Theorem cnmpopc 23825

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 2737	. 2 ⊢ ∪ (𝑂 ×t 𝐽) = ∪ (𝑂 ×t 𝐽)
2		eqid 2737	. 2 ⊢ ∪ 𝐾 = ∪ 𝐾
3		cnmpopc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		cnmpopc.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
5		iccssre 13017	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴[,]𝐶) ⊆ ℝ)
6	3, 4, 5	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐶) ⊆ ℝ)
7		cnmpopc.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))
8	6, 7	sseldd 3902	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		icccld 23664	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
10	3, 8, 9	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
11		cnmpopc.r	. . . . . . 7 ⊢ 𝑅 = (topGen‘ran (,))
12	11	fveq2i 6720	. . . . . 6 ⊢ (Clsd‘𝑅) = (Clsd‘(topGen‘ran (,)))
13	10, 12	eleqtrrdi 2849	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑅))
14		ssun1 4086	. . . . . 6 ⊢ (𝐴[,]𝐵) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
15		iccsplit 13073	. . . . . . 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 3954	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶))
18		uniretop 23660	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
19	11	unieqi 4832	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ (topGen‘ran (,))
20	18, 19	eqtr4i 2768	. . . . . 6 ⊢ ℝ = ∪ 𝑅
21	20	restcldi 22070	. . . . 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 6720	. . . 4 ⊢ (Clsd‘𝑂) = (Clsd‘(𝑅 ↾t (𝐴[,]𝐶)))
25	22, 24	eleqtrrdi 2849	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑂))
26		cnmpopc.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
27		toponuni 21811	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
28	26, 27	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29		topontop 21810	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
30		eqid 2737	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
31	30	topcld 21932	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (Clsd‘𝐽))
32	26, 29, 31	3syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝐽 ∈ (Clsd‘𝐽))
33	28, 32	eqeltrd 2838	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Clsd‘𝐽))
34		txcld 22500	. . 3 ⊢ (((𝐴[,]𝐵) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
35	25, 33, 34	syl2anc 587	. 2 ⊢ (𝜑 → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
36		icccld 23664	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
37	8, 4, 36	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
38	37, 12	eleqtrrdi 2849	. . . . 5 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑅))
39		ssun2 4087	. . . . . 6 ⊢ (𝐵[,]𝐶) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
40	39, 16	sseqtrrid 3954	. . . . 5 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶))
41	20	restcldi 22070	. . . . 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 2849	. . 3 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑂))
44		txcld 22500	. . 3 ⊢ (((𝐵[,]𝐶) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
45	43, 33, 44	syl2anc 587	. 2 ⊢ (𝜑 → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
46	16	xpeq1d 5580	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋))
47		xpundir 5618	. . . 4 ⊢ (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋))
48	46, 47	eqtrdi 2794	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)))
49		retopon 23661	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
50	11, 49	eqeltri 2834	. . . . . . 7 ⊢ 𝑅 ∈ (TopOn‘ℝ)
51		resttopon 22058	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
52	50, 6, 51	sylancr 590	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
53	23, 52	eqeltrid 2842	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ (TopOn‘(𝐴[,]𝐶)))
54		txtopon 22488	. . . . 5 ⊢ ((𝑂 ∈ (TopOn‘(𝐴[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
55	53, 26, 54	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
56		toponuni 21811	. . . 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 2779	. 2 ⊢ (𝜑 → (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) = ∪ (𝑂 ×t 𝐽))
59		cnmpopc.m	. . . . . . . . . 10 ⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵))
60	17, 6	sstrd 3911	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
61		resttopon 22058	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
62	50, 60, 61	sylancr 590	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
63	59, 62	eqeltrid 2842	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘(𝐴[,]𝐵)))
64		txtopon 22488	. . . . . . . . 9 ⊢ ((𝑀 ∈ (TopOn‘(𝐴[,]𝐵)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)))
65	63, 26, 64	syl2anc 587	. . . . . . . 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 22138	. . . . . . . . . 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 21815	. . . . . . . . 9 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
70	68, 69	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
71		elicc2 13000	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
72	3, 8, 71	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
73	72	biimpa 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
74	73	simp3d 1146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
75	74	3adant3 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → 𝑥 ≤ 𝐵)
76	75	iftrued 4447	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐷)
77	76	mpoeq3dva 7288	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷))
78	77, 66	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
79		cnf2 22146	. . . . . . . 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 2737	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
82	81	fmpo 7838	. . . . . . 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 3911	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
86		resttopon 22058	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐵[,]𝐶) ⊆ ℝ) → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
87	50, 85, 86	sylancr 590	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
88	84, 87	eqeltrid 2842	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘(𝐵[,]𝐶)))
89		txtopon 22488	. . . . . . . . 9 ⊢ ((𝑁 ∈ (TopOn‘(𝐵[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
90	88, 26, 89	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
91		elicc2 13000	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)))
92	8, 4, 91	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶)))
93	92	biimpa 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐶))
