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Theorem xkohmeo 23326

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 23171, 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 23102	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		topontop 22422	. . . . . 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 2732	. . . . . 6 ⊢ (𝐿 ↑_ko (𝐽 ×_t 𝐾)) = (𝐿 ↑_ko (𝐽 ×_t 𝐾))
10	9	xkotopon 23111	. . . . 5 ⊢ (((𝐽 ×_t 𝐾) ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑_ko (𝐽 ×_t 𝐾)) ∈ (TopOn‘((𝐽 ×_t 𝐾) Cn 𝐿)))
11	7, 8, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐿 ↑_ko (𝐽 ×_t 𝐾)) ∈ (TopOn‘((𝐽 ×_t 𝐾) Cn 𝐿)))
12		vex 3478	. . . . . . . . 9 ⊢ 𝑓 ∈ V
13		vex 3478	. . . . . . . . 9 ⊢ 𝑥 ∈ V
14	12, 13	op1std 7987	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (1^st ‘𝑧) = 𝑓)
15	12, 13	op2ndd 7988	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (2^nd ‘𝑧) = 𝑥)
16		eqidd 2733	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → 𝑦 = 𝑦)
17	14, 15, 16	oveq123d 7432	. . . . . . 7 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → ((2^nd ‘𝑧)(1^st ‘𝑧)𝑦) = (𝑥𝑓𝑦))
18	17	mpteq2dv 5250	. . . . . 6 ⊢ (𝑧 = ⟨𝑓, 𝑥⟩ → (𝑦 ∈ 𝑌 ↦ ((2^nd ‘𝑧)(1^st ‘𝑧)𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
19	18	mpompt 7524	. . . . 5 ⊢ (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦 ∈ 𝑌 ↦ ((2^nd ‘𝑧)(1^st ‘𝑧)𝑦))) = (𝑓 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))
20		txtopon 23102	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ∈ (TopOn‘(((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋)))
22		vex 3478	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
23		vex 3478	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	22, 23	op1std 7987	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (1^st ‘𝑤) = 𝑧)
25	24	fveq2d 6895	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (1^st ‘(1^st ‘𝑤)) = (1^st ‘𝑧))
26	24	fveq2d 6895	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (2^nd ‘(1^st ‘𝑤)) = (2^nd ‘𝑧))
27	22, 23	op2ndd 7988	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → (2^nd ‘𝑤) = 𝑦)
28	25, 26, 27	oveq123d 7432	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑧, 𝑦⟩ → ((2^nd ‘(1^st ‘𝑤))(1^st ‘(1^st ‘𝑤))(2^nd ‘𝑤)) = ((2^nd ‘𝑧)(1^st ‘𝑧)𝑦))
29	28	mpompt 7524	. . . . . . 7 ⊢ (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ((2^nd ‘(1^st ‘𝑤))(1^st ‘(1^st ‘𝑤))(2^nd ‘𝑤))) = (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ ((2^nd ‘𝑧)(1^st ‘𝑧)𝑦))
30		txtopon 23102	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) ∈ (TopOn‘((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)))
32		toptopon2 22427	. . . . . . . . 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 23154	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Comp ∧ 𝑦 ∈ Comp) → (𝑥 ×_t 𝑦) ∈ Comp)
37	36	txnlly 23148	. . . . . . . . 9 ⊢ ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐾 ∈ 𝑛-Locally Comp) → (𝐽 ×_t 𝐾) ∈ 𝑛-Locally Comp)
38	34, 35, 37	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ×_t 𝐾) ∈ 𝑛-Locally Comp)
39	25	mpompt 7524	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1^st ‘(1^st ‘𝑤))) = (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (1^st ‘𝑧))
40	5	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41	33	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
42		xp1st 8009	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) → (1^st ‘𝑤) ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋))
43	42	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1^st ‘𝑤) ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋))
44		xp1st 8009	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑤) ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) → (1^st ‘(1^st ‘𝑤)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐿))
45	43, 44	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1^st ‘(1^st ‘𝑤)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐿))
