Theorem xkohmeo 21618

Description: The Exponential Law for topological spaces. The "currying" function 𝐹 is a homeomorphism on function spaces when 𝐽 and 𝐾 are exponentiable spaces (by xkococn 21463, it is sufficient to assume that 𝐽, 𝐾 are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Hypotheses

Ref	Expression
xkohmeo.x	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
xkohmeo.y	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
xkohmeo.f	⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
xkohmeo.j	⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)
xkohmeo.k	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
xkohmeo.l	⊢ (𝜑 → 𝐿 ∈ Top)

Assertion

Ref	Expression
xkohmeo	⊢ (𝜑 → 𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾))Homeo((𝐿 ^ko 𝐾) ^ko 𝐽)))

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

Proof of Theorem xkohmeo

Dummy variables 𝑔 𝑡 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xkohmeo.f	. . 3 ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))
2		xkohmeo.x	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		xkohmeo.y	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		txtopon 21394	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		topontop 20718	. . . . . 6 ⊢ ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝐽 ×t 𝐾) ∈ Top)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ Top)
8		xkohmeo.l	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Top)
9		eqid 2622	. . . . . 6 ⊢ (𝐿 ^ko (𝐽 ×t 𝐾)) = (𝐿 ^ko (𝐽 ×t 𝐾))
10	9	xkotopon 21403	. . . . 5 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
11	7, 8, 10	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐿 ^ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
12		vex 3203	. . . . . . . . 9 ⊢ 𝑓 ∈ V
13		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
14	12, 13	op1std 7178	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (1st ‘𝑧) = 𝑓)
15	12, 13	op2ndd 7179	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (2nd ‘𝑧) = 𝑥)
16		eqidd 2623	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → 𝑦 = 𝑦)
17	14, 15, 16	oveq123d 6671	. . . . . . 7 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → ((2nd ‘𝑧)(1st ‘𝑧)𝑦) = (𝑥𝑓𝑦))
18	17	mpteq2dv 4745	. . . . . 6 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (𝑦 ∈ 𝑌 ↦ ((2nd ‘𝑧)(1st ‘𝑧)𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
19	18	mpt2mpt 6752	. . . . 5 ⊢ (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦 ∈ 𝑌 ↦ ((2nd ‘𝑧)(1st ‘𝑧)𝑦))) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
20		txtopon 21394	. . . . . . 7 ⊢ (((𝐿 ^ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → ((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)))
21	11, 2, 20	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)))
22		vex 3203	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
23		vex 3203	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	22, 23	op1std 7178	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (1st ‘𝑤) = 𝑧)
25	24	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (1st ‘(1st ‘𝑤)) = (1st ‘𝑧))
26	24	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd ‘(1st ‘𝑤)) = (2nd ‘𝑧))
27	22, 23	op2ndd 7179	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
28	25, 26, 27	oveq123d 6671	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤)) = ((2nd ‘𝑧)(1st ‘𝑧)𝑦))
29	28	mpt2mpt 6752	. . . . . . 7 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ ((2nd ‘𝑧)(1st ‘𝑧)𝑦))
30		txtopon 21394	. . . . . . . . 9 ⊢ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
31	21, 3, 30	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
32		eqid 2622	. . . . . . . . . 10 ⊢ ∪ 𝐿 = ∪ 𝐿
33	32	toptopon 20722	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
34	8, 33	sylib 208	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
35		xkohmeo.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)
36		xkohmeo.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
37		txcmp 21446	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Comp ∧ 𝑦 ∈ Comp) → (𝑥 ×t 𝑦) ∈ Comp)
38	37	txnlly 21440	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐾 ∈ 𝑛-Locally Comp) → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
39	35, 36, 38	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
40	25	mpt2mpt 6752	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1st ‘(1st ‘𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (1st ‘𝑧))
41	5	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
42	34	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
43		xp1st 7198	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) → (1st ‘𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
44	43	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
