ftc1cnnclem - Mathbox for Brendan Leahy

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Theorem ftc1cnnclem 37049

Description: Lemma for ftc1cnnc 37050; cf. ftc1lem4 25896. The stronger assumptions of ftc1cn 25900 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.)

**Hypotheses**
Ref	Expression
ftc1cnnc.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
ftc1cnnc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc1cnnc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc1cnnc.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc1cnnc.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
ftc1cnnc.i	⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
ftc1cnnclem.c	⊢ (𝜑 → 𝑐 ∈ (𝐴(,)𝐵))
ftc1cnnclem.h	⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
ftc1cnnclem.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ftc1cnnclem.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
ftc1cnnclem.fc	⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸))
ftc1cnnclem.x1	⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))
ftc1cnnclem.x2	⊢ (𝜑 → (abs‘(𝑋 − 𝑐)) < 𝑅)
ftc1cnnclem.y1	⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))
ftc1cnnclem.y2	⊢ (𝜑 → (abs‘(𝑌 − 𝑐)) < 𝑅)

**Assertion**
Ref	Expression
ftc1cnnclem	⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐))) < 𝐸)

Distinct variable groups: 𝑥,𝑦,𝑧,𝑡,𝐴 𝑥,𝐵,𝑦,𝑧,𝑡 𝑥,𝐹,𝑦,𝑧,𝑡 𝜑,𝑥,𝑦,𝑧,𝑡 𝑦,𝐺,𝑧 𝑥,𝑐,𝑦,𝑧,𝑡 𝑥,𝑋,𝑧,𝑡 𝑦,𝐸,𝑡 𝑦,𝐻 𝑥,𝑌,𝑡 𝑦,𝑅 Allowed substitution hints: 𝜑(𝑐) 𝐴(𝑐) 𝐵(𝑐) 𝑅(𝑥,𝑧,𝑡,𝑐) 𝐸(𝑥,𝑧,𝑐) 𝐹(𝑐) 𝐺(𝑥,𝑡,𝑐) 𝐻(𝑥,𝑧,𝑡,𝑐) 𝑋(𝑦,𝑐) 𝑌(𝑦,𝑧,𝑐)

**Proof of Theorem ftc1cnnclem**
Step	Hyp	Ref	Expression
1		ovexd 7436	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((𝐹‘𝑡) − (𝐹‘𝑐)) ∈ V)
2		ftc1cnnc.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	rexrd 11261	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		ftc1cnnc.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	4	rexrd 11261	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
6		ftc1cnnclem.x1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))
7		elicc1 13365	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ^ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵)))
8	7	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑋 ∈ (𝐴[,]𝐵)) → (𝑋 ∈ ℝ^ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))
9	8	simp2d 1140	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑋 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑋)
10	3, 5, 6, 9	syl21anc 835	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 𝑋)
11		ftc1cnnclem.y1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))
12		iccleub 13376	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑌 ∈ (𝐴[,]𝐵)) → 𝑌 ≤ 𝐵)
13	3, 5, 11, 12	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ≤ 𝐵)
14		ioossioo 13415	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝑋 ∧ 𝑌 ≤ 𝐵)) → (𝑋(,)𝑌) ⊆ (𝐴(,)𝐵))
15	3, 5, 10, 13, 14	syl22anc 836	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(,)𝑌) ⊆ (𝐴(,)𝐵))
16	15	sselda 3974	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 ∈ (𝐴(,)𝐵))
17		ftc1cnnc.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
18		cncff 24735	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
20	19	ffvelcdmda 7076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ℂ)
21	16, 20	syldan 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐹‘𝑡) ∈ ℂ)
22		ioombl 25416	. . . . . . . . . . 11 ⊢ (𝑋(,)𝑌) ∈ dom vol
23	22	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋(,)𝑌) ∈ dom vol)
24		fvexd 6896	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ V)
25	19	feqmptd 6950	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑡)))
26		ftc1cnnc.i	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
27	25, 26	eqeltrrd 2826	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑡)) ∈ 𝐿¹)
28	15, 23, 24, 27	iblss 25656	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑡)) ∈ 𝐿¹)
29		ftc1cnnclem.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑐 ∈ (𝐴(,)𝐵))
30	19, 29	ffvelcdmd 7077	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑐) ∈ ℂ)
31	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐹‘𝑐) ∈ ℂ)