46		cnf2 22760	. . . . . . . . . . . . 13 ⊢ (((𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (1^st ‘(1^st ‘𝑤)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐿)) → (1^st ‘(1^st ‘𝑤)):(𝑋 × 𝑌)⟶∪ 𝐿)
47	40, 41, 45, 46	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1^st ‘(1^st ‘𝑤)):(𝑋 × 𝑌)⟶∪ 𝐿)
48	47	feqmptd 6960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌)) → (1^st ‘(1^st ‘𝑤)) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1^st ‘(1^st ‘𝑤))‘𝑢)))
49	48	mpteq2dva 5248	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (1^st ‘(1^st ‘𝑤))) = (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1^st ‘(1^st ‘𝑤))‘𝑢))))
50	39, 49	eqtr3id 2786	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (1^st ‘𝑧)) = (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1^st ‘(1^st ‘𝑤))‘𝑢))))
51	21, 3	cnmpt1st 23179	. . . . . . . . . 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 6891	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (1^st ‘𝑡) = (1^st ‘𝑧))
53	52	cbvmptv 5261	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (1^st ‘𝑡)) = (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (1^st ‘𝑧))
54	14	mpompt 7524	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (1^st ‘𝑧)) = (𝑓 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑓)
55	11, 2	cnmpt1st 23179	. . . . . . . . . . . 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 2837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (1^st ‘𝑧)) ∈ (((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) Cn (𝐿 ↑_ko (𝐽 ×_t 𝐾))))
57	53, 56	eqeltrid 2837	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (1^st ‘𝑡)) ∈ (((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) Cn (𝐿 ↑_ko (𝐽 ×_t 𝐾))))
58	21, 3, 51, 21, 57, 52	cnmpt21 23182	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (1^st ‘𝑧)) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn (𝐿 ↑_ko (𝐽 ×_t 𝐾))))
59	50, 58	eqeltrrd 2834	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (𝑢 ∈ (𝑋 × 𝑌) ↦ ((1^st ‘(1^st ‘𝑤))‘𝑢))) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn (𝐿 ↑_ko (𝐽 ×_t 𝐾))))
60	26	mpompt 7524	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2^nd ‘(1^st ‘𝑤))) = (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (2^nd ‘𝑧))
61		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (2^nd ‘𝑡) = (2^nd ‘𝑧))
62	61	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (2^nd ‘𝑡)) = (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (2^nd ‘𝑧))
63	15	mpompt 7524	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (2^nd ‘𝑧)) = (𝑓 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑥)
64	11, 2	cnmpt2nd 23180	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑓 ∈ ((𝐽 ×_t 𝐾) Cn 𝐿), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) Cn 𝐽))
65	63, 64	eqeltrid 2837	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (2^nd ‘𝑧)) ∈ (((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) Cn 𝐽))
66	62, 65	eqeltrid 2837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (2^nd ‘𝑡)) ∈ (((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) Cn 𝐽))
67	21, 3, 51, 21, 66, 61	cnmpt21 23182	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ (2^nd ‘𝑧)) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn 𝐽))
68	60, 67	eqeltrid 2837	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2^nd ‘(1^st ‘𝑤))) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn 𝐽))
69	27	mpompt 7524	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2^nd ‘𝑤)) = (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ 𝑦)
70	21, 3	cnmpt2nd 23180	. . . . . . . . . 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 2837	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ (2^nd ‘𝑤)) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn 𝐾))
72	31, 68, 71	cnmpt1t 23176	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ⟨(2^nd ‘(1^st ‘𝑤)), (2^nd ‘𝑤)⟩) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn (𝐽 ×_t 𝐾)))
