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Theorem ftc1lem4 26002

Description: Lemma for ftc1 26005. (Contributed by Mario Carneiro, 31-Aug-2014.)

**Hypotheses**
Ref	Expression
ftc1.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
ftc1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc1.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc1.s	⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
ftc1.d	⊢ (𝜑 → 𝐷 ⊆ ℝ)
ftc1.i	⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
ftc1.c	⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
ftc1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))
ftc1.j	⊢ 𝐽 = (𝐿 ↾_t ℝ)
ftc1.k	⊢ 𝐾 = (𝐿 ↾_t 𝐷)
ftc1.l	⊢ 𝐿 = (TopOpen‘ℂ_fld)
ftc1.h	⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))
ftc1.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ftc1.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
ftc1.fc	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))
ftc1.x1	⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))
ftc1.x2	⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅)
ftc1.y1	⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))
ftc1.y2	⊢ (𝜑 → (abs‘(𝑌 − 𝐶)) < 𝑅)

**Assertion**
Ref	Expression
ftc1lem4	⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝐶))) < 𝐸)

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

**Proof of Theorem ftc1lem4**
Step	Hyp	Ref	Expression
1		ovexd 7393	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((𝐹‘𝑡) − (𝐹‘𝐶)) ∈ V)
2		ftc1.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	rexrd 11182	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		ftc1.x1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))
5		ftc1.b	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6		elicc2 13327	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵)))
7	2, 5, 6	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵)))
8	4, 7	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ ℝ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))
9	8	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ 𝑋)
10		iooss1 13296	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑋) → (𝑋(,)𝑌) ⊆ (𝐴(,)𝑌))
11	3, 9, 10	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋(,)𝑌) ⊆ (𝐴(,)𝑌))
12	5	rexrd 11182	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
13		ftc1.y1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))
14		elicc2 13327	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑌 ∈ (𝐴[,]𝐵) ↔ (𝑌 ∈ ℝ ∧ 𝐴 ≤ 𝑌 ∧ 𝑌 ≤ 𝐵)))
15	2, 5, 14	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 ∈ (𝐴[,]𝐵) ↔ (𝑌 ∈ ℝ ∧ 𝐴 ≤ 𝑌 ∧ 𝑌 ≤ 𝐵)))
16	13, 15	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ 𝐴 ≤ 𝑌 ∧ 𝑌 ≤ 𝐵))
17	16	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ≤ 𝐵)
18		iooss2 13297	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑌 ≤ 𝐵) → (𝐴(,)𝑌) ⊆ (𝐴(,)𝐵))
19	12, 17, 18	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝑌) ⊆ (𝐴(,)𝐵))
20	11, 19	sstrd 3944	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋(,)𝑌) ⊆ (𝐴(,)𝐵))
21		ftc1.s	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
22	20, 21	sstrd 3944	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋(,)𝑌) ⊆ 𝐷)
23	22	sselda 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 ∈ 𝐷)
24		ftc1.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
25		ftc1.le	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
26		ftc1.d	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
27		ftc1.i	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
28		ftc1.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
29		ftc1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))
30		ftc1.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (𝐿 ↾_t ℝ)
31		ftc1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (𝐿 ↾_t 𝐷)
32		ftc1.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (TopOpen‘ℂ_fld)
33	24, 2, 5, 25, 21, 26, 27, 28, 29, 30, 31, 32	ftc1lem3 26001	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
34	33	ffvelcdmda 7029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ ℂ)
35	23, 34	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐹‘𝑡) ∈ ℂ)
36		ioombl 25522	. . . . . . . . . . 11 ⊢ (𝑋(,)𝑌) ∈ dom vol
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋(,)𝑌) ∈ dom vol)
38		fvexd 6849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ V)
