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Theorem ftc2ditglem 26107

Description: Lemma for ftc2ditg 26108. (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 488	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
2	1	ditgpos 25918	. 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 13433	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
6	3, 4, 5	syl2anc 593	. . . . . 6 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
7		ftc2ditg.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑋[,]𝑌))
8	6, 7	sseldd 3937	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9	8	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ)
10		ftc2ditg.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝑋[,]𝑌))
11	6, 10	sseldd 3937	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
12	11	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
13		ax-resscn 11130	. . . . . . . 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 24955	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝑋[,]𝑌)–cn→ℂ) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝑋[,]𝑌)⟶ℂ)
18	17	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐹:(𝑋[,]𝑌)⟶ℂ)
19	6	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑋[,]𝑌) ⊆ ℝ)
20		iccssre 13433	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
21	8, 11, 20	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
22	21	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
23		eqid 2762	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24		tgioo4 24865	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
25	23, 24	dvres 25973	. . . . . . 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 849	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
27		iccntr 24882	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
28	8, 11, 27	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
29	28	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
30	29	reseq2d 5965	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
31	26, 30	eqtrd 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
32	3	rexrd 11232	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
33		elicc2 13415	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
34	3, 4, 33	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
35	7, 34	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌))
36	35	simp2d 1156	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≤ 𝐴)
37		iooss1 13384	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
38	32, 36, 37	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
39	4	rexrd 11232	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
40		elicc2 13415	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
41	3, 4, 40	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
42	10, 41	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌))
43	42	simp3d 1157	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≤ 𝑌)
44		iooss2 13385	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
45	39, 43, 44	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
46	38, 45	sstrd 3946	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌))
47	46	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝑌))
48		ftc2ditg.c	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
49	48	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) ∈ ((𝑋(,)𝑌)–cn→ℂ))
50		rescncf 24959	. . . . . 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 2862	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
53		cncff 24955	. . . . . . . . . . 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 6935	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
56	55	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) = (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)))
57	56	reseq1d 5964	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)))
58	47	resmptd 6029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
59	57, 58	eqtrd 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
60	31, 59	eqtrd 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
61		ioombl 25627	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
62	61	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
63		fvexd 6882	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
64		ftc2ditg.i	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
65	64	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D 𝐹) ∈ 𝐿¹)
66	56, 65	eqeltrrd 2863	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
67	47, 62, 63, 66	iblss 25867	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
68	60, 67	eqeltrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
69		iccss2 13421	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑋[,]𝑌) ∧ 𝐵 ∈ (𝑋[,]𝑌)) → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
70	7, 10, 69	syl2anc 593	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
71		rescncf 24959	. . . . . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
74	9, 12, 1, 52, 68, 73	ftc2 26106	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
75	31	fveq1d 6869	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
76		fvres 6886	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
77	75, 76	sylan9eq 2817	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
78	77	itgeq2dv 25844	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
79	9	rexrd 11232	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
80	12	rexrd 11232	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
81		ubicc2 13469	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
82		lbicc2 13468	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83		fvres 6886	. . . . . 6 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
84		fvres 6886	. . . . . 6 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
85	83, 84	oveqan12d 7415	. . . . 5 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
86	81, 82, 85	syl2anc 593	. . . 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 1390	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
88	74, 78, 87	3eqtr3d 2805	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
89	2, 88	eqtrd 2797	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 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 Vcvv 3454 ⊆ wss 3904 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5647 ran crn 5648 ↾ cres 5649 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 ℝ^*cxr 11215 ≤ cle 11217 − cmin 11414 (,)cioo 13349 [,]cicc 13352 TopOpenctopn 17450 topGenctg 17466 ℂ_fldccnfld 21424 intcnt 23077 –cn→ccncf 24938 volcvol 25525 𝐿¹cibl 25679 ∫citg 25680 ⨜cdit 25908 D cdv 25925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cc 10392 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-symdif 4205 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-ofr 7661 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-dju 9859 df-card 9897 df-acn 9900 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-clim 15515 df-rlim 15516 df-sum 15714 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-fbas 21421 df-fg 21422 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cld 23079 df-ntr 23080 df-cls 23081 df-nei 23158 df-lp 23196 df-perf 23197 df-cn 23287 df-cnp 23288 df-haus 23375 df-cmp 23447 df-tx 23622 df-hmeo 23815 df-fil 23906 df-fm 23998 df-flim 23999 df-flf 24000 df-xms 24380 df-ms 24381 df-tms 24382 df-cncf 24940 df-ovol 25526 df-vol 25527 df-mbf 25681 df-itg1 25682 df-itg2 25683 df-ibl 25684 df-itg 25685 df-0p 25732 df-ditg 25909 df-limc 25928 df-dv 25929

This theorem is referenced by: ftc2ditg 26108

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