32		fconstmpt 5728	. . . . . . . . . 10 ⊢ ((𝑋(,)𝑌) × {(𝐹‘𝑐)}) = (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑐))
33		mblvol 25381	. . . . . . . . . . . . 13 ⊢ ((𝑋(,)𝑌) ∈ dom vol → (vol‘(𝑋(,)𝑌)) = (vol*‘(𝑋(,)𝑌)))
34	22, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ (vol‘(𝑋(,)𝑌)) = (vol*‘(𝑋(,)𝑌))
35		ioossicc 13407	. . . . . . . . . . . . . 14 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
36	35	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌))
37		iccssre 13403	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
38	2, 4, 37	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
39	38, 6	sseldd 3975	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℝ)
40	38, 11	sseldd 3975	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ ℝ)
41		iccmbl 25417	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ∈ dom vol)
42	39, 40, 41	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋[,]𝑌) ∈ dom vol)
43		mblss 25382	. . . . . . . . . . . . . 14 ⊢ ((𝑋[,]𝑌) ∈ dom vol → (𝑋[,]𝑌) ⊆ ℝ)
44	42, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
45		mblvol 25381	. . . . . . . . . . . . . . 15 ⊢ ((𝑋[,]𝑌) ∈ dom vol → (vol‘(𝑋[,]𝑌)) = (vol*‘(𝑋[,]𝑌)))
46	42, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (vol‘(𝑋[,]𝑌)) = (vol*‘(𝑋[,]𝑌)))
47		iccvolcl 25418	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (vol‘(𝑋[,]𝑌)) ∈ ℝ)
48	39, 40, 47	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (vol‘(𝑋[,]𝑌)) ∈ ℝ)
49	46, 48	eqeltrrd 2826	. . . . . . . . . . . . 13 ⊢ (𝜑 → (vol*‘(𝑋[,]𝑌)) ∈ ℝ)
50		ovolsscl 25337	. . . . . . . . . . . . 13 ⊢ (((𝑋(,)𝑌) ⊆ (𝑋[,]𝑌) ∧ (𝑋[,]𝑌) ⊆ ℝ ∧ (vol‘(𝑋[,]𝑌)) ∈ ℝ) → (vol‘(𝑋(,)𝑌)) ∈ ℝ)
51	36, 44, 49, 50	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (𝜑 → (vol*‘(𝑋(,)𝑌)) ∈ ℝ)
52	34, 51	eqeltrid 2829	. . . . . . . . . . 11 ⊢ (𝜑 → (vol‘(𝑋(,)𝑌)) ∈ ℝ)
53		iblconst 25669	. . . . . . . . . . 11 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ (𝐹‘𝑐) ∈ ℂ) → ((𝑋(,)𝑌) × {(𝐹‘𝑐)}) ∈ 𝐿¹)
54	23, 52, 30, 53	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋(,)𝑌) × {(𝐹‘𝑐)}) ∈ 𝐿¹)
55	32, 54	eqeltrrid 2830	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑐)) ∈ 𝐿¹)
56		eqid 2724	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
57	56	subcn 24704	. . . . . . . . . . . 12 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
58	57	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
59	19, 15	feqresmpt 6951	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)𝑌)) = (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑡)))
60		rescncf 24739	. . . . . . . . . . . . 13 ⊢ ((𝑋(,)𝑌) ⊆ (𝐴(,)𝐵) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → (𝐹 ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ)))
61	15, 17, 60	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
62	59, 61	eqeltrrd 2826	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑡)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
63		ioossre 13382	. . . . . . . . . . . . . 14 ⊢ (𝑋(,)𝑌) ⊆ ℝ
64		ax-resscn 11163	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
65	63, 64	sstri 3983	. . . . . . . . . . . . 13 ⊢ (𝑋(,)𝑌) ⊆ ℂ
66		ssid 3996	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
67		cncfmptc 24754	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑐) ∈ ℂ ∧ (𝑋(,)𝑌) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑐)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
68	65, 66, 67	mp3an23 1449	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑐) ∈ ℂ → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑐)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
69	30, 68	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑐)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
70	56, 58, 62, 69	cncfmpt2f 24757	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
71		cnmbf 25510	. . . . . . . . . 10 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (𝑡 ∈ (𝑋(,)𝑌) ↦ ((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ ((𝑋(,)𝑌)–cn→ℂ)) → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ MblFn)
72	22, 70, 71	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ MblFn)
