Theorem ftc1anclem7 33823

Description: Lemma for ftc1anc 33825. (Contributed by Brendan Leahy, 13-May-2018.)

Hypotheses

Ref	Expression
ftc1anc.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
ftc1anc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc1anc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc1anc.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc1anc.s	⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
ftc1anc.d	⊢ (𝜑 → 𝐷 ⊆ ℝ)
ftc1anc.i	⊢ (𝜑 → 𝐹 ∈ 𝐿1)
ftc1anc.f	⊢ (𝜑 → 𝐹:𝐷⟶ℂ)

Assertion

Ref

Expression

ftc1anclem7

⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)))) < ((𝑦 / 2) + (𝑦 / 2)))

Distinct variable groups:   𝑓,𝑔,𝑟,𝑡,𝑢,𝑤,𝑥,𝑦,𝐴   𝐵,𝑓,𝑔,𝑟,𝑡,𝑢,𝑤,𝑥,𝑦   𝐷,𝑓,𝑔,𝑟,𝑡,𝑢,𝑤,𝑥,𝑦   𝑓,𝐹,𝑔,𝑟,𝑡,𝑢,𝑤,𝑥,𝑦   𝜑,𝑓,𝑔,𝑟,𝑡,𝑢,𝑤,𝑥,𝑦   𝑓,𝐺,𝑔,𝑟,𝑢,𝑤,𝑦

Allowed substitution hints:   𝐺(𝑥,𝑡)

