Theorem cnmpopc 24820

Description: Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)

Hypotheses

Ref	Expression
cnmpopc.r	⊢ 𝑅 = (topGen‘ran (,))
cnmpopc.m	⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵))
cnmpopc.n	⊢ 𝑁 = (𝑅 ↾t (𝐵[,]𝐶))
cnmpopc.o	⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶))
cnmpopc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
cnmpopc.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
cnmpopc.b	⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))
cnmpopc.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpopc.q	⊢ ((𝜑 ∧ (𝑥 = 𝐵 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐸)
cnmpopc.d	⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
cnmpopc.e	⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))

Assertion

Ref	Expression
cnmpopc	⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem cnmpopc

Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ ∪ (𝑂 ×t 𝐽) = ∪ (𝑂 ×t 𝐽)
2		eqid 2729	. 2 ⊢ ∪ 𝐾 = ∪ 𝐾
3		cnmpopc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		cnmpopc.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
5		iccssre 13332	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴[,]𝐶) ⊆ ℝ)
6	3, 4, 5	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐶) ⊆ ℝ)
7		cnmpopc.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))
8	6, 7	sseldd 3936	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		icccld 24652	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
10	3, 8, 9	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
11		cnmpopc.r	. . . . . . 7 ⊢ 𝑅 = (topGen‘ran (,))
12	11	fveq2i 6825	. . . . . 6 ⊢ (Clsd‘𝑅) = (Clsd‘(topGen‘ran (,)))
13	10, 12	eleqtrrdi 2839	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑅))
14		ssun1 4129	. . . . . 6 ⊢ (𝐴[,]𝐵) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
15		iccsplit 13388	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
16	3, 4, 7, 15	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
17	14, 16	sseqtrrid 3979	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶))
18		uniretop 24648	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
19	11	unieqi 4870	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ (topGen‘ran (,))
20	18, 19	eqtr4i 2755	. . . . . 6 ⊢ ℝ = ∪ 𝑅
21	20	restcldi 23058	. . . . 5 ⊢ (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐴[,]𝐵) ∈ (Clsd‘𝑅) ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶)) → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶))))
22	6, 13, 17, 21	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶))))
23		cnmpopc.o	. . . . 5 ⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶))
24	23	fveq2i 6825	. . . 4 ⊢ (Clsd‘𝑂) = (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))
25	22, 24	eleqtrrdi 2839	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑂))
26		cnmpopc.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
27		toponuni 22799	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
28	26, 27	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29		topontop 22798	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
30		eqid 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
31	30	topcld 22920	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (Clsd‘𝐽))
32	26, 29, 31	3syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝐽 ∈ (Clsd‘𝐽))
33	28, 32	eqeltrd 2828	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Clsd‘𝐽))
34		txcld 23488	. . 3 ⊢ (((𝐴[,]𝐵) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
35	25, 33, 34	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
36		icccld 24652	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
37	8, 4, 36	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
38	37, 12	eleqtrrdi 2839	. . . . 5 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑅))
39		ssun2 4130	. . . . . 6 ⊢ (𝐵[,]𝐶) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
40	39, 16	sseqtrrid 3979	. . . . 5 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶))
41	20	restcldi 23058	. . . . 5 ⊢ (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐵[,]𝐶) ∈ (Clsd‘𝑅) ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶)) → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶))))
42	6, 38, 40, 41	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶))))
43	42, 24	eleqtrrdi 2839	. . 3 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑂))
44		txcld 23488	. . 3 ⊢ (((𝐵[,]𝐶) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
45	43, 33, 44	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
46	16	xpeq1d 5648	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋))
47		xpundir 5689	. . . 4 ⊢ (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋))
48	46, 47	eqtrdi 2780	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)))
49		retopon 24649	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
50	11, 49	eqeltri 2824	. . . . . . 7 ⊢ 𝑅 ∈ (TopOn‘ℝ)
51		resttopon 23046	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
52	50, 6, 51	sylancr 587	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
53	23, 52	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (TopOn‘(𝐴[,]𝐶)))
54		txtopon 23476	. . . . 5 ⊢ ((𝑂 ∈ (TopOn‘(𝐴[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