45		xp1st 7198	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) → (1st ‘(1st ‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
46	44, 45	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st ‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
47		cnf2 21053	. . . . . . . . . . . . 13 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (1st ‘(1st ‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (1st ‘(1st ‘𝑤)):(𝑋 × 𝑌)⟶∪ 𝐿)
48	41, 42, 46, 47	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st ‘𝑤)):(𝑋 × 𝑌)⟶∪ 𝐿)
49	48	feqmptd 6249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st ‘𝑤)) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st ‘𝑤))‘𝑢)))
50	49	mpteq2dva 4744	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1st ‘(1st ‘𝑤))) = (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st ‘𝑤))‘𝑢))))
51	40, 50	syl5eqr 2670	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (1st ‘𝑧)) = (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st ‘𝑤))‘𝑢))))
52	21, 3	cnmpt1st 21471	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ 𝑧) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn ((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽)))
53		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (1st ‘𝑡) = (1st ‘𝑧))
54	53	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑧))
55	14	mpt2mpt 6752	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑓)
56	11, 2	cnmpt1st 21471	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑓) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
57	55, 56	syl5eqel 2705	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑧)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
58	54, 57	syl5eqel 2705	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑡)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
59	21, 3, 52, 21, 58, 53	cnmpt21 21474	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (1st ‘𝑧)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
60	51, 59	eqeltrrd 2702	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st ‘𝑤))‘𝑢))) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
61	26	mpt2mpt 6752	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st ‘𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (2nd ‘𝑧))
62		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (2nd ‘𝑡) = (2nd ‘𝑧))
63	62	cbvmptv 4750	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑧))
64	15	mpt2mpt 6752	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑥)
65	11, 2	cnmpt2nd 21472	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
66	64, 65	syl5eqel 2705	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑧)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
67	63, 66	syl5eqel 2705	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑡)) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
68	21, 3, 52, 21, 67, 62	cnmpt21 21474	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (2nd ‘𝑧)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
69	61, 68	syl5eqel 2705	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st ‘𝑤))) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
70	27	mpt2mpt 6752	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘𝑤)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ 𝑦)
71	21, 3	cnmpt2nd 21472	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
72	70, 71	syl5eqel 2705	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘𝑤)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
73	31, 69, 72	cnmpt1t 21468	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn (𝐽 ×t 𝐾)))
74		fveq2 6191	. . . . . . . . 9 ⊢ (𝑢 = ⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩ → ((1st ‘(1st ‘𝑤))‘𝑢) = ((1st ‘(1st ‘𝑤))‘⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩))
75		df-ov 6653	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤)) = ((1st ‘(1st ‘𝑤))‘⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩)
76	74, 75	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑢 = ⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩ → ((1st ‘(1st ‘𝑤))‘𝑢) = ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤)))
77	31, 5, 34, 39, 60, 73, 76	cnmptk1p 21488	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤))) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐿))
78	29, 77	syl5eqelr 2706	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ ((2nd ‘𝑧)(1st ‘𝑧)𝑦)) ∈ ((((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐿))
79	21, 3, 78	cnmpt2k 21491	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦 ∈ 𝑌 ↦ ((2nd ‘𝑧)(1st ‘𝑧)𝑦))) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko 𝐾)))
80	19, 79	syl5eqelr 2706	. . . 4 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) ∈ (((𝐿 ^ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ^ko 𝐾)))
