ftc2ditglem - Metamath Proof Explorer

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Theorem ftc2ditglem 26365

Description: Lemma for ftc2ditg 26366. (Contributed by Mario Carneiro, 3-Sep-2014.)

**Hypotheses**
Ref	Expression
ftc2ditg.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
ftc2ditg.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
ftc2ditg.a	⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
ftc2ditg.b	⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
ftc2ditg.c	⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
ftc2ditg.i	⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
ftc2ditg.f	⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))

**Assertion**
Ref	Expression
ftc2ditglem	⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Distinct variable groups: 𝑡,𝐴 𝑡,𝐵 𝑡,𝐹 𝜑,𝑡 𝑡,𝑋 𝑡,𝑌

**Proof of Theorem ftc2ditglem**
Step	Hyp	Ref	Expression
1		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
2	1	ditgpos 26176	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
3		ftc2ditg.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
4		ftc2ditg.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ)
5		iccssre 13560	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
6	3, 4, 5	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
7		ftc2ditg.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
8	6, 7	sseldd 3932	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
10		ftc2ditg.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
11	6, 10	sseldd 3932	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
12	11	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
13		ax-resscn 11257	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
14	13	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ℝ ⊆ ℂ)
15		ftc2ditg.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ))
16		cncff 25214	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
17	15, 16	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝑋[,]𝑌)⟶ℂ)
18	17	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
19	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑋[,]𝑌) ⊆ ℝ)
20		iccssre 13560	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
21	8, 11, 20	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
22	21	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
23		eqid 2761	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24		tgioo4 25124	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
25	23, 24	dvres 26231	. . . . . . 7 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:(𝑋[,]𝑌)⟶ℂ) ∧ ((𝑋[,]𝑌) ⊆ ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
26	14, 18, 19, 22, 25	syl22anc 852	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
27		iccntr 25141	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
28	8, 11, 27	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
29	28	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
30	29	reseq2d 5970	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
31	26, 30	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
32	3	rexrd 11359	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
33		elicc2 13542	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
34	3, 4, 33	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
35	7, 34	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))
36	35	simp2d 1161	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴)
37		iooss1 13511	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
38	32, 36, 37	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
39	4	rexrd 11359	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
40		elicc2 13542	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
41	3, 4, 40	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
42	10, 41	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))
43	42	simp3d 1162	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌)
44		iooss2 13512	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
45	39, 43, 44	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
46	38, 45	sstrd 3941	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌))
47	46	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌))
48		ftc2ditg.c	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
49	48	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
50		rescncf 25218	. . . . . 6 ⊢ ((𝐴(,)𝐵) ⊆ (𝑋(,)𝑌) → ((ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ)))
51	47, 49, 50	sylc 66	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
52	31, 51	eqeltrd 2861	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
53		cncff 25214	. . . . . . . . . . 11 ⊢ ((ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ) → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
54	48, 53	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
55	54	feqmptd 6953	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
56	55	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
57	56	reseq1d 5969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)))
58	47	resmptd 6032	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
59	57, 58	eqtrd 2796	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
60	31, 59	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
61		ioombl 25886	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
62	61	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
63		fvexd 6900	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
64		ftc2ditg.i	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
65	64	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) ∈ 𝐿¹)
66	56, 65	eqeltrrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
67	47, 62, 63, 66	iblss 26125	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
68	60, 67	eqeltrd 2861	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
69		iccss2 13548	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑋[,]𝑌) ∧ 𝐵 ∈ (𝑋[,]𝑌)) → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
70	7, 10, 69	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
71		rescncf 25218	. . . . . 6 ⊢ ((𝐴[,]𝐵) ⊆ (𝑋[,]𝑌) → (𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
72	70, 15, 71	sylc 66	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
73	72	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
74	9, 12, 1, 52, 68, 73	ftc2 26364	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
75	31	fveq1d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
76		fvres 6904	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
77	75, 76	sylan9eq 2816	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
78	77	itgeq2dv 26102	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
79	9	rexrd 11359	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
80	12	rexrd 11359	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
81		ubicc2 13596	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
82		lbicc2 13595	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83		fvres 6904	. . . . . 6 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
84		fvres 6904	. . . . . 6 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
85	83, 84	oveqan12d 7439	. . . . 5 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
86	81, 82, 85	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
87	79, 80, 1, 86	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
88	74, 78, 87	3eqtr3d 2804	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
89	2, 88	eqtrd 2796	1 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 ran crn 5652 ↾ cres 5653 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 ℝ^*cxr 11342 ≤ cle 11344 − cmin 11541 (,)cioo 13476 [,]cicc 13479 TopOpenctopn 17592 topGenctg 17608 ℂ_fldccnfld 21678 intcnt 23335 –cn→ccncf 25197 volcvol 25784 𝐿¹cibl 25938 ∫citg 25939 ⨜cdit 26166 D cdv 26183

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cc 10513 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-omul 8481 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-ovol 25785 df-vol 25786 df-mbf 25940 df-itg1 25941 df-itg2 25942 df-ibl 25943 df-itg 25944 df-0p 25991 df-ditg 26167 df-limc 26186 df-dv 26187

This theorem is used by: ftc2ditg 26366

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