55	53, 26, 54	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
56		toponuni 22799	. . . 4 ⊢ ((𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)) → ((𝐴[,]𝐶) × 𝑋) = ∪ (𝑂 ×t 𝐽))
57	55, 56	syl 17	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = ∪ (𝑂 ×t 𝐽))
58	48, 57	eqtr3d 2766	. 2 ⊢ (𝜑 → (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) = ∪ (𝑂 ×t 𝐽))
59		cnmpopc.m	. . . . . . . . . 10 ⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵))
60	17, 6	sstrd 3946	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
61		resttopon 23046	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
62	50, 60, 61	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
63	59, 62	eqeltrid 2832	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘(𝐴[,]𝐵)))
64		txtopon 23476	. . . . . . . . 9 ⊢ ((𝑀 ∈ (TopOn‘(𝐴[,]𝐵)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)))
65	63, 26, 64	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)))
66		cnmpopc.d	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
67		cntop2 23126	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾) → 𝐾 ∈ Top)
68	66, 67	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
69		toptopon2 22803	. . . . . . . . 9 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
70	68, 69	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
71		elicc2 13314	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
72	3, 8, 71	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
73	72	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
74	73	simp3d 1144	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
75	74	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → 𝑥 ≤ 𝐵)
76	75	iftrued 4484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐷)
77	76	mpoeq3dva 7426	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷))
78	77, 66	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
79		cnf2 23134	. . . . . . . 8 ⊢ (((𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾)) → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾)
80	65, 70, 78, 79	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾)
81		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
82	81	fmpo 8003	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾)
83	80, 82	sylibr 234	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
84		cnmpopc.n	. . . . . . . . . 10 ⊢ 𝑁 = (𝑅 ↾t (𝐵[,]𝐶))
85	40, 6	sstrd 3946	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
86		resttopon 23046	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐵[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
87	50, 85, 86	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
88	84, 87	eqeltrid 2832	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘(𝐵[,]𝐶)))
89		txtopon 23476	. . . . . . . . 9 ⊢ ((𝑁 ∈ (TopOn‘(𝐵[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
90	88, 26, 89	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
91		elicc2 13314	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)))
92	8, 4, 91	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)))
93	92	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶))
94	93	simp2d 1143	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ≤ 𝑥)
95	94	biantrud 531	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥)))
96	93	simp1d 1142	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ∈ ℝ)
97	8	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ∈ ℝ)
98	96, 97	letri3d 11258	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 = 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥)))
99	95, 98	bitr4d 282	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ 𝑥 = 𝐵))
100	99	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝐵 ↔ 𝑥 = 𝐵))
101		cnmpopc.q	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 = 𝐵 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐸)
102	101	ancom2s 650	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → 𝐷 = 𝐸)
103	102	ifeq1d 4496	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = if(𝑥 ≤ 𝐵, 𝐸, 𝐸))
104		ifid 4517	. . . . . . . . . . . . . . 15 ⊢ if(𝑥 ≤ 𝐵, 𝐸, 𝐸) = 𝐸
105	103, 104	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
106	105	expr 456	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸))
107	106	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸))
108	100, 107	sylbid 240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸))
109		iffalse 4485	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
110	108, 109	pm2.61d1 180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
111	110	mpoeq3dva 7426	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸))
112		cnmpopc.e	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
113	111, 112	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
114		cnf2 23134	. . . . . . . 8 ⊢ (((𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾)) → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾)
115	90, 70, 113, 114	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾)
116		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
117	116	fmpo 8003	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾)
118	115, 117	sylibr 234	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