39	33	feqmptd 6902	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝐹‘𝑡)))
40	39, 27	eqeltrrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ 𝐷 ↦ (𝐹‘𝑡)) ∈ 𝐿¹)
41	22, 37, 38, 40	iblss 25762	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝑡)) ∈ 𝐿¹)
42	21, 28	sseldd 3934	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
43	33, 42	ffvelcdmd 7030	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
44	43	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐹‘𝐶) ∈ ℂ)
45		fconstmpt 5686	. . . . . . . . . 10 ⊢ ((𝑋(,)𝑌) × {(𝐹‘𝐶)}) = (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝐶))
46		mblvol 25487	. . . . . . . . . . . . 13 ⊢ ((𝑋(,)𝑌) ∈ dom vol → (vol‘(𝑋(,)𝑌)) = (vol*‘(𝑋(,)𝑌)))
47	36, 46	ax-mp 5	. . . . . . . . . . . 12 ⊢ (vol‘(𝑋(,)𝑌)) = (vol*‘(𝑋(,)𝑌))
48		ioossicc 13349	. . . . . . . . . . . . . 14 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
49	48	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌))
50		iccssre 13345	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
51	2, 5, 50	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
52	51, 4	sseldd 3934	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℝ)
53	51, 13	sseldd 3934	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ ℝ)
54		iccmbl 25523	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋[,]𝑌) ∈ dom vol)
55	52, 53, 54	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋[,]𝑌) ∈ dom vol)
56		mblss 25488	. . . . . . . . . . . . . 14 ⊢ ((𝑋[,]𝑌) ∈ dom vol → (𝑋[,]𝑌) ⊆ ℝ)
57	55, 56	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋[,]𝑌) ⊆ ℝ)
58		mblvol 25487	. . . . . . . . . . . . . . 15 ⊢ ((𝑋[,]𝑌) ∈ dom vol → (vol‘(𝑋[,]𝑌)) = (vol*‘(𝑋[,]𝑌)))
59	55, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (vol‘(𝑋[,]𝑌)) = (vol*‘(𝑋[,]𝑌)))
60		iccvolcl 25524	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (vol‘(𝑋[,]𝑌)) ∈ ℝ)
61	52, 53, 60	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (vol‘(𝑋[,]𝑌)) ∈ ℝ)
62	59, 61	eqeltrrd 2837	. . . . . . . . . . . . 13 ⊢ (𝜑 → (vol*‘(𝑋[,]𝑌)) ∈ ℝ)
63		ovolsscl 25443	. . . . . . . . . . . . 13 ⊢ (((𝑋(,)𝑌) ⊆ (𝑋[,]𝑌) ∧ (𝑋[,]𝑌) ⊆ ℝ ∧ (vol‘(𝑋[,]𝑌)) ∈ ℝ) → (vol‘(𝑋(,)𝑌)) ∈ ℝ)
64	49, 57, 62, 63	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (vol*‘(𝑋(,)𝑌)) ∈ ℝ)
65	47, 64	eqeltrid 2840	. . . . . . . . . . 11 ⊢ (𝜑 → (vol‘(𝑋(,)𝑌)) ∈ ℝ)
66		iblconst 25775	. . . . . . . . . . 11 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℂ) → ((𝑋(,)𝑌) × {(𝐹‘𝐶)}) ∈ 𝐿¹)
67	37, 65, 43, 66	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋(,)𝑌) × {(𝐹‘𝐶)}) ∈ 𝐿¹)
68	45, 67	eqeltrrid 2841	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐹‘𝐶)) ∈ 𝐿¹)
69	35, 41, 44, 68	iblsub 25779	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ ((𝐹‘𝑡) − (𝐹‘𝐶))) ∈ 𝐿¹)
70	1, 69	itgcl 25741	. . . . . . 7 ⊢ (𝜑 → ∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 ∈ ℂ)
71	70	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 ∈ ℂ)
72	53, 52	resubcld 11565	. . . . . . . 8 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℝ)
73	72	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℝ)
74	73	recnd 11160	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℂ)
75	52, 53	posdifd 11724	. . . . . . . 8 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 0 < (𝑌 − 𝑋)))
76	75	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (𝑌 − 𝑋))
77	76	gt0ne0d 11701	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ≠ 0)
78	71, 74, 77	divcld 11917	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 / (𝑌 − 𝑋)) ∈ ℂ)
79	43	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝐹‘𝐶) ∈ ℂ)
80		ltle 11221	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌))
81	52, 53, 80	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌))
82	81	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌)
83	24, 2, 5, 25, 21, 26, 27, 33, 4, 13	ftc1lem1 25998	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡)
84	82, 83	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡)
85	35, 44	npcand 11496	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (((𝐹‘𝑡) − (𝐹‘𝐶)) + (𝐹‘𝐶)) = (𝐹‘𝑡))
86	85	itgeq2dv 25739	. . . . . . . . . 10 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(((𝐹‘𝑡) − (𝐹‘𝐶)) + (𝐹‘𝐶)) d𝑡 = ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡)
87	35, 44	subcld 11492	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((𝐹‘𝑡) − (𝐹‘𝐶)) ∈ ℂ)