73	21, 28, 31, 55, 72	iblsubnc 37039	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ 𝐿¹)
74	1, 73	itgcl 25635	. . . . . . 7 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 ∈ ℂ)
75	74	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 ∈ ℂ)
76	40, 39	resubcld 11639	. . . . . . . 8 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℝ)
77	76	recnd 11239	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℂ)
78	77	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℂ)
79	39, 40	posdifd 11798	. . . . . . . 8 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 0 < (𝑌 − 𝑋)))
80	79	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (𝑌 − 𝑋))
81	80	gt0ne0d 11775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ≠ 0)
82	75, 78, 81	divcld 11987	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋)) ∈ ℂ)
83	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝐹‘𝑐) ∈ ℂ)
84		ltle 11299	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌))
85	39, 40, 84	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌))
86	85	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌)
87		ftc1cnnc.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
88		ftc1cnnc.le	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
89		ssidd 3997	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
90		ioossre 13382	. . . . . . . . . . 11 ⊢ (𝐴(,)𝐵) ⊆ ℝ
91	90	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
92	87, 2, 4, 88, 89, 91, 26, 19, 6, 11	ftc1lem1 25892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡)
93	86, 92	syldan 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡)
94	21, 31	npcand 11572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (((𝐹‘𝑡) − (𝐹‘𝑐)) + (𝐹‘𝑐)) = (𝐹‘𝑡))
95	94	itgeq2dv 25633	. . . . . . . . . 10 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(((𝐹‘𝑡) − (𝐹‘𝑐)) + (𝐹‘𝑐)) d𝑡 = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡)
96	21, 31	subcld 11568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((𝐹‘𝑡) − (𝐹‘𝑐)) ∈ ℂ)
97	94	mpteq2dva 5238	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (((𝐹‘𝑡) − (𝐹‘𝑐)) + (𝐹‘𝑐))) = (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑡)))
98	97, 59	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (((𝐹‘𝑡) − (𝐹‘𝑐)) + (𝐹‘𝑐))) = (𝐹 ↾ (𝑋(,)𝑌)))
99		iblmbf 25619	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ 𝐿¹ → 𝐹 ∈ MblFn)
100	26, 99	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ MblFn)
101		mbfres 25495	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑋(,)𝑌) ∈ dom vol) → (𝐹 ↾ (𝑋(,)𝑌)) ∈ MblFn)
102	100, 22, 101	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)𝑌)) ∈ MblFn)
103	98, 102	eqeltrd 2825	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (((𝐹‘𝑡) − (𝐹‘𝑐)) + (𝐹‘𝑐))) ∈ MblFn)
104	96, 73, 31, 55, 103	itgaddnc 37038	. . . . . . . . . 10 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(((𝐹‘𝑡) − (𝐹‘𝑐)) + (𝐹‘𝑐)) d𝑡 = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡))
105	95, 104	eqtr3d 2766	. . . . . . . . 9 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡 = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡))
106	105	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡 = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡))
107		itgconst 25670	. . . . . . . . . . . 12 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ (𝐹‘𝑐) ∈ ℂ) → ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡 = ((𝐹‘𝑐) · (vol‘(𝑋(,)𝑌))))
108	23, 52, 30, 107	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡 = ((𝐹‘𝑐) · (vol‘(𝑋(,)𝑌))))
109	108	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡 = ((𝐹‘𝑐) · (vol‘(𝑋(,)𝑌))))
110	39	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝑋 ∈ ℝ)
111	40	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝑌 ∈ ℝ)
112		ovolioo 25419	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ ∧ 𝑋 ≤ 𝑌) → (vol*‘(𝑋(,)𝑌)) = (𝑌 − 𝑋))
113	110, 111, 86, 112	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (vol*‘(𝑋(,)𝑌)) = (𝑌 − 𝑋))
114	34, 113	eqtrid 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (vol‘(𝑋(,)𝑌)) = (𝑌 − 𝑋))
115	114	oveq2d 7417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝑐) · (vol‘(𝑋(,)𝑌))) = ((𝐹‘𝑐) · (𝑌 − 𝑋)))
