Theorem xkohmeo 22874

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 22719, 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 22650	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		topontop 21970	. . . . . 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 2738	. . . . . 6 ⊢ (𝐿 ↑ko (𝐽 ×t 𝐾)) = (𝐿 ↑ko (𝐽 ×t 𝐾))
10	9	xkotopon 22659	. . . . 5 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
11	7, 8, 10	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐿 ↑ko (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 ×t 𝐾) Cn 𝐿)))
12		vex 3426	. . . . . . . . 9 ⊢ 𝑓 ∈ V
13		vex 3426	. . . . . . . . 9 ⊢ 𝑥 ∈ V
14	12, 13	op1std 7814	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (1st ‘𝑧) = 𝑓)
15	12, 13	op2ndd 7815	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (2nd ‘𝑧) = 𝑥)
16		eqidd 2739	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → 𝑦 = 𝑦)
17	14, 15, 16	oveq123d 7276	. . . . . . 7 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → ((2nd ‘𝑧)(1st ‘𝑧)𝑦) = (𝑥𝑓𝑦))
18	17	mpteq2dv 5172	. . . . . 6 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (𝑦 ∈ 𝑌 ↦ ((2nd ‘𝑧)(1st ‘𝑧)𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
19	18	mpompt 7366	. . . . 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 22650	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → ((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) ∈ (TopOn‘(((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋)))
22		vex 3426	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
23		vex 3426	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	22, 23	op1std 7814	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (1st ‘𝑤) = 𝑧)
25	24	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (1st ‘(1st ‘𝑤)) = (1st ‘𝑧))
26	24	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd ‘(1st ‘𝑤)) = (2nd ‘𝑧))
27	22, 23	op2ndd 7815	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
28	25, 26, 27	oveq123d 7276	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤)) = ((2nd ‘𝑧)(1st ‘𝑧)𝑦))
29	28	mpompt 7366	. . . . . . 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 22650	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) ∈ (TopOn‘((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
32		toptopon2 21975	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
33	8, 32	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
34		xkohmeo.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)
35		xkohmeo.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)
36		txcmp 22702	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Comp ∧ 𝑦 ∈ Comp) → (𝑥 ×t 𝑦) ∈ Comp)
37	36	txnlly 22696	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐾 ∈ 𝑛-Locally Comp) → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
38	34, 35, 37	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ 𝑛-Locally Comp)
39	25	mpompt 7366	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1st ‘(1st ‘𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (1st ‘𝑧))
40	5	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41	33	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
42		xp1st 7836	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) → (1st ‘𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
43	42	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋))
44		xp1st 7836	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑤) ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) → (1st ‘(1st ‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
45	43, 44	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st ‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
46		cnf2 22308	. . . . . . . . . . . . 13 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (1st ‘(1st ‘𝑤)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (1st ‘(1st ‘𝑤)):(𝑋 × 𝑌)⟶∪ 𝐿)
47	40, 41, 45, 46	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st ‘𝑤)):(𝑋 × 𝑌)⟶∪ 𝐿)
48	47	feqmptd 6819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1st ‘(1st ‘𝑤)) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1st ‘(1st ‘𝑤))‘𝑢)))
49	48	mpteq2dva 5170	. . . . . . . . . 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))))
50	39, 49	eqtr3id 2793	. . . . . . . . 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))))
51	21, 3	cnmpt1st 22727	. . . . . . . . . 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)))
52		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (1st ‘𝑡) = (1st ‘𝑧))
53	52	cbvmptv 5183	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑧))
54	14	mpompt 7366	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑓)
55	11, 2	cnmpt1st 22727	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑓) ∈ (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ↑ko (𝐽 ×t 𝐾))))
56	54, 55	eqeltrid 2843	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑧)) ∈ (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ↑ko (𝐽 ×t 𝐾))))
57	53, 56	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (1st ‘𝑡)) ∈ (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ↑ko (𝐽 ×t 𝐾))))
58	21, 3, 51, 21, 57, 52	cnmpt21 22730	. . . . . . . . 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))))
59	50, 58	eqeltrrd 2840	. . . . . . . 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))))
60	26	mpompt 7366	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st ‘𝑤))) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (2nd ‘𝑧))
61		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (2nd ‘𝑡) = (2nd ‘𝑧))
