Theorem cnmptk2 23912

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 6889	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
2		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥𝑘
3	1, 2	nffv 6888	. . . 4 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)
4		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
5		nfmpt1 5204	. . . . . . 7 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌 ↦ 𝐴)
6	4, 5	nfmpt 5203	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
7		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑦𝑤
8	6, 7	nffv 6888	. . . . 5 ⊢ Ⅎ𝑦((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)
9		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑦𝑘
10	8, 9	nffv 6888	. . . 4 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)
11		nfcv 2922	. . . 4 ⊢ Ⅎ𝑤(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)
12		nfcv 2922	. . . 4 ⊢ Ⅎ𝑘(((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)
13		fveq2 6878	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥))
14	13	fveq1d 6880	. . . . 5 ⊢ (𝑤 = 𝑥 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘))
15		fveq2 6878	. . . . 5 ⊢ (𝑘 = 𝑦 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑘) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦))
16	14, 15	sylan9eq 2815	. . . 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 7507	. . 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 23699	. . . . . . . . . . . . 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 23138	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
25	23, 24	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Top)
26		eqid 2760	. . . . . . . . . . . . 13 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
27	26	xkotopon 23826	. . . . . . . . . . . 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 23474	. . . . . . . . . . 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 7106	. . . . . . . . 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 2760	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))
35	34	fvmpt2 6998	. . . . . . . 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 6880	. . . . . 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 23474	. . . . . . . . 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 7106	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)
45		eqid 2760	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑦 ∈ 𝑌 ↦ 𝐴)
46	45	fvmpt2 6998	. . . . . . 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 2795	. . . . 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 7490	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
51	17, 50	eqtrid 2807	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))
52	19, 39	cnmpt1st 23894	. . . 4 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑤) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
53	19, 39, 52, 29	cnmpt21f 23898	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ↑ko 𝐾)))
54	19, 39	cnmpt2nd 23895	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ 𝑘) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
55		eqid 2760	. . . . 5 ⊢ (𝐾 Cn 𝐿) = (𝐾 Cn 𝐿)
56		toponuni 23139	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
57	39, 56	syl 18	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝐾)
58		mpoeq12 7486	. . . . 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 2760	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
61		eqid 2760	. . . . . 6 ⊢ (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧)) = (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ ∪ 𝐾 ↦ (𝑓‘𝑧))
62	60, 61	xkofvcn 23910	. . . . 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 2860	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐾 Cn 𝐿), 𝑧 ∈ 𝑌 ↦ (𝑓‘𝑧)) ∈ (((𝐿 ↑ko 𝐾) ×t 𝐾) Cn 𝐿))
65		fveq1 6877	. . . 4 ⊢ (𝑓 = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) → (𝑓‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧))
66		fveq2 6878	. . . 4 ⊢ (𝑧 = 𝑘 → (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑧) = (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘))
67	65, 66	sylan9eq 2815	. . 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 23900	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝑋, 𝑘 ∈ 𝑌 ↦ (((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤)‘𝑘)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
69	51, 68	eqeltrrd 2861	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 2145  ∪ cuni 4867   ↦ cmpt 5186  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  Topctop 23118  TopOnctopon 23135   Cn ccn 23449  Compccmp 23611  𝑛-Locally cnlly 23691   ×_t ctx 23786   ↑_ko cxko 23787

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-1o 8455  df-2o 8456  df-map 8828  df-ixp 8905  df-en 8953  df-dom 8954  df-fin 8956  df-fi 9381  df-rest 17507  df-topgen 17528  df-pt 17529  df-top 23119  df-topon 23136  df-bases 23171  df-ntr 23245  df-nei 23323  df-cn 23452  df-cnp 23453  df-cmp 23612  df-nlly 23693  df-tx 23788  df-xko 23789

This theorem is used by:  xkocnv  24040