73		fveq2 6891	. . . . . . . . 9 ⊢ (𝑢 = ⟨(2^nd ‘(1^st ‘𝑤)), (2^nd ‘𝑤)⟩ → ((1^st ‘(1^st ‘𝑤))‘𝑢) = ((1^st ‘(1^st ‘𝑤))‘⟨(2^nd ‘(1^st ‘𝑤)), (2^nd ‘𝑤)⟩))
74		df-ov 7414	. . . . . . . . 9 ⊢ ((2^nd ‘(1^st ‘𝑤))(1^st ‘(1^st ‘𝑤))(2^nd ‘𝑤)) = ((1^st ‘(1^st ‘𝑤))‘⟨(2^nd ‘(1^st ‘𝑤)), (2^nd ‘𝑤)⟩)
75	73, 74	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑢 = ⟨(2^nd ‘(1^st ‘𝑤)), (2^nd ‘𝑤)⟩ → ((1^st ‘(1^st ‘𝑤))‘𝑢) = ((2^nd ‘(1^st ‘𝑤))(1^st ‘(1^st ‘𝑤))(2^nd ‘𝑤)))
76	31, 5, 33, 38, 59, 72, 75	cnmptk1p 23196	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ((((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) × 𝑌) ↦ ((2^nd ‘(1^st ‘𝑤))(1^st ‘(1^st ‘𝑤))(2^nd ‘𝑤))) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn 𝐿))
77	29, 76	eqeltrrid 2838	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋), 𝑦 ∈ 𝑌 ↦ ((2^nd ‘𝑧)(1^st ‘𝑧)𝑦)) ∈ ((((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) ×_t 𝐾) Cn 𝐿))
78	21, 3, 77	cnmpt2k 23199	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (((𝐽 ×_t 𝐾) Cn 𝐿) × 𝑋) ↦ (𝑦 ∈ 𝑌 ↦ ((2^nd ‘𝑧)(1^st ‘𝑧)𝑦))) ∈ (((𝐿 ↑_ko (𝐽 ×_t 𝐾)) ×_t 𝐽) Cn (𝐿 ↑_ko 𝐾)))
79	19, 78	eqeltrrid 2838	. . . 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 23199	. . 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 2837	. 2 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ↑_ko (𝐽 ×_t 𝐾)) Cn ((𝐿 ↑_ko 𝐾) ↑_ko 𝐽)))
82	2, 3, 1, 34, 35, 8	xkocnv 23325	. . . 4 ⊢ (𝜑 → ^◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
83	13, 23	op1std 7987	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑧) = 𝑥)
84	83	fveq2d 6895	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑔‘(1^st ‘𝑧)) = (𝑔‘𝑥))
85	13, 23	op2ndd 7988	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑧) = 𝑦)
86	84, 85	fveq12d 6898	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧)) = ((𝑔‘𝑥)‘𝑦))
87	86	mpompt 7524	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))
88	87	mpteq2i 5253	. . . 4 ⊢ (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧)))) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))
89	82, 88	eqtr4di 2790	. . 3 ⊢ (𝜑 → ^◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧)))))
90		nllytop 22984	. . . . . 6 ⊢ (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
91	34, 90	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
92		nllytop 22984	. . . . . . 7 ⊢ (𝐾 ∈ 𝑛-Locally Comp → 𝐾 ∈ Top)
93	35, 92	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
94		xkotop 23099	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑_ko 𝐾) ∈ Top)
95	93, 8, 94	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐿 ↑_ko 𝐾) ∈ Top)
96		eqid 2732	. . . . . 6 ⊢ ((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) = ((𝐿 ↑_ko 𝐾) ↑_ko 𝐽)
97	96	xkotopon 23111	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐿 ↑_ko 𝐾) ∈ Top) → ((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ↑_ko 𝐾))))
98	91, 95, 97	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ∈ (TopOn‘(𝐽 Cn (𝐿 ↑_ko 𝐾))))
99		vex 3478	. . . . . . . . 9 ⊢ 𝑔 ∈ V
100	99, 22	op1std 7987	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (1^st ‘𝑤) = 𝑔)
101	99, 22	op2ndd 7988	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (2^nd ‘𝑤) = 𝑧)
102	101	fveq2d 6895	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (1^st ‘(2^nd ‘𝑤)) = (1^st ‘𝑧))
103	100, 102	fveq12d 6898	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤))) = (𝑔‘(1^st ‘𝑧)))
104	101	fveq2d 6895	. . . . . . 7 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (2^nd ‘(2^nd ‘𝑤)) = (2^nd ‘𝑧))
105	103, 104	fveq12d 6898	. . . . . 6 ⊢ (𝑤 = ⟨𝑔, 𝑧⟩ → (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘(2^nd ‘(2^nd ‘𝑤))) = ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧)))
106	105	mpompt 7524	. . . . 5 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘(2^nd ‘(2^nd ‘𝑤)))) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧)))