Proof of Theorem ftc1anclem7

Step	Hyp	Ref	Expression
1		i1ff 23663	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
2	1	ffvelrnda 6502	. . . . . . . . . 10 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ∈ ℝ)
3	2	recnd 10270	. . . . . . . . 9 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ∈ ℂ)
4		ax-icn 10197	. . . . . . . . . 10 ⊢ i ∈ ℂ
5		i1ff 23663	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ dom ∫1 → 𝑔:ℝ⟶ℝ)
6	5	ffvelrnda 6502	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑔‘𝑥) ∈ ℝ)
7	6	recnd 10270	. . . . . . . . . 10 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑔‘𝑥) ∈ ℂ)
8		mulcl 10222	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ (𝑔‘𝑥) ∈ ℂ) → (i · (𝑔‘𝑥)) ∈ ℂ)
9	4, 7, 8	sylancr 567	. . . . . . . . 9 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (i · (𝑔‘𝑥)) ∈ ℂ)
10		addcl 10220	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (i · (𝑔‘𝑥)) ∈ ℂ) → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ)
11	3, 9, 10	syl2an 575	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ)) → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ)
12	11	anandirs 650	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ)
13		reex 10229	. . . . . . . . 9 ⊢ ℝ ∈ V
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ℝ ∈ V)
15	2	adantlr 686	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ∈ ℝ)
16		ovexd 6825	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (i · (𝑔‘𝑥)) ∈ V)
17	1	feqmptd 6391	. . . . . . . . 9 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑥 ∈ ℝ ↦ (𝑓‘𝑥)))
18	17	adantr 466	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 𝑓 = (𝑥 ∈ ℝ ↦ (𝑓‘𝑥)))
19	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑔 ∈ dom ∫1 → ℝ ∈ V)
20	4	a1i 11	. . . . . . . . . 10 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → i ∈ ℂ)
21		fconstmpt 5303	. . . . . . . . . . 11 ⊢ (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i)
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑔 ∈ dom ∫1 → (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i))
23	5	feqmptd 6391	. . . . . . . . . 10 ⊢ (𝑔 ∈ dom ∫1 → 𝑔 = (𝑥 ∈ ℝ ↦ (𝑔‘𝑥)))
24	19, 20, 6, 22, 23	offval2 7061	. . . . . . . . 9 ⊢ (𝑔 ∈ dom ∫1 → ((ℝ × {i}) ∘𝑓 · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔‘𝑥))))
25	24	adantl 467	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((ℝ × {i}) ∘𝑓 · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔‘𝑥))))
26	14, 15, 16, 18, 25	offval2 7061	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑓 ∘𝑓 + ((ℝ × {i}) ∘𝑓 · 𝑔)) = (𝑥 ∈ ℝ ↦ ((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))
27		absf 14285	. . . . . . . . 9 ⊢ abs:ℂ⟶ℝ
28	27	a1i 11	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → abs:ℂ⟶ℝ)
29	28	feqmptd 6391	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → abs = (𝑡 ∈ ℂ ↦ (abs‘𝑡)))
30		fveq2 6332	. . . . . . 7 ⊢ (𝑡 = ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) → (abs‘𝑡) = (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))
31	12, 26, 29, 30	fmptco 6539	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs ∘ (𝑓 ∘𝑓 + ((ℝ × {i}) ∘𝑓 · 𝑔))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥))))))
32		ftc1anclem3 33819	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs ∘ (𝑓 ∘𝑓 + ((ℝ × {i}) ∘𝑓 · 𝑔))) ∈ dom ∫1)
33	31, 32	eqeltrrd 2851	. . . . 5 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥))))) ∈ dom ∫1)
34		ioombl 23553	. . . . 5 ⊢ (𝑢(,)𝑤) ∈ dom vol
35		fveq2 6332	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑓‘𝑥) = (𝑓‘𝑡))
36		fveq2 6332	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
37	36	oveq2d 6809	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (i · (𝑔‘𝑥)) = (i · (𝑔‘𝑡)))
38	35, 37	oveq12d 6811	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) = ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))
39	38	fveq2d 6336	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))) = (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
40		eqid 2771	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥))))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))
41		fvex 6342	. . . . . . . . . 10 ⊢ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ V
42	39, 40, 41	fvmpt 6424	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))‘𝑡) = (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
43	42	eqcomd 2777	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) = ((𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))‘𝑡))
44	43	ifeq1d 4243	. . . . . . 7 ⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) = if(𝑡 ∈ (𝑢(,)𝑤), ((𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))‘𝑡), 0))
45	44	mpteq2ia 4874	. . . . . 6 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))‘𝑡), 0))
46	45	i1fres 23692	. . . . 5 ⊢ (((𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥))))) ∈ dom ∫1 ∧ (𝑢(,)𝑤) ∈ dom vol) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∈ dom ∫1)
47	33, 34, 46	sylancl 566	. . . 4 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∈ dom ∫1)
48		breq2 4790	. . . . . . 7 ⊢ ((abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) → (0 ≤ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ↔ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)))
49		breq2 4790	. . . . . . 7 ⊢ (0 = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)))
50		elioore 12410	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑢(,)𝑤) → 𝑡 ∈ ℝ)
51		eleq1w 2833	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ ℝ ↔ 𝑡 ∈ ℝ))
52	51	anbi2d 606	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) ↔ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)))
53	38	eleq1d 2835	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ ↔ ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ))
54	52, 53	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → ((((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ) ↔ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)))
55	54, 12	chvarv 2425	. . . . . . . . 9 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
56	55	absge0d 14391	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
57	50, 56	sylan2 572	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
58		0le0 11312	. . . . . . . 8 ⊢ 0 ≤ 0
59	58	a1i 11	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ 0)
60	48, 49, 57, 59	ifbothda 4262	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))
61	60	ralrimivw 3116	. . . . 5 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))
62		ax-resscn 10195	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
63	62	a1i 11	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ℝ ⊆ ℂ)
64		c0ex 10236	. . . . . . . . . 10 ⊢ 0 ∈ V
65	41, 64	ifex 4295	. . . . . . . . 9 ⊢ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ∈ V
66		eqid 2771	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))
67	65, 66	fnmpti 6162	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) Fn ℝ
68	67	a1i 11	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) Fn ℝ)
