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Theorem ftc2ditglem 26106

Description: Lemma for ftc2ditg 26107. (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 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
2	1	ditgpos 25911	. 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 13489	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
6	3, 4, 5	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
7		ftc2ditg.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
8	6, 7	sseldd 4009	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
10		ftc2ditg.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
11	6, 10	sseldd 4009	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
13		ax-resscn 11241	. . . . . . . 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 24938	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝑋[,]𝑌)⟶ℂ)
18	17	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
19	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑋[,]𝑌) ⊆ ℝ)
20		iccssre 13489	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
21	8, 11, 20	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
23		eqid 2740	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24	23	tgioo2 24844	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
25	23, 24	dvres 25966	. . . . . . 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 838	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
27		iccntr 24862	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
28	8, 11, 27	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
30	29	reseq2d 6009	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
31	26, 30	eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
32	3	rexrd 11340	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
33		elicc2 13472	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
34	3, 4, 33	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
35	7, 34	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))
36	35	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴)
37		iooss1 13442	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
38	32, 36, 37	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
39	4	rexrd 11340	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
40		elicc2 13472	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
41	3, 4, 40	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
42	10, 41	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))
43	42	simp3d 1144	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌)
44		iooss2 13443	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
45	39, 43, 44	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
46	38, 45	sstrd 4019	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌))
47	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌))
48		ftc2ditg.c	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
49	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
50		rescncf 24942	. . . . . 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 65	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
52	31, 51	eqeltrd 2844	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
53		cncff 24938	. . . . . . . . . . 11 ⊢ ((ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ) → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
54	48, 53	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐹):(𝑋(,)𝑌)⟶ℂ)
55	54	feqmptd 6990	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
57	56	reseq1d 6008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)))
58	47	resmptd 6069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
59	57, 58	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
60	31, 59	eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
61		ioombl 25619	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
62	61	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
63		fvexd 6935	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
64		ftc2ditg.i	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
65	64	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) ∈ 𝐿¹)
66	56, 65	eqeltrrd 2845	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
67	47, 62, 63, 66	iblss 25860	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
68	60, 67	eqeltrd 2844	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
69		iccss2 13478	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑋[,]𝑌) ∧ 𝐵 ∈ (𝑋[,]𝑌)) → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
70	7, 10, 69	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
71		rescncf 24942	. . . . . 6 ⊢ ((𝐴[,]𝐵) ⊆ (𝑋[,]𝑌) → (𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
72	70, 15, 71	sylc 65	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
73	72	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
74	9, 12, 1, 52, 68, 73	ftc2 26105	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
75	31	fveq1d 6922	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
76		fvres 6939	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
77	75, 76	sylan9eq 2800	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
78	77	itgeq2dv 25837	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
79	9	rexrd 11340	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
80	12	rexrd 11340	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
81		ubicc2 13525	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
82		lbicc2 13524	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83		fvres 6939	. . . . . 6 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
84		fvres 6939	. . . . . 6 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
85	83, 84	oveqan12d 7467	. . . . 5 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
86	81, 82, 85	syl2anc 583	. . . 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 1371	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
88	74, 78, 87	3eqtr3d 2788	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
89	2, 88	eqtrd 2780	1 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ⨜[𝐴 → 𝐵]((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 ↾ cres 5702 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 ℝ^*cxr 11323 ≤ cle 11325 − cmin 11520 (,)cioo 13407 [,]cicc 13410 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 intcnt 23046 –cn→ccncf 24921 volcvol 25517 𝐿¹cibl 25671 ∫citg 25672 ⨜cdit 25901 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-symdif 4272 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-ditg 25902 df-limc 25921 df-dv 25922

This theorem is referenced by: ftc2ditg 26107

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