81	11, 2, 80	cnmpt2k 21491	. . 3 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) ∈ ((𝐿 ^ko (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
82	1, 81	syl5eqel 2705	. 2 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
83	2, 3, 1, 35, 36, 8	xkocnv 21617	. . . 4 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
84	13, 23	op1std 7178	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
85	84	fveq2d 6195	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑔‘(1st ‘𝑧)) = (𝑔‘𝑥))
86	13, 23	op2ndd 7179	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
87	85, 86	fveq12d 6197	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)) = ((𝑔‘𝑥)‘𝑦))
88	87	mpt2mpt 6752	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
89	88	mpteq2i 4741	. . . 4 ⊢ (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
90	83, 89	syl6eqr 2674	. . 3 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)))))
91		nllytop 21276	. . . . . 6 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
92	35, 91	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
93		nllytop 21276	. . . . . . 7 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
94	36, 93	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
95		xkotop 21391	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ Top)
96	94, 8, 95	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐿 ^ko 𝐾) ∈ Top)
97		eqid 2622	. . . . . 6 ⊢ ((𝐿 ^ko 𝐾) ^ko 𝐽) = ((𝐿 ^ko 𝐾) ^ko 𝐽)
98	97	xkotopon 21403	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐿 ^ko 𝐾) ∈ Top) → ((𝐿 ^ko 𝐾) ^ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ^ko 𝐾))))
99	92, 96, 98	syl2anc 693	. . . 4 ⊢ (𝜑 → ((𝐿 ^ko 𝐾) ^ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ^ko 𝐾))))
100		vex 3203	. . . . . . . . 9 ⊢ 𝑔 ∈ V
101	100, 22	op1std 7178	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (1st ‘𝑤) = 𝑔)
102	100, 22	op2ndd 7179	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd ‘𝑤) = 𝑧)
103	102	fveq2d 6195	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (1st ‘(2nd ‘𝑤)) = (1st ‘𝑧))
104	101, 103	fveq12d 6197	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) = (𝑔‘(1st ‘𝑧)))
105	102	fveq2d 6195	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd ‘(2nd ‘𝑤)) = (2nd ‘𝑧))
106	104, 105	fveq12d 6197	. . . . . 6 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘(2nd ‘(2nd ‘𝑤))) = ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)))
107	106	mpt2mpt 6752	. . . . 5 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘(2nd ‘(2nd ‘𝑤)))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)))
108		txtopon 21394	. . . . . . 7 ⊢ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ^ko 𝐾))) ∧ (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) → (((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))))
109	99, 5, 108	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))))
110	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → 𝐾 ∈ (TopOn‘𝑌))
111	34	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → 𝐿 ∈ (TopOn‘∪ 𝐿))
112	2	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → 𝐽 ∈ (TopOn‘𝑋))
113		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
114	113	xkotopon 21403	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
115	94, 8, 114	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
116	115	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
117		xp1st 7198	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) → (1st ‘𝑤) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
118	117	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘𝑤) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
119		cnf2 21053	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (1st ‘𝑤) ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (1st ‘𝑤):𝑋⟶(𝐾 Cn 𝐿))
120	112, 116, 118, 119	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘𝑤):𝑋⟶(𝐾 Cn 𝐿))
121		xp2nd 7199	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) → (2nd ‘𝑤) ∈ (𝑋 × 𝑌))
122	121	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (2nd ‘𝑤) ∈ (𝑋 × 𝑌))
123		xp1st 7198	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑤) ∈ (𝑋 × 𝑌) → (1st ‘(2nd ‘𝑤)) ∈ 𝑋)
124	122, 123	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘(2nd ‘𝑤)) ∈ 𝑋)
125	120, 124	ffvelrnd 6360	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) ∈ (𝐾 Cn 𝐿))
126		cnf2 21053	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) ∈ (𝐾 Cn 𝐿)) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))):𝑌⟶∪ 𝐿)
127	110, 111, 125, 126	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))):𝑌⟶∪ 𝐿)
128	127	feqmptd 6249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) = (𝑦 ∈ 𝑌 ↦ (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘𝑦)))
129	128	mpteq2dva 4744	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))) = (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦 ∈ 𝑌 ↦ (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘𝑦))))