119		ralun 4149	. . . . . 6 ⊢ ((∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ∧ ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾) → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
120	83, 118, 119	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
121	120, 16	raleqtrrdv 3293	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
122		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
123	122	fmpo 8003	. . . 4 ⊢ (∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾)
124	121, 123	sylib 218	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾)
125	57	feq2d 6636	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):∪ (𝑂 ×t 𝐽)⟶∪ 𝐾))
126	124, 125	mpbid 232	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):∪ (𝑂 ×t 𝐽)⟶∪ 𝐾)
127		ssid 3958	. . . 4 ⊢ 𝑋 ⊆ 𝑋
128		resmpo 7469	. . . 4 ⊢ (((𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ 𝑋 ⊆ 𝑋) → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)))
129	17, 127, 128	sylancl 586	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)))
130		retop 24647	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
131	11, 130	eqeltri 2824	. . . . . . . . 9 ⊢ 𝑅 ∈ Top
132		ovex 7382	. . . . . . . . 9 ⊢ (𝐴[,]𝐶) ∈ V
133		resttop 23045	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐶) ∈ V) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top)
134	131, 132, 133	mp2an 692	. . . . . . . 8 ⊢ (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top
135	23, 134	eqeltri 2824	. . . . . . 7 ⊢ 𝑂 ∈ Top
136	135	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ Top)
137		ovexd 7384	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ V)
138		txrest 23516	. . . . . 6 ⊢ (((𝑂 ∈ Top ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ ((𝐴[,]𝐵) ∈ V ∧ 𝑋 ∈ (Clsd‘𝐽))) → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)))
139	136, 26, 137, 33, 138	syl22anc 838	. . . . 5 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)))
140	131	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Top)
141		ovexd 7384	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐶) ∈ V)
142		restabs 23050	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵)))
143	140, 17, 141, 142	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵)))
144	23	oveq1i 7359	. . . . . . 7 ⊢ (𝑂 ↾t (𝐴[,]𝐵)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵))
145	143, 144, 59	3eqtr4g 2789	. . . . . 6 ⊢ (𝜑 → (𝑂 ↾t (𝐴[,]𝐵)) = 𝑀)
146	28	oveq2d 7365	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t 𝑋) = (𝐽 ↾t ∪ 𝐽))
147	30	restid 17337	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ↾t ∪ 𝐽) = 𝐽)
148	26, 147	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t ∪ 𝐽) = 𝐽)
149	146, 148	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾t 𝑋) = 𝐽)
150	145, 149	oveq12d 7367	. . . . 5 ⊢ (𝜑 → ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)) = (𝑀 ×t 𝐽))
151	139, 150	eqtrd 2764	. . . 4 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = (𝑀 ×t 𝐽))
152	151	oveq1d 7364	. . 3 ⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾) = ((𝑀 ×t 𝐽) Cn 𝐾))
153	78, 129, 152	3eltr4d 2843	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾))
154		resmpo 7469	. . . 4 ⊢ (((𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ 𝑋 ⊆ 𝑋) → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)))
155	40, 127, 154	sylancl 586	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)))
156		ovexd 7384	. . . . . 6 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ V)
157		txrest 23516	. . . . . 6 ⊢ (((𝑂 ∈ Top ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ ((𝐵[,]𝐶) ∈ V ∧ 𝑋 ∈ (Clsd‘𝐽))) → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)))
158	136, 26, 156, 33, 157	syl22anc 838	. . . . 5 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)))
159		restabs 23050	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶)))
160	140, 40, 141, 159	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶)))
161	23	oveq1i 7359	. . . . . . 7 ⊢ (𝑂 ↾t (𝐵[,]𝐶)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶))
162	160, 161, 84	3eqtr4g 2789	. . . . . 6 ⊢ (𝜑 → (𝑂 ↾t (𝐵[,]𝐶)) = 𝑁)
163	162, 149	oveq12d 7367	. . . . 5 ⊢ (𝜑 → ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)) = (𝑁 ×t 𝐽))
164	158, 163	eqtrd 2764	. . . 4 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = (𝑁 ×t 𝐽))
165	164	oveq1d 7364	. . 3 ⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾) = ((𝑁 ×t 𝐽) Cn 𝐾))
166	113, 155, 165	3eltr4d 2843	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾))
167	1, 2, 35, 45, 58, 126, 153, 166	paste 23179	1 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∪ cun 3901   ⊆ wss 3903  ifcif 4476  ∪ cuni 4858   class class class wbr 5092   × cxp 5617  ran crn 5620   ↾ cres 5621  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  ℝcr 11008   ≤ cle 11150  (,)cioo 13248  [,]cicc 13251   ↾_t crest 17324  topGenctg 17341  Topctop 22778  TopOnctopon 22795  Clsdccld 22901   Cn ccn 23109   ×_t ctx 23445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fi 9301  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-ioo 13252  df-icc 13255  df-rest 17326  df-topgen 17347  df-top 22779  df-topon 22796  df-bases 22831  df-cld 22904  df-cn 23112  df-tx 23447

This theorem is referenced by:  htpycc  24877  pcocn  24915  pcohtpylem  24917  pcopt  24920  pcopt2  24921  pcoass  24922  pcorevlem  24924