94	93	simp2d 1145	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ≤ 𝑥)
95	94	biantrud 535	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ≤ 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥)))
96	93	simp1d 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ∈ ℝ)
97	8	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ∈ ℝ)
98	96, 97	letri3d 10974	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 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 4458	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = if(𝑥 ≤ 𝐵, 𝐸, 𝐸))
104		ifid 4479	. . . . . . . . . . . . . . 15 ⊢ if(𝑥 ≤ 𝐵, 𝐸, 𝐸) = 𝐸
105	103, 104	eqtrdi 2794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑋 ∧ 𝑥 = 𝐵)) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
106	105	expr 460	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 = 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸))
107	106	3adant2 1133	. . . . . . . . . . . 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 4448	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ≤ 𝐵 → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
110	108, 109	pm2.61d1 183	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦 ∈ 𝑋) → if(𝑥 ≤ 𝐵, 𝐷, 𝐸) = 𝐸)
111	110	mpoeq3dva 7288	. . . . . . . . 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 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
114		cnf2 22146	. . . . . . . 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 2737	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
117	116	fmpo 7838	. . . . . . 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 4106	. . . . . 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 587	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
121	16	raleqdv 3325	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾 ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾))
122	120, 121	mpbird 260	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦 ∈ 𝑋 if(𝑥 ≤ 𝐵, 𝐷, 𝐸) ∈ ∪ 𝐾)
123		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸))
124	123	fmpo 7838	. . . 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)
125	122, 124	sylib 221	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶∪ 𝐾)
126	57	feq2d 6531	. . 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))
127	125, 126	mpbid 235	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)):∪ (𝑂 ×t 𝐽)⟶∪ 𝐾)
128		ssid 3923	. . . 4 ⊢ 𝑋 ⊆ 𝑋
129		resmpo 7330	. . . 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)))
130	17, 128, 129	sylancl 589	. . 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)))
131		retop 23659	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
132	11, 131	eqeltri 2834	. . . . . . . . 9 ⊢ 𝑅 ∈ Top
133		ovex 7246	. . . . . . . . 9 ⊢ (𝐴[,]𝐶) ∈ V
134		resttop 22057	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐶) ∈ V) → (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top)
135	132, 133, 134	mp2an 692	. . . . . . . 8 ⊢ (𝑅 ↾t (𝐴[,]𝐶)) ∈ Top
136	23, 135	eqeltri 2834	. . . . . . 7 ⊢ 𝑂 ∈ Top
137	136	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ Top)
138		ovexd 7248	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ∈ V)
139		txrest 22528	. . . . . 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)))
140	137, 26, 138, 33, 139	syl22anc 839	. . . . 5 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)))
141	132	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Top)
142		ovexd 7248	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐶) ∈ V)
143		restabs 22062	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵)))
144	141, 17, 142, 143	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅 ↾t (𝐴[,]𝐵)))
145	23	oveq1i 7223	. . . . . . 7 ⊢ (𝑂 ↾t (𝐴[,]𝐵)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵))
146	144, 145, 59	3eqtr4g 2803	. . . . . 6 ⊢ (𝜑 → (𝑂 ↾t (𝐴[,]𝐵)) = 𝑀)
147	28	oveq2d 7229	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t 𝑋) = (𝐽 ↾t ∪ 𝐽))
148	30	restid 16938	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ↾t ∪ 𝐽) = 𝐽)
149	26, 148	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t ∪ 𝐽) = 𝐽)
150	147, 149	eqtrd 2777	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾t 𝑋) = 𝐽)
151	146, 150	oveq12d 7231	. . . . 5 ⊢ (𝜑 → ((𝑂 ↾t (𝐴[,]𝐵)) ×t (𝐽 ↾t 𝑋)) = (𝑀 ×t 𝐽))
152	140, 151	eqtrd 2777	. . . 4 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = (𝑀 ×t 𝐽))
153	152	oveq1d 7228	. . 3 ⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾) = ((𝑀 ×t 𝐽) Cn 𝐾))
154	78, 130, 153	3eltr4d 2853	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾))
155		resmpo 7330	. . . 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)))
156	40, 128, 155	sylancl 589	. . 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)))
157		ovexd 7248	. . . . . 6 ⊢ (𝜑 → (𝐵[,]𝐶) ∈ V)
158		txrest 22528	. . . . . 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)))
159	137, 26, 157, 33, 158	syl22anc 839	. . . . 5 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)))
160		restabs 22062	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶)))
161	141, 40, 142, 160	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅 ↾t (𝐵[,]𝐶)))
162	23	oveq1i 7223	. . . . . . 7 ⊢ (𝑂 ↾t (𝐵[,]𝐶)) = ((𝑅 ↾t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶))
163	161, 162, 84	3eqtr4g 2803	. . . . . 6 ⊢ (𝜑 → (𝑂 ↾t (𝐵[,]𝐶)) = 𝑁)
164	163, 150	oveq12d 7231	. . . . 5 ⊢ (𝜑 → ((𝑂 ↾t (𝐵[,]𝐶)) ×t (𝐽 ↾t 𝑋)) = (𝑁 ×t 𝐽))
165	159, 164	eqtrd 2777	. . . 4 ⊢ (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = (𝑁 ×t 𝐽))
166	165	oveq1d 7228	. . 3 ⊢ (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾) = ((𝑁 ×t 𝐽) Cn 𝐾))
167	113, 156, 166	3eltr4d 2853	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾))
168	1, 2, 35, 45, 58, 127, 154, 167	paste 22191	1 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408   ∪ cun 3864   ⊆ wss 3866  ifcif 4439  ∪ cuni 4819   class class class wbr 5053   × cxp 5549  ran crn 5552   ↾ cres 5553  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  ℝcr 10728   ≤ cle 10868  (,)cioo 12935  [,]cicc 12938   ↾_t crest 16925  topGenctg 16942  Topctop 21790  TopOnctopon 21807  Clsdccld 21913   Cn ccn 22121   ×_t ctx 22457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-ioo 12939  df-icc 12942  df-rest 16927  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-cld 21916  df-cn 22124  df-tx 22459

This theorem is referenced by:  htpycc  23877  pcocn  23914  pcohtpylem  23916  pcopt  23919  pcopt2  23920  pcoass  23921  pcorevlem  23923