116	109, 115	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡 = ((𝐹‘𝑐) · (𝑌 − 𝑋)))
117	116	oveq2d 7417	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ∫(𝑋(,)𝑌)(𝐹‘𝑐) d𝑡) = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ((𝐹‘𝑐) · (𝑌 − 𝑋))))
118	93, 106, 117	3eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ((𝐹‘𝑐) · (𝑌 − 𝑋))))
119	118	oveq1d 7416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) = ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ((𝐹‘𝑐) · (𝑌 − 𝑋))) / (𝑌 − 𝑋)))
120	83, 78	mulcld 11231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝑐) · (𝑌 − 𝑋)) ∈ ℂ)
121	75, 120, 78, 81	divdird 12025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 + ((𝐹‘𝑐) · (𝑌 − 𝑋))) / (𝑌 − 𝑋)) = ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋)) + (((𝐹‘𝑐) · (𝑌 − 𝑋)) / (𝑌 − 𝑋))))
122	83, 78, 81	divcan4d 11993	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐹‘𝑐) · (𝑌 − 𝑋)) / (𝑌 − 𝑋)) = (𝐹‘𝑐))
123	122	oveq2d 7417	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋)) + (((𝐹‘𝑐) · (𝑌 − 𝑋)) / (𝑌 − 𝑋))) = ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋)) + (𝐹‘𝑐)))
124	119, 121, 123	3eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) = ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋)) + (𝐹‘𝑐)))
125	82, 83, 124	mvrraddd 11623	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐)) = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋)))
126	125	fveq2d 6885	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐))) = (abs‘(∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋))))
127	75, 78, 81	absdivd 15399	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘(∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡 / (𝑌 − 𝑋))) = ((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) / (abs‘(𝑌 − 𝑋))))
128	76	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℝ)
129		0re 11213	. . . . . . 7 ⊢ 0 ∈ ℝ
130		ltle 11299	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝑌 − 𝑋) ∈ ℝ) → (0 < (𝑌 − 𝑋) → 0 ≤ (𝑌 − 𝑋)))
131	129, 128, 130	sylancr 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (0 < (𝑌 − 𝑋) → 0 ≤ (𝑌 − 𝑋)))
132	80, 131	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ (𝑌 − 𝑋))
133	128, 132	absidd 15366	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘(𝑌 − 𝑋)) = (𝑌 − 𝑋))
134	133	oveq2d 7417	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) / (abs‘(𝑌 − 𝑋))) = ((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) / (𝑌 − 𝑋)))
135	126, 127, 134	3eqtrd 2768	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐))) = ((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) / (𝑌 − 𝑋)))
136	75	abscld 15380	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) ∈ ℝ)
137	96	abscld 15380	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ ℝ)
138		cncfss 24741	. . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
139	64, 66, 138	mp2an 689	. . . . . . . . . . 11 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
140		abscncf 24743	. . . . . . . . . . 11 ⊢ abs ∈ (ℂ–cn→ℝ)
141	139, 140	sselii 3971	. . . . . . . . . 10 ⊢ abs ∈ (ℂ–cn→ℂ)
142	141	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℂ))
143	142, 70	cncfmpt1f 24756	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
144		cnmbf 25510	. . . . . . . 8 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (𝑡 ∈ (𝑋(,)𝑌) ↦ (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ((𝑋(,)𝑌)–cn→ℂ)) → (𝑡 ∈ (𝑋(,)𝑌) ↦ (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ MblFn)
145	22, 143, 144	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ MblFn)
146	1, 73, 145	iblabsnc 37042	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ 𝐿¹)
147	137, 146	itgrecl 25649	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡 ∈ ℝ)
148	147	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡 ∈ ℝ)
149		ftc1cnnclem.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
150	149	rpred 13013	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ)
151	76, 150	remulcld 11241	. . . . 5 ⊢ (𝜑 → ((𝑌 − 𝑋) · 𝐸) ∈ ℝ)
152	151	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝑌 − 𝑋) · 𝐸) ∈ ℝ)