69	63, 68	0pledm 23660	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (0𝑝 ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ↔ (ℝ × {0}) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))))
70	64	a1i 11	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ∈ V)
71	65	a1i 11	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ∈ V)
72		fconstmpt 5303	. . . . . . . 8 ⊢ (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
73	72	a1i 11	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
74		eqidd 2772	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)))
75	14, 70, 71, 73, 74	ofrfval2 7062	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((ℝ × {0}) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)))
76	69, 75	bitrd 268	. . . . 5 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (0𝑝 ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)))
77	61, 76	mpbird 247	. . . 4 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0𝑝 ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)))
78		itg2itg1 23723	. . . . 5 ⊢ (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∈ dom ∫1 ∧ 0𝑝 ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) = (∫1‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))))
79		itg1cl 23672	. . . . . 6 ⊢ ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∈ dom ∫1 → (∫1‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
80	79	adantr 466	. . . . 5 ⊢ (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∈ dom ∫1 ∧ 0𝑝 ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) → (∫1‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
81	78, 80	eqeltrd 2850	. . . 4 ⊢ (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∈ dom ∫1 ∧ 0𝑝 ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
82	47, 77, 81	syl2anc 565	. . 3 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
83	82	ad6antlr 714	. 2 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
84		simplll 750	. . . . 5 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)))
85		ftc1anc.a	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ∈ ℝ)
86	85	rexrd 10291	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
87		ftc1anc.b	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ ℝ)
88	87	rexrd 10291	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
89	86, 88	jca 495	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))
90		df-icc 12387	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑡 ∈ ℝ* ∣ (𝑥 ≤ 𝑡 ∧ 𝑡 ≤ 𝑦)})
91	90	elixx3g 12393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑢 ∈ ℝ*) ∧ (𝐴 ≤ 𝑢 ∧ 𝑢 ≤ 𝐵)))
92	91	simprbi 478	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝑢 ∧ 𝑢 ≤ 𝐵))
93	92	simpld 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝐴[,]𝐵) → 𝐴 ≤ 𝑢)
94	90	elixx3g 12393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐴 ≤ 𝑤 ∧ 𝑤 ≤ 𝐵)))
95	94	simprbi 478	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝑤 ∧ 𝑤 ≤ 𝐵))
96	95	simprd 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝐴[,]𝐵) → 𝑤 ≤ 𝐵)
97	93, 96	anim12i 592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴 ≤ 𝑢 ∧ 𝑤 ≤ 𝐵))
98		ioossioo 12471	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝑢 ∧ 𝑤 ≤ 𝐵)) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
99	89, 97, 98	syl2an 575	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
100		ftc1anc.s	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
101	100	adantr 466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ 𝐷)
102	99, 101	sstrd 3762	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ 𝐷)
103	102	3adantr3 1176	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (𝑢(,)𝑤) ⊆ 𝐷)
104	103	sselda 3752	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡 ∈ 𝐷)
105		ftc1anc.f	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
106	105	ffvelrnda 6502	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ ℂ)
107	106	adantlr 686	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ ℂ)
108	104, 107	syldan 571	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹‘𝑡) ∈ ℂ)
109	108	adantllr 690	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹‘𝑡) ∈ ℂ)
110	55	adantll 685	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
111	50, 110	sylan2 572	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
112	111	adantlr 686	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
113	109, 112	subcld 10594	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
114	113	abscld 14383	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ)
115	114	rexrd 10291	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
116	113	absge0d 14391	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
117		elxrge0 12488	. . . . . . . . 9 ⊢ ((abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
118	115, 116, 117	sylanbrc 564	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ (0[,]+∞))
119		0e0iccpnf 12490	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
120	119	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
121	118, 120	ifclda 4259	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ (0[,]+∞))
122	121	adantr 466	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ (0[,]+∞))
123		eqid 2771	. . . . . 6 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))
124	122, 123	fmptd 6527	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)):ℝ⟶(0[,]+∞))
125	84, 124	sylan 561	. . . 4 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)):ℝ⟶(0[,]+∞))
126		rpre 12042	. . . . . 6 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
127	126	rehalfcld 11481	. . . . 5 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
128	127	ad2antlr 698	. . . 4 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (𝑦 / 2) ∈ ℝ)
129		simpll 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) → (𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)))
130	102	sselda 3752	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡 ∈ 𝐷)
131	130	adantllr 690	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡 ∈ 𝐷)
132	106	adantlr 686	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ ℂ)
133		ftc1anc.d	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
134	133	sselda 3752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ ℝ)
135	134	adantlr 686	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ ℝ)
136	135, 110	syldan 571	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
137	132, 136	subcld 10594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → ((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
138	137	abscld 14383	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ)
139	138	rexrd 10291	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
140	139	adantlr 686	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
141	131, 140	syldan 571	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
142	137	absge0d 14391	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → 0 ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