107		txtopon 23102	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → (((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) ∈ (TopOn‘((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))))
109	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → 𝐾 ∈ (TopOn‘𝑌))
110	33	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → 𝐿 ∈ (TopOn‘∪ 𝐿))
111		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝐿 ↑_ko 𝐾) = (𝐿 ↑_ko 𝐾)
112	111	xkotopon 23111	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑_ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
113	93, 8, 112	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ↑_ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
114		xp1st 8009	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) → (1^st ‘𝑤) ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)))
115		cnf2 22760	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑_ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (1^st ‘𝑤) ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾))) → (1^st ‘𝑤):𝑋⟶(𝐾 Cn 𝐿))
116	2, 113, 114, 115	syl2an3an 1422	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → (1^st ‘𝑤):𝑋⟶(𝐾 Cn 𝐿))
117		xp2nd 8010	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) → (2^nd ‘𝑤) ∈ (𝑋 × 𝑌))
118	117	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → (2^nd ‘𝑤) ∈ (𝑋 × 𝑌))
119		xp1st 8009	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝑤) ∈ (𝑋 × 𝑌) → (1^st ‘(2^nd ‘𝑤)) ∈ 𝑋)
120	118, 119	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → (1^st ‘(2^nd ‘𝑤)) ∈ 𝑋)
121	116, 120	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤))) ∈ (𝐾 Cn 𝐿))
122		cnf2 22760	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤))) ∈ (𝐾 Cn 𝐿)) → ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤))):𝑌⟶∪ 𝐿)
123	109, 110, 121, 122	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤))):𝑌⟶∪ 𝐿)
124	123	feqmptd 6960	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤))) = (𝑦 ∈ 𝑌 ↦ (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘𝑦)))
125	124	mpteq2dva 5248	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))) = (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦 ∈ 𝑌 ↦ (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘𝑦))))
126	100	mpompt 7524	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1^st ‘𝑤)) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔)
127	116	feqmptd 6960	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌))) → (1^st ‘𝑤) = (𝑥 ∈ 𝑋 ↦ ((1^st ‘𝑤)‘𝑥)))
128	127	mpteq2dva 5248	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1^st ‘𝑤)) = (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥 ∈ 𝑋 ↦ ((1^st ‘𝑤)‘𝑥))))
129	126, 128	eqtr3id 2786	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑔) = (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥 ∈ 𝑋 ↦ ((1^st ‘𝑤)‘𝑥))))
130	98, 5	cnmpt1st 23179	. . . . . . . . 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 2834	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑥 ∈ 𝑋 ↦ ((1^st ‘𝑤)‘𝑥))) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn ((𝐿 ↑_ko 𝐾) ↑_ko 𝐽)))
132	102	mpompt 7524	. . . . . . . . 9 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1^st ‘(2^nd ‘𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1^st ‘𝑧))
133	98, 5	cnmpt2nd 23180	. . . . . . . . . 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 5261	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ (1^st ‘𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1^st ‘𝑧))
135	83	mpompt 7524	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1^st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
136	2, 3	cnmpt1st 23179	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
137	135, 136	eqeltrid 2837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (1^st ‘𝑧)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
138	134, 137	eqeltrid 2837	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (1^st ‘𝑡)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