153	74	cjcld 15140	. . . . . . . . 9 ⊢ (𝜑 → (∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) ∈ ℂ)
154		cncfmptc 24754	. . . . . . . . . 10 ⊢ (((∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) ∈ ℂ ∧ (𝑋(,)𝑌) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (𝑋(,)𝑌) ↦ (∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
155	65, 66, 154	mp3an23 1449	. . . . . . . . 9 ⊢ ((∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) ∈ ℂ → (𝑥 ∈ (𝑋(,)𝑌) ↦ (∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
156	153, 155	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ (∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡)) ∈ ((𝑋(,)𝑌)–cn→ℂ))
157		nfcv 2895	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝐹‘𝑡) − (𝐹‘𝑐))
158		nfcsb1v 3910	. . . . . . . . . 10 ⊢ Ⅎ𝑡⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐))
159		csbeq1a 3899	. . . . . . . . . 10 ⊢ (𝑡 = 𝑥 → ((𝐹‘𝑡) − (𝐹‘𝑐)) = ⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐)))
160	157, 158, 159	cbvmpt 5249	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋(,)𝑌) ↦ ((𝐹‘𝑡) − (𝐹‘𝑐))) = (𝑥 ∈ (𝑋(,)𝑌) ↦ ⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐)))
161	160, 70	eqeltrrid 2830	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
162	156, 161	mulcncf 25296	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) · ⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
163		cnmbf 25510	. . . . . . 7 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (𝑥 ∈ (𝑋(,)𝑌) ↦ ((∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) · ⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ((𝑋(,)𝑌)–cn→ℂ)) → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) · ⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ MblFn)
164	22, 162, 163	sylancr 586	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ ((∗‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) · ⦋𝑥 / 𝑡⦌((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ MblFn)
165	96, 73, 145, 164	itgabsnc 37047	. . . . 5 ⊢ (𝜑 → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) ≤ ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡)
166	165	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) ≤ ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡)
167		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
168	150	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℝ)
169		fconstmpt 5728	. . . . . . . . . 10 ⊢ ((𝑋(,)𝑌) × {𝐸}) = (𝑡 ∈ (𝑋(,)𝑌) ↦ 𝐸)
170	149	rpcnd 13015	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℂ)
171		iblconst 25669	. . . . . . . . . . 11 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ 𝐸 ∈ ℂ) → ((𝑋(,)𝑌) × {𝐸}) ∈ 𝐿¹)
172	23, 52, 170, 171	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋(,)𝑌) × {𝐸}) ∈ 𝐿¹)
173	169, 172	eqeltrrid 2830	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ 𝐸) ∈ 𝐿¹)
174		cncfmptc 24754	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℂ ∧ (𝑋(,)𝑌) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑡 ∈ (𝑋(,)𝑌) ↦ 𝐸) ∈ ((𝑋(,)𝑌)–cn→ℂ))
175	65, 66, 174	mp3an23 1449	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℂ → (𝑡 ∈ (𝑋(,)𝑌) ↦ 𝐸) ∈ ((𝑋(,)𝑌)–cn→ℂ))
176	170, 175	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ 𝐸) ∈ ((𝑋(,)𝑌)–cn→ℂ))
177	56, 58, 176, 143	cncfmpt2f 24757	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
178		cnmbf 25510	. . . . . . . . . 10 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))))) ∈ ((𝑋(,)𝑌)–cn→ℂ)) → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))))) ∈ MblFn)
179	22, 177, 178	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))))) ∈ MblFn)
180	168, 173, 137, 146, 179	iblsubnc 37039	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))))) ∈ 𝐿¹)
181	180	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))))) ∈ 𝐿¹)
182		ftc1cnnclem.fc	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸))
183	182	ralrimiva 3138	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸))
184	183	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ∀𝑦 ∈ (𝐴(,)𝐵)((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸))