88	87, 69, 44, 68	itgadd 25782	. . . . . . . . . 10 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(((𝐹‘𝑡) − (𝐹‘𝐶)) + (𝐹‘𝐶)) d𝑡 = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 + ∫(𝑋(,)𝑌)(𝐹‘𝐶) d𝑡))
89	86, 88	eqtr3d 2773	. . . . . . . . 9 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡 = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 + ∫(𝑋(,)𝑌)(𝐹‘𝐶) d𝑡))
90	89	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐹‘𝑡) d𝑡 = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 + ∫(𝑋(,)𝑌)(𝐹‘𝐶) d𝑡))
91		itgconst 25776	. . . . . . . . . . . 12 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ (𝐹‘𝐶) ∈ ℂ) → ∫(𝑋(,)𝑌)(𝐹‘𝐶) d𝑡 = ((𝐹‘𝐶) · (vol‘(𝑋(,)𝑌))))
92	37, 65, 43, 91	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐹‘𝐶) d𝑡 = ((𝐹‘𝐶) · (vol‘(𝑋(,)𝑌))))
93	92	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐹‘𝐶) d𝑡 = ((𝐹‘𝐶) · (vol‘(𝑋(,)𝑌))))
94	52	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝑋 ∈ ℝ)
95	53	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝑌 ∈ ℝ)
96		ovolioo 25525	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ ∧ 𝑋 ≤ 𝑌) → (vol*‘(𝑋(,)𝑌)) = (𝑌 − 𝑋))
97	94, 95, 82, 96	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (vol*‘(𝑋(,)𝑌)) = (𝑌 − 𝑋))
98	47, 97	eqtrid 2783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (vol‘(𝑋(,)𝑌)) = (𝑌 − 𝑋))
99	98	oveq2d 7374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝐶) · (vol‘(𝑋(,)𝑌))) = ((𝐹‘𝐶) · (𝑌 − 𝑋)))
100	93, 99	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐹‘𝐶) d𝑡 = ((𝐹‘𝐶) · (𝑌 − 𝑋)))
101	100	oveq2d 7374	. . . . . . . 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))))
102	84, 90, 101	3eqtrd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐺‘𝑌) − (𝐺‘𝑋)) = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 + ((𝐹‘𝐶) · (𝑌 − 𝑋))))
103	102	oveq1d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) = ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 + ((𝐹‘𝐶) · (𝑌 − 𝑋))) / (𝑌 − 𝑋)))
104	79, 74	mulcld 11152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝐶) · (𝑌 − 𝑋)) ∈ ℂ)
105	71, 104, 74, 77	divdird 11955	. . . . . 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))))
106	79, 74, 77	divcan4d 11923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐹‘𝐶) · (𝑌 − 𝑋)) / (𝑌 − 𝑋)) = (𝐹‘𝐶))
107	106	oveq2d 7374	. . . . . 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)))
108	103, 105, 107	3eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) = ((∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 / (𝑌 − 𝑋)) + (𝐹‘𝐶)))
109	78, 79, 108	mvrraddd 11549	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝐶)) = (∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 / (𝑌 − 𝑋)))
110	109	fveq2d 6838	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝐶))) = (abs‘(∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡 / (𝑌 − 𝑋))))
111	71, 74, 77	absdivd 15381	. . 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))))
112		0re 11134	. . . . . . 7 ⊢ 0 ∈ ℝ
113		ltle 11221	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝑌 − 𝑋) ∈ ℝ) → (0 < (𝑌 − 𝑋) → 0 ≤ (𝑌 − 𝑋)))
114	112, 73, 113	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (0 < (𝑌 − 𝑋) → 0 ≤ (𝑌 − 𝑋)))
115	76, 114	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ (𝑌 − 𝑋))
116	73, 115	absidd 15346	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘(𝑌 − 𝑋)) = (𝑌 − 𝑋))
117	116	oveq2d 7374	. . 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)))
118	110, 111, 117	3eqtrd 2775	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝐶))) = ((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) / (𝑌 − 𝑋)))
119	70	abscld 15362	. . . . 5 ⊢ (𝜑 → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) ∈ ℝ)
120	119	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) ∈ ℝ)
121	87	abscld 15362	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) ∈ ℝ)
122	1, 69	iblabs 25786	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) ∈ 𝐿¹)
123	121, 122	itgrecl 25755	. . . . 5 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡 ∈ ℝ)
124	123	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡 ∈ ℝ)
125		ftc1.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
126	125	rpred 12949	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ)
127	72, 126	remulcld 11162	. . . . 5 ⊢ (𝜑 → ((𝑌 − 𝑋) · 𝐸) ∈ ℝ)
128	127	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝑌 − 𝑋) · 𝐸) ∈ ℝ)
