p.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
20		cnmptk1p.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
21		nllytop 22824	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
22	20, 21	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Top)
23		cnmptk1p.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
24		topontop 22262	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
26		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
27	26	xkotopon 22951	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
28	22, 25, 27	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
29		cnmptk2.a	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
30		cnf2 22600	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
31	19, 28, 29, 30	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
32	31	fvmptelcdm 7061	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
33	32	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
34		eqid 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
35	34	fvmpt2 6959	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥) = (𝑦 ∈ 𝑌 ↦ 𝐴))
36	18, 33, 35	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥) = (𝑦 ∈ 𝑌 ↦ 𝐴))
37	36	fveq1d 6844	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑦))
38		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
39		cnmptk1p.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
40	39	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
41	23	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍))
42		cnf2 22600	. . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
43	40, 41, 32, 42	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
44	43	fvmptelcdm 7061	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)
45		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
46	45	fvmpt2 6959	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑦) = 𝐴)
47	38, 44, 46	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑦) = 𝐴)
48	37, 47	eqtrd 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = 𝐴)
49	48	3impa 1110	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = 𝐴)
50	49	mpoeq3dva 7434	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
51	17, 50	eqtrid 2788	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
52	19, 39	cnmpt1st 23019	. . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
53	19, 39, 52, 29	cnmpt21f 23023	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ↑ko 𝐾)))
54	19, 39	cnmpt2nd 23020	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
55		eqid 2736	. . . . 5 ⊢ (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56		toponuni 22263	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
57	39, 56	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
58		mpoeq12 7430	. . . . 5 ⊢ (((𝐾 Cn 𝐿) = (𝐾 Cn 𝐿) ∧ 𝑌 = ∪ 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)))
59	55, 57, 58	sylancr 587	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)))
60		eqid 2736	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
61		eqid 2736	. . . . . 6 ⊢ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧))
62	60, 61	xkofvcn 23035	. . . . 5 ⊢ ((𝐾 ∈ 𝑛-Locally Comp ∧ 𝐿 ∈ Top) → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
63	20, 25, 62	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
64	59, 63	eqeltrd 2838	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
65		fveq1 6841	. . . 4 ⊢ (𝑓 = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) → (𝑓‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧))
66		fveq2 6842	. . . 4 ⊢ (𝑧 = 𝑘 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘))
67	65, 66	sylan9eq 2796	. . 3 ⊢ ((𝑓 = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) ∧ 𝑧 = 𝑘) → (𝑓‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘))
68	19, 39, 53, 54, 28, 39, 64, 67	cnmpt22 23025	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
69	51, 68	eqeltrrd 2839	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∪ cuni 4865   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Compccmp 22737  𝑛-Locally cnlly 22816   ×_t ctx 22911   ↑_ko cxko 22912

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-pt 17326  df-top 22243  df-topon 22260  df-bases 22296  df-ntr 22371  df-nei 22449  df-cn 22578  df-cnp 22579  df-cmp 22738  df-nlly 22818  df-tx 22913  df-xko 22914

This theorem is referenced by:  xkocnv  23165