185	90, 29	sselid 3972	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑐 ∈ ℝ)
186		ftc1cnnclem.r	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
187	186	rpred 13013	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℝ)
188	185, 187	resubcld 11639	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑐 − 𝑅) ∈ ℝ)
189	188	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝑐 − 𝑅) ∈ ℝ)
190	39	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑋 ∈ ℝ)
191		elioore 13351	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (𝑋(,)𝑌) → 𝑡 ∈ ℝ)
192	191	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 ∈ ℝ)
193		ftc1cnnclem.x2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (abs‘(𝑋 − 𝑐)) < 𝑅)
194	39, 185, 187	absdifltd 15377	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((abs‘(𝑋 − 𝑐)) < 𝑅 ↔ ((𝑐 − 𝑅) < 𝑋 ∧ 𝑋 < (𝑐 + 𝑅))))
195	193, 194	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑐 − 𝑅) < 𝑋 ∧ 𝑋 < (𝑐 + 𝑅)))
196	195	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑐 − 𝑅) < 𝑋)
197	196	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝑐 − 𝑅) < 𝑋)
198		eliooord 13380	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (𝑋(,)𝑌) → (𝑋 < 𝑡 ∧ 𝑡 < 𝑌))
199	198	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝑋 < 𝑡 ∧ 𝑡 < 𝑌))
200	199	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑋 < 𝑡)
201	189, 190, 192, 197, 200	lttrd 11372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝑐 − 𝑅) < 𝑡)
202	40	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑌 ∈ ℝ)
203	185, 187	readdcld 11240	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑐 + 𝑅) ∈ ℝ)
204	203	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝑐 + 𝑅) ∈ ℝ)
205	199	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 < 𝑌)
206		ftc1cnnclem.y2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (abs‘(𝑌 − 𝑐)) < 𝑅)
207	40, 185, 187	absdifltd 15377	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((abs‘(𝑌 − 𝑐)) < 𝑅 ↔ ((𝑐 − 𝑅) < 𝑌 ∧ 𝑌 < (𝑐 + 𝑅))))
208	206, 207	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑐 − 𝑅) < 𝑌 ∧ 𝑌 < (𝑐 + 𝑅)))
209	208	simprd 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 < (𝑐 + 𝑅))
210	209	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑌 < (𝑐 + 𝑅))
211	192, 202, 204, 205, 210	lttrd 11372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 < (𝑐 + 𝑅))
212	185	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑐 ∈ ℝ)
213	187	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑅 ∈ ℝ)
214	192, 212, 213	absdifltd 15377	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((abs‘(𝑡 − 𝑐)) < 𝑅 ↔ ((𝑐 − 𝑅) < 𝑡 ∧ 𝑡 < (𝑐 + 𝑅))))
215	201, 211, 214	mpbir2and 710	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (abs‘(𝑡 − 𝑐)) < 𝑅)
216		fvoveq1 7424	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (abs‘(𝑦 − 𝑐)) = (abs‘(𝑡 − 𝑐)))
217	216	breq1d 5148	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → ((abs‘(𝑦 − 𝑐)) < 𝑅 ↔ (abs‘(𝑡 − 𝑐)) < 𝑅))
218	217	imbrov2fvoveq 7426	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → (((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸) ↔ ((abs‘(𝑡 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) < 𝐸)))
219	218	rspcv 3600	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (∀𝑦 ∈ (𝐴(,)𝐵)((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸) → ((abs‘(𝑡 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) < 𝐸)))
220	16, 184, 215, 219	syl3c 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) < 𝐸)
221		difrp 13009	. . . . . . . . . 10 ⊢ (((abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) ∈ ℝ ∧ 𝐸 ∈ ℝ) → ((abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) < 𝐸 ↔ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ℝ⁺))
222	137, 168, 221	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) < 𝐸 ↔ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ℝ⁺))
223	220, 222	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ℝ⁺)
224	223	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 < 𝑌) ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) ∈ ℝ⁺)
225	177	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐))))) ∈ ((𝑋(,)𝑌)–cn→ℂ))