62	61	cbvmptv 5183	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑡)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑧))
63	15	mpompt 7366	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑧)) = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑥)
64	11, 2	cnmpt2nd 22728	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
65	63, 64	eqeltrid 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑧)) ∈ (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
66	62, 65	eqeltrid 2843	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) ↦ (2nd ‘𝑡)) ∈ (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn 𝐽))
67	21, 3, 51, 21, 66, 61	cnmpt21 22730	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (2nd ‘𝑧)) ∈ ((((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
68	60, 67	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘(1st ‘𝑤))) ∈ ((((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐽))
69	27	mpompt 7366	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘𝑤)) = (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ 𝑦)
70	21, 3	cnmpt2nd 22728	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
71	69, 70	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2nd ‘𝑤)) ∈ ((((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) ×t 𝐾) Cn 𝐾))
72	31, 68, 71	cnmpt1t 22724	. . . . . . . 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)))
73		fveq2 6756	. . . . . . . . 9 ⊢ (𝑢 = ⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩ → ((1st ‘(1st ‘𝑤))‘𝑢) = ((1st ‘(1st ‘𝑤))‘⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩))
74		df-ov 7258	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤)) = ((1st ‘(1st ‘𝑤))‘⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩)
75	73, 74	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑢 = ⟨(2nd ‘(1st ‘𝑤)), (2nd ‘𝑤)⟩ → ((1st ‘(1st ‘𝑤))‘𝑢) = ((2nd ‘(1st ‘𝑤))(1st ‘(1st ‘𝑤))(2nd ‘𝑤)))
76	31, 5, 33, 38, 59, 72, 75	cnmptk1p 22744	. . . . . . 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))
77	29, 76	eqeltrrid 2844	. . . . . 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))
78	21, 3, 77	cnmpt2k 22747	. . . . 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)))
79	19, 78	eqeltrrid 2844	. . . 4 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) ∈ (((𝐿 ↑ko (𝐽 ×t 𝐾)) ×t 𝐽) Cn (𝐿 ↑ko 𝐾)))
80	11, 2, 79	cnmpt2k 22747	. . 3 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) ∈ ((𝐿 ↑ko (𝐽 ×t 𝐾)) Cn ((𝐿 ↑ko 𝐾) ↑ko 𝐽)))
81	1, 80	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ↑ko (𝐽 ×t 𝐾)) Cn ((𝐿 ↑ko 𝐾) ↑ko 𝐽)))
82	2, 3, 1, 34, 35, 8	xkocnv 22873	. . . 4 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
83	13, 23	op1std 7814	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
84	83	fveq2d 6760	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑔‘(1st ‘𝑧)) = (𝑔‘𝑥))
85	13, 23	op2ndd 7815	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
86	84, 85	fveq12d 6763	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)) = ((𝑔‘𝑥)‘𝑦))
87	86	mpompt 7366	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
88	87	mpteq2i 5175	. . . 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)))
89	82, 88	eqtr4di 2797	. . 3 ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)))))
90		nllytop 22532	. . . . . 6 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
91	34, 90	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
92		nllytop 22532	. . . . . . 7 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
93	35, 92	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
94		xkotop 22647	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ Top)
95	93, 8, 94	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ Top)
96		eqid 2738	. . . . . 6 ⊢ ((𝐿 ↑ko 𝐾) ↑ko 𝐽) = ((𝐿 ↑ko 𝐾) ↑ko 𝐽)
97	96	xkotopon 22659	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐿 ↑ko 𝐾) ∈ Top) → ((𝐿 ↑ko 𝐾) ↑ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ↑ko 𝐾))))
98	91, 95, 97	syl2anc 583	. . . 4 ⊢ (𝜑 → ((𝐿 ↑ko 𝐾) ↑ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ↑ko 𝐾))))
99		vex 3426	. . . . . . . . 9 ⊢ 𝑔 ∈ V
100	99, 22	op1std 7814	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (1st ‘𝑤) = 𝑔)
101	99, 22	op2ndd 7815	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd ‘𝑤) = 𝑧)
102	101	fveq2d 6760	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (1st ‘(2nd ‘𝑤)) = (1st ‘𝑧))
103	100, 102	fveq12d 6763	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) = (𝑔‘(1st ‘𝑧)))
104	101	fveq2d 6760	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (2nd ‘(2nd ‘𝑤)) = (2nd ‘𝑧))
105	103, 104	fveq12d 6763	. . . . . 6 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘(2nd ‘(2nd ‘𝑤))) = ((𝑔‘(1st ‘𝑧))‘(2nd ‘𝑧)))
106	105	mpompt 7366	. . . . 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)))
107		txtopon 22650	. . . . . . 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))))
108	98, 5, 107	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (((𝐿 ↑ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))))
109	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → 𝐾 ∈ (TopOn‘𝑌))
110	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → 𝐿 ∈ (TopOn‘∪ 𝐿))
111		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾)
112	111	xkotopon 22659	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
113	93, 8, 112	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
