Theorem cnmptk2 23894

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 6896	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
2		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥𝑘
3	1, 2	nffv 6895	. . . 4 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)
4		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
5		nfmpt1 5212	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌 ↦ 𝐴)
6	4, 5	nfmpt 5211	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
7		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑦𝑤
8	6, 7	nffv 6895	. . . . 5 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
9		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦𝑘
10	8, 9	nffv 6895	. . . 4 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)
11		nfcv 2927	. . . 4 ⊢ Ⅎ𝑤(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)
12		nfcv 2927	. . . 4 ⊢ Ⅎ𝑘(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)
13		fveq2 6885	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥))
14	13	fveq1d 6887	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘))
15		fveq2 6885	. . . . 5 ⊢ (𝑘 = 𝑦 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦))
16	14, 15	sylan9eq 2820	. . . 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 7513	. . 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 781	. . . . . . . 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 23681	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
22	20, 21	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Top)
23		cnmptk1p.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
24		topontop 23120	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
25	23, 24	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
26		eqid 2765	. . . . . . . . . . . . 13 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
27	26	xkotopon 23808	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
28	22, 25, 27	syl2anc 596	. . . . . . . . . . 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 23456	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)):𝑋⟶(𝐾 Cn 𝐿))
32	31	fvmptelcdm 7112	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
33	32	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿))
34		eqid 2765	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
35	34	fvmpt2 7005	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥) = (𝑦 ∈ 𝑌 ↦ 𝐴))
36	18, 33, 35	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥) = (𝑦 ∈ 𝑌 ↦ 𝐴))
37	36	fveq1d 6887	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑦))
38		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
39		cnmptk1p.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
40	39	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
41	23	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘𝑍))
42		cnf2 23456	. . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
43	40, 41, 32, 42	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝐴):𝑌⟶𝑍)
44	43	fvmptelcdm 7112	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)
45		eqid 2765	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
46	45	fvmpt2 7005	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑦) = 𝐴)
47	38, 44, 46	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑦) = 𝐴)
48	37, 47	eqtrd 2800	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = 𝐴)
49	48	3impa 1127	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦) = 𝐴)
50	49	mpoeq3dva 7496	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
51	17, 50	eqtrid 2812	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
52	19, 39	cnmpt1st 23876	. . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
53	19, 39, 52, 29	cnmpt21f 23880	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ↑ko 𝐾)))
54	19, 39	cnmpt2nd 23877	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
55		eqid 2765	. . . . 5 ⊢ (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56		toponuni 23121	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
57	39, 56	syl 18	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
58		mpoeq12 7492	. . . . 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 599	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)))
60		eqid 2765	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
61		eqid 2765	. . . . . 6 ⊢ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧))
62	60, 61	xkofvcn 23892	. . . . 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 596	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
64	59, 63	eqeltrd 2865	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
65		fveq1 6884	. . . 4 ⊢ (𝑓 = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) → (𝑓‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧))
66		fveq2 6885	. . . 4 ⊢ (𝑧 = 𝑘 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘))
67	65, 66	sylan9eq 2820	. . 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 23882	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
69	51, 68	eqeltrrd 2866	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∪ cuni 4874   ↦ cmpt 5194  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  Topctop 23100  TopOnctopon 23117   Cn ccn 23431  Compccmp 23593  𝑛-Locally cnlly 23673   ×_t ctx 23768   ↑_ko cxko 23769

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-1o 8459  df-2o 8460  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-fin 8953  df-fi 9378  df-rest 17497  df-topgen 17518  df-pt 17519  df-top 23101  df-topon 23118  df-bases 23153  df-ntr 23227  df-nei 23305  df-cn 23434  df-cnp 23435  df-cmp 23594  df-nlly 23675  df-tx 23770  df-xko 23771

This theorem is used by:  xkocnv  24022