226	167, 181, 224, 225	itggt0cn 37048	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < ∫(𝑋(,)𝑌)(𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) d𝑡)
227	168, 173, 137, 146, 179	itgsubnc 37040	. . . . . . . 8 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) d𝑡 = (∫(𝑋(,)𝑌)𝐸 d𝑡 − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡))
228	227	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) d𝑡 = (∫(𝑋(,)𝑌)𝐸 d𝑡 − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡))
229		itgconst 25670	. . . . . . . . . . 11 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ 𝐸 ∈ ℂ) → ∫(𝑋(,)𝑌)𝐸 d𝑡 = (𝐸 · (vol‘(𝑋(,)𝑌))))
230	23, 52, 170, 229	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → ∫(𝑋(,)𝑌)𝐸 d𝑡 = (𝐸 · (vol‘(𝑋(,)𝑌))))
231	230	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)𝐸 d𝑡 = (𝐸 · (vol‘(𝑋(,)𝑌))))
232	114	oveq2d 7417	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝐸 · (vol‘(𝑋(,)𝑌))) = (𝐸 · (𝑌 − 𝑋)))
233	170, 77	mulcomd 11232	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 · (𝑌 − 𝑋)) = ((𝑌 − 𝑋) · 𝐸))
234	233	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝐸 · (𝑌 − 𝑋)) = ((𝑌 − 𝑋) · 𝐸))
235	231, 232, 234	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)𝐸 d𝑡 = ((𝑌 − 𝑋) · 𝐸))
236	235	oveq1d 7416	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (∫(𝑋(,)𝑌)𝐸 d𝑡 − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡) = (((𝑌 − 𝑋) · 𝐸) − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡))
237	228, 236	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝑐)))) d𝑡 = (((𝑌 − 𝑋) · 𝐸) − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡))
238	226, 237	breqtrd 5164	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (((𝑌 − 𝑋) · 𝐸) − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡))
239	147, 151	posdifd 11798	. . . . . 6 ⊢ (𝜑 → (∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡 < ((𝑌 − 𝑋) · 𝐸) ↔ 0 < (((𝑌 − 𝑋) · 𝐸) − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡)))
240	239	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 0 < (((𝑌 − 𝑋) · 𝐸) − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡)) → ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡 < ((𝑌 − 𝑋) · 𝐸))
241	238, 240	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝑐))) d𝑡 < ((𝑌 − 𝑋) · 𝐸))
242	136, 148, 152, 166, 241	lelttrd 11369	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) < ((𝑌 − 𝑋) · 𝐸))
243	150	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝐸 ∈ ℝ)
244		ltdivmul 12086	. . . 4 ⊢ (((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) ∈ ℝ ∧ 𝐸 ∈ ℝ ∧ ((𝑌 − 𝑋) ∈ ℝ ∧ 0 < (𝑌 − 𝑋))) → (((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) / (𝑌 − 𝑋)) < 𝐸 ↔ (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) < ((𝑌 − 𝑋) · 𝐸)))
245	136, 243, 128, 80, 244	syl112anc 1371	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) / (𝑌 − 𝑋)) < 𝐸 ↔ (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) < ((𝑌 − 𝑋) · 𝐸)))
246	242, 245	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝑐)) d𝑡) / (𝑌 − 𝑋)) < 𝐸)
247	135, 246	eqbrtrd 5160	1 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐))) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⦋csb 3885 ∖ cdif 3937 ⊆ wss 3940 {csn 4620 class class class wbr 5138 ↦ cmpt 5221 × cxp 5664 dom cdm 5666 ↾ cres 5668 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 ℝcr 11105 0cc0 11106 + caddc 11109 · cmul 11111 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 − cmin 11441 / cdiv 11868 ℝ⁺crp 12971 (,)cioo 13321 [,]cicc 13324 ∗ccj 15040 abscabs 15178 TopOpenctopn 17366 ℂ_fldccnfld 21228 Cn ccn 23050 ×_t ctx 23386 –cn→ccncf 24718 vol*covol 25313 volcvol 25314 MblFncmbf 25465 𝐿¹cibl 25468 ∫citg 25469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-symdif 4234 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cn 23053 df-cnp 23054 df-cmp 23213 df-tx 23388 df-hmeo 23581 df-xms 24148 df-ms 24149 df-tms 24150 df-cncf 24720 df-ovol 25315 df-vol 25316 df-mbf 25470 df-itg1 25471 df-itg2 25472 df-ibl 25473 df-itg 25474 df-0p 25521

This theorem is referenced by: ftc1cnnc 37050

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