114		xp1st 7836	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌)) → (1st ‘𝑤) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
115		cnf2 22308	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (1st ‘𝑤) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (1st ‘𝑤):𝑋⟶(𝐾 Cn 𝐿))
116	2, 113, 114, 115	syl2an3an 1420	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘𝑤):𝑋⟶(𝐾 Cn 𝐿))
117		xp2nd 7837	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌)) → (2nd ‘𝑤) ∈ (𝑋 × 𝑌))
118	117	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → (2nd ‘𝑤) ∈ (𝑋 × 𝑌))
119		xp1st 7836	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑤) ∈ (𝑋 × 𝑌) → (1st ‘(2nd ‘𝑤)) ∈ 𝑋)
120	118, 119	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘(2nd ‘𝑤)) ∈ 𝑋)
121	116, 120	ffvelrnd 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) ∈ (𝐾 Cn 𝐿))
122		cnf2 22308	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) ∈ (𝐾 Cn 𝐿)) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))):𝑌⟶∪ 𝐿)
123	109, 110, 121, 122	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))):𝑌⟶∪ 𝐿)
124	123	feqmptd 6819	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))) = (𝑦 ∈ 𝑌 ↦ (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘𝑦)))
125	124	mpteq2dva 5170	. . . . . . 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))))
126	100	mpompt 7366	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘𝑤)) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔)
127	116	feqmptd 6819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌))) → (1st ‘𝑤) = (𝑥 ∈ 𝑋 ↦ ((1st ‘𝑤)‘𝑥)))
128	127	mpteq2dva 5170	. . . . . . . . . 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))))
129	126, 128	eqtr3id 2793	. . . . . . . . 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))))
130	98, 5	cnmpt1st 22727	. . . . . . . . 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)))
131	129, 130	eqeltrrd 2840	. . . . . . . 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)))
132	102	mpompt 7366	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd ‘𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧))
133	98, 5	cnmpt2nd 22728	. . . . . . . . . 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)))
134	52	cbvmptv 5183	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧))
135	83	mpompt 7366	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
136	2, 3	cnmpt1st 22727	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
137	135, 136	eqeltrid 2843	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
138	134, 137	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
139	98, 5, 133, 5, 138, 52	cnmpt21 22730	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) ∈ ((((𝐿 ↑ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
140	132, 139	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1st ‘(2nd ‘𝑤))) ∈ ((((𝐿 ↑ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐽))
141		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = (1st ‘(2nd ‘𝑤)) → ((1st ‘𝑤)‘𝑥) = ((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤))))
142	108, 2, 113, 34, 131, 140, 141	cnmptk1p 22744	. . . . . . 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)))
143	125, 142	eqeltrrd 2840	. . . . . 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)))
144	104	mpompt 7366	. . . . . . 7 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd ‘𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧))
145	61	cbvmptv 5183	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧))
146	85	mpompt 7366	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
147	2, 3	cnmpt2nd 22728	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
148	146, 147	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
149	145, 148	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑡)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
150	98, 5, 133, 5, 149, 61	cnmpt21 22730	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) ∈ ((((𝐿 ↑ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
151	144, 150	eqeltrid 2843	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2nd ‘(2nd ‘𝑤))) ∈ ((((𝐿 ↑ko 𝐾) ↑ko 𝐽) ×t (𝐽 ×t 𝐾)) Cn 𝐾))
152		fveq2 6756	. . . . . 6 ⊢ (𝑦 = (2nd ‘(2nd ‘𝑤)) → (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘𝑦) = (((1st ‘𝑤)‘(1st ‘(2nd ‘𝑤)))‘(2nd ‘(2nd ‘𝑤))))
153	108, 3, 33, 35, 143, 151, 152	cnmptk1p 22744	. . . . 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))
154	106, 153	eqeltrrid 2844	. . . 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))
155	98, 5, 154	cnmpt2k 22747	. . 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))))
156	89, 155	eqeltrd 2839	. 2 ⊢ (𝜑 → ◡𝐹 ∈ (((𝐿 ↑ko 𝐾) ↑ko 𝐽) Cn (𝐿 ↑ko (𝐽 ×t 𝐾))))
157		ishmeo 22818	. 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)))))
158	81, 156, 157	sylanbrc 582	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ↑ko (𝐽 ×t 𝐾))Homeo((𝐿 ↑ko 𝐾) ↑ko 𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564  ∪ cuni 4836   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802  2^nd c2nd 7803  Topctop 21950  TopOnctopon 21967   Cn ccn 22283  Compccmp 22445  𝑛-Locally cnlly 22524   ×_t ctx 22619   ↑_ko cxko 22620  Homeochmeo 22812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-map 8575  df-ixp 8644  df-en 8692  df-dom 8693  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-pt 17072  df-top 21951  df-topon 21968  df-bases 22004  df-ntr 22079  df-nei 22157  df-cn 22286  df-cnp 22287  df-cmp 22446  df-nlly 22526  df-tx 22621  df-xko 22622  df-hmeo 22814

This theorem is referenced by: (None)