139	98, 5, 133, 5, 138, 52	cnmpt21 23182	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (1^st ‘𝑧)) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn 𝐽))
140	132, 139	eqeltrid 2837	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (1^st ‘(2^nd ‘𝑤))) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn 𝐽))
141		fveq2 6891	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘(2^nd ‘𝑤)) → ((1^st ‘𝑤)‘𝑥) = ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤))))
142	108, 2, 113, 34, 131, 140, 141	cnmptk1p 23196	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ ((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn (𝐿 ↑_ko 𝐾)))
143	125, 142	eqeltrrd 2834	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (𝑦 ∈ 𝑌 ↦ (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘𝑦))) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn (𝐿 ↑_ko 𝐾)))
144	104	mpompt 7524	. . . . . . 7 ⊢ (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2^nd ‘(2^nd ‘𝑤))) = (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2^nd ‘𝑧))
145	61	cbvmptv 5261	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) ↦ (2^nd ‘𝑡)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2^nd ‘𝑧))
146	85	mpompt 7524	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2^nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
147	2, 3	cnmpt2nd 23180	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
148	146, 147	eqeltrid 2837	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ (2^nd ‘𝑧)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
149	145, 148	eqeltrid 2837	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋 × 𝑌) ↦ (2^nd ‘𝑡)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
150	98, 5, 133, 5, 149, 61	cnmpt21 23182	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ (2^nd ‘𝑧)) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn 𝐾))
151	144, 150	eqeltrid 2837	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (2^nd ‘(2^nd ‘𝑤))) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn 𝐾))
152		fveq2 6891	. . . . . 6 ⊢ (𝑦 = (2^nd ‘(2^nd ‘𝑤)) → (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘𝑦) = (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘(2^nd ‘(2^nd ‘𝑤))))
153	108, 3, 33, 35, 143, 151, 152	cnmptk1p 23196	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ ((𝐽 Cn (𝐿 ↑_ko 𝐾)) × (𝑋 × 𝑌)) ↦ (((1^st ‘𝑤)‘(1^st ‘(2^nd ‘𝑤)))‘(2^nd ‘(2^nd ‘𝑤)))) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn 𝐿))
154	106, 153	eqeltrrid 2838	. . . 4 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)), 𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧))) ∈ ((((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) ×_t (𝐽 ×_t 𝐾)) Cn 𝐿))
155	98, 5, 154	cnmpt2k 23199	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐽 Cn (𝐿 ↑_ko 𝐾)) ↦ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((𝑔‘(1^st ‘𝑧))‘(2^nd ‘𝑧)))) ∈ (((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) Cn (𝐿 ↑_ko (𝐽 ×_t 𝐾))))
156	89, 155	eqeltrd 2833	. 2 ⊢ (𝜑 → ^◡𝐹 ∈ (((𝐿 ↑_ko 𝐾) ↑_ko 𝐽) Cn (𝐿 ↑_ko (𝐽 ×_t 𝐾))))
157		ishmeo 23270	. 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 583	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ↑_ko (𝐽 ×_t 𝐾))Homeo((𝐿 ↑_ko 𝐾) ↑_ko 𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⟨cop 4634 ∪ cuni 4908 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7975 2^nd c2nd 7976 Topctop 22402 TopOnctopon 22419 Cn ccn 22735 Compccmp 22897 𝑛-Locally cnlly 22976 ×_t ctx 23071 ↑_ko cxko 23072 Homeochmeo 23264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-fin 8945 df-fi 9408 df-rest 17370 df-topgen 17391 df-pt 17392 df-top 22403 df-topon 22420 df-bases 22456 df-ntr 22531 df-nei 22609 df-cn 22738 df-cnp 22739 df-cmp 22898 df-nlly 22978 df-tx 23073 df-xko 23074 df-hmeo 23266

This theorem is referenced by: (None)

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