ftc2ditglem - Metamath Proof Explorer

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Theorem ftc2ditglem 26006

Description: Lemma for ftc2ditg 26007. (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 25811	. 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 13343	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ⊆ ℝ)
6	3, 4, 5	syl2anc 584	. . . . . 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 480	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ)
13		ax-resscn 11081	. . . . . . . 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 24840	. . . . . . . . 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 13343	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
21	8, 11, 20	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) ⊆ ℝ)
23		eqid 2734	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24		tgioo4 24747	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
25	23, 24	dvres 25866	. . . . . . 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 24764	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
28	8, 11, 27	syl2anc 584	. . . . . . . 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 5936	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
31	26, 30	eqtrd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
32	3	rexrd 11180	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
33		elicc2 13325	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐴 ∈ (𝑋[,]𝑌) ↔ (𝐴 ∈ ℝ ∧ 𝑋 ≤ 𝐴 ∧ 𝐴 ≤ 𝑌)))
34	3, 4, 33	syl2anc 584	. . . . . . . . . . 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 13294	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 ≤ 𝐴) → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
38	32, 36, 37	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝑋(,)𝐵))
39	4	rexrd 11180	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ ℝ^*)
40		elicc2 13325	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝐵 ∈ (𝑋[,]𝑌) ↔ (𝐵 ∈ ℝ ∧ 𝑋 ≤ 𝐵 ∧ 𝐵 ≤ 𝑌)))
41	3, 4, 40	syl2anc 584	. . . . . . . . . . 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 13295	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℝ^* ∧ 𝐵 ≤ 𝑌) → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
45	39, 43, 44	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑋(,)𝐵) ⊆ (𝑋(,)𝑌))
46	38, 45	sstrd 3942	. . . . . . 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 24844	. . . . . 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 2834	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
53		cncff 24840	. . . . . . . . . . 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 6900	. . . . . . . . 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 5935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)))
58	47	resmptd 5997	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
59	57, 58	eqtrd 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
60	31, 59	eqtrd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
61		ioombl 25520	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
62	61	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
63		fvexd 6847	. . . . . 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 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
67	47, 62, 63, 66	iblss 25760	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
68	60, 67	eqeltrd 2834	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
69		iccss2 13331	. . . . . . 7 ⊢ ((𝐴 ∈ (𝑋[,]𝑌) ∧ 𝐵 ∈ (𝑋[,]𝑌)) → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
70	7, 10, 69	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝑋[,]𝑌))
71		rescncf 24844	. . . . . 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 26005	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
75	31	fveq1d 6834	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
76		fvres 6851	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
77	75, 76	sylan9eq 2789	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≤ 𝐵) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
78	77	itgeq2dv 25737	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
79	9	rexrd 11180	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
80	12	rexrd 11180	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
81		ubicc2 13379	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
82		lbicc2 13378	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
83		fvres 6851	. . . . . 6 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
84		fvres 6851	. . . . . 6 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
85	83, 84	oveqan12d 7375	. . . . 5 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
86	81, 82, 85	syl2anc 584	. . . 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 1373	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
88	74, 78, 87	3eqtr3d 2777	. 2 ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
89	2, 88	eqtrd 2769	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 1086 = wceq 1541 ∈ wcel 2113 Vcvv 3438 ⊆ wss 3899 class class class wbr 5096 ↦ cmpt 5177 dom cdm 5622 ran crn 5623 ↾ cres 5624 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 ℝ^*cxr 11163 ≤ cle 11165 − cmin 11362 (,)cioo 13259 [,]cicc 13262 TopOpenctopn 17339 topGenctg 17355 ℂ_fldccnfld 21307 intcnt 22959 –cn→ccncf 24823 volcvol 25418 𝐿¹cibl 25572 ∫citg 25573 ⨜cdit 25801 D cdv 25818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cc 10343 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4203 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-disj 5064 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-acn 9852 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-rlim 15410 df-sum 15608 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-cmp 23329 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-ovol 25419 df-vol 25420 df-mbf 25574 df-itg1 25575 df-itg2 25576 df-ibl 25577 df-itg 25578 df-0p 25625 df-ditg 25802 df-limc 25821 df-dv 25822

This theorem is referenced by: ftc2ditg 26007

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