129	87, 69	itgabs 25792	. . . . 5 ⊢ (𝜑 → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) ≤ ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡)
130	129	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) ≤ ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡)
131	76, 98	breqtrrd 5126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (vol‘(𝑋(,)𝑌)))
132	126	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝐸 ∈ ℝ)
133		fconstmpt 5686	. . . . . . . . . 10 ⊢ ((𝑋(,)𝑌) × {𝐸}) = (𝑡 ∈ (𝑋(,)𝑌) ↦ 𝐸)
134	126	recnd 11160	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℂ)
135		iblconst 25775	. . . . . . . . . . 11 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ 𝐸 ∈ ℂ) → ((𝑋(,)𝑌) × {𝐸}) ∈ 𝐿¹)
136	37, 65, 134, 135	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋(,)𝑌) × {𝐸}) ∈ 𝐿¹)
137	133, 136	eqeltrrid 2841	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ 𝐸) ∈ 𝐿¹)
138	132, 137, 121, 122	iblsub 25779	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶))))) ∈ 𝐿¹)
139	138	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑡 ∈ (𝑋(,)𝑌) ↦ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶))))) ∈ 𝐿¹)
140	26, 42	sseldd 3934	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ ℝ)
141		ftc1.r	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
142	141	rpred 12949	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℝ)
143	140, 142	resubcld 11565	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
144	143	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐶 − 𝑅) ∈ ℝ)
145	52	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑋 ∈ ℝ)
146	22, 26	sstrd 3944	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋(,)𝑌) ⊆ ℝ)
147	146	sselda 3933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 ∈ ℝ)
148		ftc1.x2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (abs‘(𝑋 − 𝐶)) < 𝑅)
149	52, 140, 142	absdifltd 15359	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((abs‘(𝑋 − 𝐶)) < 𝑅 ↔ ((𝐶 − 𝑅) < 𝑋 ∧ 𝑋 < (𝐶 + 𝑅))))
150	148, 149	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐶 − 𝑅) < 𝑋 ∧ 𝑋 < (𝐶 + 𝑅)))
151	150	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝑋)
152	151	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐶 − 𝑅) < 𝑋)
153		eliooord 13321	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (𝑋(,)𝑌) → (𝑋 < 𝑡 ∧ 𝑡 < 𝑌))
154	153	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝑋 < 𝑡 ∧ 𝑡 < 𝑌))
155	154	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑋 < 𝑡)
156	144, 145, 147, 152, 155	lttrd 11294	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐶 − 𝑅) < 𝑡)
157	53	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑌 ∈ ℝ)
158	140, 142	readdcld 11161	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
159	158	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐶 + 𝑅) ∈ ℝ)
160	154	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 < 𝑌)
161		ftc1.y2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (abs‘(𝑌 − 𝐶)) < 𝑅)
162	53, 140, 142	absdifltd 15359	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((abs‘(𝑌 − 𝐶)) < 𝑅 ↔ ((𝐶 − 𝑅) < 𝑌 ∧ 𝑌 < (𝐶 + 𝑅))))
163	161, 162	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐶 − 𝑅) < 𝑌 ∧ 𝑌 < (𝐶 + 𝑅)))
164	163	simprd 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 < (𝐶 + 𝑅))
165	164	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑌 < (𝐶 + 𝑅))
166	147, 157, 159, 160, 165	lttrd 11294	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑡 < (𝐶 + 𝑅))
167	140	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝐶 ∈ ℝ)
168	142	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → 𝑅 ∈ ℝ)
169	147, 167, 168	absdifltd 15359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((abs‘(𝑡 − 𝐶)) < 𝑅 ↔ ((𝐶 − 𝑅) < 𝑡 ∧ 𝑡 < (𝐶 + 𝑅))))
170	156, 166, 169	mpbir2and 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (abs‘(𝑡 − 𝐶)) < 𝑅)
171		fvoveq1 7381	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → (abs‘(𝑦 − 𝐶)) = (abs‘(𝑡 − 𝐶)))
172	171	breq1d 5108	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → ((abs‘(𝑦 − 𝐶)) < 𝑅 ↔ (abs‘(𝑡 − 𝐶)) < 𝑅))
173	172	imbrov2fvoveq 7383	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → (((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸) ↔ ((abs‘(𝑡 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) < 𝐸)))
174		ftc1.fc	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))