130	101	mpt2mpt 6752	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘𝑤)) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔)
131	120	feqmptd 6249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘𝑤) = (𝑥 ∈ 𝑋 ↦ ((1st ‘𝑤)‘𝑥)))
132	131	mpteq2dva 4744	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘𝑤)) = (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥 ∈ 𝑋 ↦ ((1st ‘𝑤)‘𝑥))))
133	130, 132	syl5eqr 2670	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) = (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥 ∈ 𝑋 ↦ ((1st ‘𝑤)‘𝑥))))
134	99, 5	cnmpt1st 21471	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
135	133, 134	eqeltrrd 2702	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥 ∈ 𝑋 ↦ ((1st ‘𝑤)‘𝑥))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)))
136	103	mpt2mpt 6752	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd ‘𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧))
137	99, 5	cnmpt2nd 21472	. . . . . . . . . 10 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐽 ×t 𝐾)))
138	53	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧))
139	84	mpt2mpt 6752	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
140	2, 3	cnmpt1st 21471	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
141	139, 140	syl5eqel 2705	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
142	138, 141	syl5eqel 2705	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
143	99, 5, 137, 5, 142, 53	cnmpt21 21474	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
144	136, 143	syl5eqel 2705	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd ‘𝑤))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
145		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = (1st ‘(2nd ‘𝑤)) → ((1st ‘𝑤)‘𝑥) = ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))))
146	109, 2, 115, 35, 135, 144, 145	cnmptk1p 21488	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐿 ^ko 𝐾)))
147	129, 146	eqeltrrd 2702	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦 ∈ 𝑌 ↦ (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘𝑦))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn (𝐿 ^ko 𝐾)))
148	105	mpt2mpt 6752	. . . . . . 7 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd ‘𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧))
149	62	cbvmptv 4750	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧))
150	86	mpt2mpt 6752	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
151	2, 3	cnmpt2nd 21472	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
152	150, 151	syl5eqel 2705	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
153	149, 152	syl5eqel 2705	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
154	99, 5, 137, 5, 153, 62	cnmpt21 21474	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
155	148, 154	syl5eqel 2705	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd ‘𝑤))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
156		fveq2 6191	. . . . . 6 ⊢ (𝑦 = (2nd ‘(2nd ‘𝑤)) → (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘𝑦) = (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘(2nd ‘(2nd ‘𝑤))))
157	109, 3, 34, 36, 147, 155, 156	cnmptk1p 21488	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ^ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘(2nd ‘(2nd ‘𝑤)))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐿))
158	107, 157	syl5eqelr 2706	. . . 4 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧))) ∈ ((((𝐿 ^ko 𝐾) ^ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐿))
159	99, 5, 158	cnmpt2k 21491	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ^ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)))) ∈ (((𝐿 ^ko 𝐾) ^ko 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
160	90, 159	eqeltrd 2701	. 2 ⊢ (𝜑 → ◡𝐹 ∈ (((𝐿 ^ko 𝐾) ^ko 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾))))
161		ishmeo 21562	. 2 ⊢ (𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾))Homeo((𝐿 ^ko 𝐾) ^ko 𝐽)) ↔ (𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾)) Cn ((𝐿 ^ko 𝐾) ^ko 𝐽)) ∧ ◡𝐹 ∈ (((𝐿 ^ko 𝐾) ^ko 𝐽) Cn (𝐿 ^ko (𝐽 ×t 𝐾)))))
162	82, 160, 161	sylanbrc 698	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ^ko (𝐽 ×t 𝐾))Homeo((𝐿 ^ko 𝐾) ^ko 𝐽)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ⟨cop 4183  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  Topctop 20698  TopOnctopon 20715   Cn ccn 21028  Compccmp 21189  𝑛-Locally cnlly 21268   ×_t ctx 21363   ^_ko cxko 21364  Homeochmeo 21556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-pt 16105  df-top 20699  df-topon 20716  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-cnp 21032  df-cmp 21190  df-nlly 21270  df-tx 21365  df-xko 21366  df-hmeo 21558

This theorem is referenced by: (None)