Theorem cnmptk2 22837

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 6785	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
2		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥𝑘
3	1, 2	nffv 6784	. . . 4 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)
4		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
5		nfmpt1 5182	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌 ↦ 𝐴)
6	4, 5	nfmpt 5181	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
7		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦𝑤
8	6, 7	nffv 6784	. . . . 5 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
9		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑦𝑘
10	8, 9	nffv 6784	. . . 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 6774	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥))
14	13	fveq1d 6776	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘))
15		fveq2 6774	. . . . 5 ⊢ (𝑘 = 𝑦 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦))
16	14, 15	sylan9eq 2798	. . . 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 7369	. . 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 766	. . . . . . . 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 22624	. . . . . . . . . . . . 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 22062	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
26		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
27	26	xkotopon 22751	. . . . . . . . . . . 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 22400	. . . . . . . . . . 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 1370	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
32	31	fvmptelrn 6987	. . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
35	34	fvmpt2 6886	. . . . . . . 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 6776	. . . . . 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 22400	. . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
43	40, 41, 32, 42	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
44	43	fvmptelrn 6987	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)
45		eqid 2738	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
46	45	fvmpt2 6886	. . . . . . 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 2778	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = 𝐴)
49	48	3impa 1109	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = 𝐴)
50	49	mpoeq3dva 7352	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
51	17, 50	eqtrid 2790	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
52	19, 39	cnmpt1st 22819	. . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
53	19, 39, 52, 29	cnmpt21f 22823	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ↑ko 𝐾)))
54	19, 39	cnmpt2nd 22820	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
55		eqid 2738	. . . . 5 ⊢ (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56		toponuni 22063	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
57	39, 56	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
58		mpoeq12 7348	. . . . 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 2738	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
61		eqid 2738	. . . . . 6 ⊢ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧))
62	60, 61	xkofvcn 22835	. . . . 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 2839	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
65		fveq1 6773	. . . 4 ⊢ (𝑓 = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) → (𝑓‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧))
66		fveq2 6774	. . . 4 ⊢ (𝑧 = 𝑘 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘))
67	65, 66	sylan9eq 2798	. . 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 22825	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
69	51, 68	eqeltrrd 2840	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∪ cuni 4839   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Topctop 22042  TopOnctopon 22059   Cn ccn 22375  Compccmp 22537  𝑛-Locally cnlly 22616   ×_t ctx 22711   ↑_ko cxko 22712

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-pt 17155  df-top 22043  df-topon 22060  df-bases 22096  df-ntr 22171  df-nei 22249  df-cn 22378  df-cnp 22379  df-cmp 22538  df-nlly 22618  df-tx 22713  df-xko 22714

This theorem is referenced by:  xkocnv  22965