Theorem cnmpopc 25048

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 2765	. 2 ⊢ ∪ (𝑂 ×t 𝐽) = ∪ (𝑂 ×t 𝐽)
2		eqid 2765	. 2 ⊢ ∪ 𝐾 = ∪ 𝐾
3		cnmpopc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		cnmpopc.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
5		iccssre 13447	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴[,]𝐶) ⊆ ℝ)
6	3, 4, 5	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐶) ⊆ ℝ)
7		cnmpopc.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))
8	6, 7	sseldd 3940	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		icccld 24884	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
10	3, 8, 9	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
11		cnmpopc.r	. . . . . . 7 ⊢ 𝑅 = (topGen‘ran (,))
12	11	fveq2i 6874	. . . . . 6 ⊢ (Clsd‘𝑅) = (Clsd‘(topGen‘ran (,)))
13	10, 12	eleqtrrdi 2876	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑅))
14		ssun1 4133	. . . . . 6 ⊢ (𝐴[,]𝐵) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
15		iccsplit 13503	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
16	3, 4, 7, 15	syl3anc 1394	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
17	14, 16	sseqtrrid 3982	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶))
18		uniretop 24880	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
19	11	unieqi 4880	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ (topGen‘ran (,))
20	18, 19	eqtr4i 2791	. . . . . 6 ⊢ ℝ = ∪ 𝑅
21	20	restcldi 23291	. . . . 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 1394	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶))))
23		cnmpopc.o	. . . . 5 ⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶))
24	23	fveq2i 6874	. . . 4 ⊢ (Clsd‘𝑂) = (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))
25	22, 24	eleqtrrdi 2876	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑂))
26		cnmpopc.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
27		toponuni 23032	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
28	26, 27	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29		topontop 23031	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
30		eqid 2765	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
31	30	topcld 23153	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (Clsd‘𝐽))
32	26, 29, 31	3syl 19	. . . 4 ⊢ (𝜑 → ∪ 𝐽 ∈ (Clsd‘𝐽))
33	28, 32	eqeltrd 2865	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Clsd‘𝐽))
34		txcld 23721	. . 3 ⊢ (((𝐴[,]𝐵) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
35	25, 33, 34	syl2anc 595	. 2 ⊢ (𝜑 → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
36		icccld 24884	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
37	8, 4, 36	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
38	37, 12	eleqtrrdi 2876	. . . . 5 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑅))
39		ssun2 4134	. . . . . 6 ⊢ (𝐵[,]𝐶) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
40	39, 16	sseqtrrid 3982	. . . . 5 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶))
41	20	restcldi 23291	. . . . 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 1394	. . . 4 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅 ↾t (𝐴[,]𝐶))))
43	42, 24	eleqtrrdi 2876	. . 3 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑂))
44		txcld 23721	. . 3 ⊢ (((𝐵[,]𝐶) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
45	43, 33, 44	syl2anc 595	. 2 ⊢ (𝜑 → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
46	16	xpeq1d 5681	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋))
47		xpundir 5722	. . . 4 ⊢ (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋))
48	46, 47	eqtrdi 2816	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)))
49		retopon 24881	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
50	11, 49	eqeltri 2861	. . . . . . 7 ⊢ 𝑅 ∈ (TopOn‘ℝ)
51		resttopon 23279	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
52	50, 6, 51	sylancr 598	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
53	23, 52	eqeltrid 2869	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (TopOn‘(𝐴[,]𝐶)))
54		txtopon 23709	. . . . 5 ⊢ ((𝑂 ∈ (TopOn‘(𝐴[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
55	53, 26, 54	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
56		toponuni 23032	. . . 4 ⊢ ((𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)) → ((𝐴[,]𝐶) × 𝑋) = ∪ (𝑂 ×t 𝐽))
57	55, 56	syl 18	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = ∪ (𝑂 ×t 𝐽))
58	48, 57	eqtr3d 2802	. 2 ⊢ (𝜑 → (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) = ∪ (𝑂 ×t 𝐽))
59		cnmpopc.m	. . . . . . . . . 10 ⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵))
60	17, 6	sstrd 3949	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
61		resttopon 23279	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
62	50, 60, 61	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
63	59, 62	eqeltrid 2869	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘(𝐴[,]𝐵)))
64		txtopon 23709	. . . . . . . . 9 ⊢ ((𝑀 ∈ (TopOn‘(𝐴[,]𝐵)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)))
65	63, 26, 64	syl2anc 595	. . . . . . . 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 23359	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾) → 𝐾 ∈ Top)
68	66, 67	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
69		toptopon2 23036	. . . . . . . . 9 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
70	68, 69	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
71		elicc2 13429	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
72	3, 8, 71	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
73	72	biimpa 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
74	73	simp3d 1160	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
75	74	3adant3 1148	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → 𝑥 ≤ 𝐵)
76	75	iftrued 4491	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐷)
77	76	mpoeq3dva 7477	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷))
78	77, 66	eqeltrd 2865	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
79		cnf2 23367	. . . . . . . 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 1394	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶∪ 𝐾)
81		eqid 2765	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
82	81	fmpo 8053	. . . . . . 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 237	. . . . . 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 3949	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
86		resttopon 23279	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐵[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
87	50, 85, 86	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
88	84, 87	eqeltrid 2869	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘(𝐵[,]𝐶)))
89		txtopon 23709	. . . . . . . . 9 ⊢ ((𝑁 ∈ (TopOn‘(𝐵[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