175	174	ralrimiva 3128	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))
176	175	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝐸))
177	173, 176, 23	rspcdva 3577	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((abs‘(𝑡 − 𝐶)) < 𝑅 → (abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) < 𝐸))
178	170, 177	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) < 𝐸)
179		difrp 12945	. . . . . . . . . 10 ⊢ (((abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) ∈ ℝ ∧ 𝐸 ∈ ℝ) → ((abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) < 𝐸 ↔ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) ∈ ℝ⁺))
180	121, 132, 179	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → ((abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) < 𝐸 ↔ (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) ∈ ℝ⁺))
181	178, 180	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) ∈ ℝ⁺)
182	181	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 < 𝑌) ∧ 𝑡 ∈ (𝑋(,)𝑌)) → (𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) ∈ ℝ⁺)
183	131, 139, 182	itggt0 25801	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < ∫(𝑋(,)𝑌)(𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) d𝑡)
184	132, 137, 121, 122	itgsub 25783	. . . . . . . 8 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) d𝑡 = (∫(𝑋(,)𝑌)𝐸 d𝑡 − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡))
185	184	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))
186		itgconst 25776	. . . . . . . . . . 11 ⊢ (((𝑋(,)𝑌) ∈ dom vol ∧ (vol‘(𝑋(,)𝑌)) ∈ ℝ ∧ 𝐸 ∈ ℂ) → ∫(𝑋(,)𝑌)𝐸 d𝑡 = (𝐸 · (vol‘(𝑋(,)𝑌))))
187	37, 65, 134, 186	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ∫(𝑋(,)𝑌)𝐸 d𝑡 = (𝐸 · (vol‘(𝑋(,)𝑌))))
188	187	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)𝐸 d𝑡 = (𝐸 · (vol‘(𝑋(,)𝑌))))
189	98	oveq2d 7374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝐸 · (vol‘(𝑋(,)𝑌))) = (𝐸 · (𝑌 − 𝑋)))
190	72	recnd 11160	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℂ)
191	134, 190	mulcomd 11153	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 · (𝑌 − 𝑋)) = ((𝑌 − 𝑋) · 𝐸))
192	191	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝐸 · (𝑌 − 𝑋)) = ((𝑌 − 𝑋) · 𝐸))
193	188, 189, 192	3eqtrd 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)𝐸 d𝑡 = ((𝑌 − 𝑋) · 𝐸))
194	193	oveq1d 7373	. . . . . . 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))
195	185, 194	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(𝐸 − (abs‘((𝐹‘𝑡) − (𝐹‘𝐶)))) d𝑡 = (((𝑌 − 𝑋) · 𝐸) − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡))
196	183, 195	breqtrd 5124	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (((𝑌 − 𝑋) · 𝐸) − ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡))
197	123, 127	posdifd 11724	. . . . . 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)))
198	197	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))
199	196, 198	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ∫(𝑋(,)𝑌)(abs‘((𝐹‘𝑡) − (𝐹‘𝐶))) d𝑡 < ((𝑌 − 𝑋) · 𝐸))
200	120, 124, 128, 130, 199	lelttrd 11291	. . 3 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) < ((𝑌 − 𝑋) · 𝐸))
201	71	abscld 15362	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) ∈ ℝ)
202	126	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 𝐸 ∈ ℝ)
203		ltdivmul 12017	. . . 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)))
204	201, 202, 73, 76, 203	syl112anc 1376	. . 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)))
205	200, 204	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((abs‘∫(𝑋(,)𝑌)((𝐹‘𝑡) − (𝐹‘𝐶)) d𝑡) / (𝑌 − 𝑋)) < 𝐸)
206	118, 205	eqbrtrd 5120	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 Vcvv 3440 ∖ cdif 3898 ⊆ wss 3901 {csn 4580 class class class wbr 5098 ↦ cmpt 5179 × cxp 5622 dom cdm 5624 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 + caddc 11029 · cmul 11031 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 / cdiv 11794 ℝ⁺crp 12905 (,)cioo 13261 [,]cicc 13264 abscabs 15157 ↾_t crest 17340 TopOpenctopn 17341 ℂ_fldccnfld 21309 CnP ccnp 23169 vol*covol 25419 volcvol 25420 𝐿¹cibl 25574 ∫citg 25575

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-symdif 4205 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cn 23171 df-cnp 23172 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-itg 25580 df-0p 25627

This theorem is referenced by: ftc1lem5 26003

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