90	88, 26, 89	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
91		elicc2 13429	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)))
92	8, 4, 91	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)))
93	92	biimpa 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶))
94	93	simp2d 1159	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ≤ 𝑥)
95	94	biantrud 540	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥)))
96	93	simp1d 1158	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ∈ ℝ)
97	8	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ∈ ℝ)
98	96, 97	letri3d 11340	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 = 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥)))
99	95, 98	bitr4d 285	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ 𝑥 = 𝐵))
100	99	3adant3 1148	. . . . . . . . . . . 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 662	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → 𝐷 = 𝐸)
103	102	ifeq1d 4503	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = if(𝑥 ≤ 𝐵, 𝐸, 𝐸))
104		ifid 4524	. . . . . . . . . . . . . . 15 ⊢ if(𝑥 ≤ 𝐵, 𝐸, 𝐸) = 𝐸
105	103, 104	eqtrdi 2816	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
106	105	expr 461	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸))
107	106	3adant2 1147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸))
108	100, 107	sylbid 243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → (𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸))
109		iffalse 4492	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
110	108, 109	pm2.61d1 182	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
111	110	mpoeq3dva 7477	. . . . . . . . 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 2865	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
114		cnf2 23367	. . . . . . . 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 1394	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶∪ 𝐾)
116		eqid 2765	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
117	116	fmpo 8053	. . . . . . 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 237	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
119		ralun 4153	. . . . . 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 595	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
121	120, 16	raleqtrrdv 3327	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
122		eqid 2765	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
123	122	fmpo 8053	. . . 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 221	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾)
125	57	feq2d 6679	. . 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 235	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):∪ (𝑂 ×t 𝐽)⟶∪ 𝐾)
127		ssid 3961	. . . 4 ⊢ 𝑋 ⊆ 𝑋
128		resmpo 7520	. . . 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 597	. . 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 24879	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
131	11, 130	eqeltri 2861	. . . . . . . . 9 ⊢ 𝑅 ∈ Top
132		ovex 7433	. . . . . . . . 9 ⊢ (𝐴[,]𝐶) ∈ V
133		resttop 23278	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐶) ∈ V) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top)
134	131, 132, 133	mp2an 704	. . . . . . . 8 ⊢ (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top
135	23, 134	eqeltri 2861	. . . . . . 7 ⊢ 𝑂 ∈ Top
136	135	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ Top)
137		ovexd 7435	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ V)
138		txrest 23749	. . . . . 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 851	. . . . 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 7435	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐶) ∈ V)
142		restabs 23283	. . . . . . . 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 1394	. . . . . . 7 ⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵)))
144	23	oveq1i 7410	. . . . . . 7 ⊢ (𝑂 ↾t (𝐴[,]𝐵)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵))
145	143, 144, 59	3eqtr4g 2825	. . . . . 6 ⊢ (𝜑 → (𝑂 ↾t (𝐴[,]𝐵)) = 𝑀)
146	28	oveq2d 7416	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t 𝑋) = (𝐽 ↾t ∪ 𝐽))
147	30	restid 17476	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ↾t ∪ 𝐽) = 𝐽)
148	26, 147	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t ∪ 𝐽) = 𝐽)
149	146, 148	eqtrd 2800	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾t 𝑋) = 𝐽)
150	145, 149	oveq12d 7418	. . . . 5 ⊢ (𝜑 → ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)) = (𝑀 ×t 𝐽))
151	139, 150	eqtrd 2800	. . . 4 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = (𝑀 ×t 𝐽))
152	151	oveq1d 7415	. . 3 ⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾) = ((𝑀 ×t 𝐽) Cn 𝐾))
153	78, 129, 152	3eltr4d 2880	. 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 7520	. . . 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 597	. . 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 7435	. . . . . 6 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ V)
157		txrest 23749	. . . . . 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 851	. . . . 5 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)))
159		restabs 23283	. . . . . . . 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 1394	. . . . . . 7 ⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶)))
161	23	oveq1i 7410	. . . . . . 7 ⊢ (𝑂 ↾t (𝐵[,]𝐶)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶))
162	160, 161, 84	3eqtr4g 2825	. . . . . 6 ⊢ (𝜑 → (𝑂 ↾t (𝐵[,]𝐶)) = 𝑁)
163	162, 149	oveq12d 7418	. . . . 5 ⊢ (𝜑 → ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)) = (𝑁 ×t 𝐽))
164	158, 163	eqtrd 2800	. . . 4 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = (𝑁 ×t 𝐽))
165	164	oveq1d 7415	. . 3 ⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾) = ((𝑁 ×t 𝐽) Cn 𝐾))
166	113, 155, 165	3eltr4d 2880	. 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 23412	1 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ∪ cun 3905   ⊆ wss 3907  ifcif 4483  ∪ cuni 4868   class class class wbr 5105   × cxp 5650  ran crn 5653   ↾ cres 5654  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ℝcr 11087   ≤ cle 11232  (,)cioo 13363  [,]cicc 13366   ↾_t crest 17463  topGenctg 17480  Topctop 23011  TopOnctopon 23028  Clsdccld 23134   Cn ccn 23342   ×_t ctx 23678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fi 9359  df-sup 9390  df-inf 9391  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-n0 12496  df-z 12583  df-uz 12854  df-q 12964  df-ioo 13367  df-icc 13370  df-rest 17465  df-topgen 17486  df-top 23012  df-topon 23029  df-bases 23064  df-cld 23137  df-cn 23345  df-tx 23680

This theorem is referenced by:  htpycc  25100  pcocn  25137  pcohtpylem  25139  pcopt  25142  pcopt2  25143  pcoass  25144  pcorevlem  25146