143	142	adantlr 686	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ 𝐷) → 0 ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
144	131, 143	syldan 571	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
145	141, 144, 117	sylanbrc 564	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ (0[,]+∞))
146	119	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
147	145, 146	ifclda 4259	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ (0[,]+∞))
148	147	adantr 466	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ (0[,]+∞))
149	148, 123	fmptd 6527	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)):ℝ⟶(0[,]+∞))
150		itg2cl 23719	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)):ℝ⟶(0[,]+∞) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ∈ ℝ*)
151	149, 150	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ∈ ℝ*)
152	129, 151	sylan 561	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ∈ ℝ*)
153		0cnd 10235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝑡 ∈ 𝐷) → 0 ∈ ℂ)
154	106, 153	ifclda 4259	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) ∈ ℂ)
155		subcl 10482	. . . . . . . . . . . . . . . 16 ⊢ ((if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) ∈ ℂ ∧ ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ) → (if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
156	154, 55, 155	syl2an 575	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)) → (if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
157	156	anassrs 458	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
158	157	abscld 14383	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ)
159	158	rexrd 10291	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
160	157	absge0d 14391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
161		elxrge0 12488	. . . . . . . . . . . 12 ⊢ ((abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ (0[,]+∞) ↔ ((abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ* ∧ 0 ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
162	159, 160, 161	sylanbrc 564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ (0[,]+∞))
163		eqid 2771	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))) = (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
164	162, 163	fmptd 6527	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))):ℝ⟶(0[,]+∞))
165		itg2cl 23719	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))):ℝ⟶(0[,]+∞) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) ∈ ℝ*)
166	164, 165	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) ∈ ℝ*)
167	166	ad3antrrr 701	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) ∈ ℝ*)
168		rphalfcl 12061	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
169	168	rpxrd 12076	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ*)
170	169	ad2antlr 698	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑦 / 2) ∈ ℝ*)
171	164	adantr 466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))):ℝ⟶(0[,]+∞))
172		breq1 4789	. . . . . . . . . . . . 13 ⊢ ((abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) → ((abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
173		breq1 4789	. . . . . . . . . . . . 13 ⊢ (0 = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) → (0 ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
174	138	leidd 10796	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
175		iftrue 4231	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ 𝐷 → if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) = (𝐹‘𝑡))
176	175	fvoveq1d 6815	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ 𝐷 → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) = (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
177	176	adantl 467	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) = (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
178	174, 177	breqtrrd 4814	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
179	178	adantlr 686	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
180	131, 179	syldan 571	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
181	180	adantlr 686	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
182	160	adantlr 686	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
183	182	adantr 466	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
184	172, 173, 181, 183	ifbothda 4262	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
185	184	ralrimiva 3115	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
186	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
187		fvex 6342	. . . . . . . . . . . . . . 15 ⊢ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ V
188	187, 64	ifex 4295	. . . . . . . . . . . . . 14 ⊢ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ V
189	188	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ V)
190		fvexd 6344	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ V)
191		eqidd 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)))
192		eqidd 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))) = (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
193	186, 189, 190, 191, 192	ofrfval2 7062	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
194	193	ad2antrr 697	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
195	185, 194	mpbird 247	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))))
196		itg2le 23726	. . . . . . . . . 10 ⊢ (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))))
197	149, 171, 195, 196	syl3anc 1476	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))))
198	129, 197	sylan 561	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))))
199		simpllr 752	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2))
200	152, 167, 170, 198, 199	xrlelttrd 12196	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < (𝑦 / 2))
201		xrltle 12187	. . . . . . . 8 ⊢ (((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ∈ ℝ* ∧ (𝑦 / 2) ∈ ℝ*) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < (𝑦 / 2) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (𝑦 / 2)))
202	152, 170, 201	syl2anc 565	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < (𝑦 / 2) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (𝑦 / 2)))
203	200, 202	mpd 15	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (𝑦 / 2))
204	203	adantllr 690	. . . . 5 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (𝑦 / 2))
205	204	3adantr3 1176	. . . 4 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (𝑦 / 2))
206		itg2lecl 23725	. . . 4 ⊢ (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑦 / 2) ∈ ℝ ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ≤ (𝑦 / 2)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ∈ ℝ)
207	125, 128, 205, 206	syl3anc 1476	. . 3 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ∈ ℝ)
208	207	adantr 466	. 2 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) ∈ ℝ)
209	127	ad3antlr 702	. 2 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (𝑦 / 2) ∈ ℝ)
210	82	adantr 466	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
211		2rp 12040	. . . . . . . . 9 ⊢ 2 ∈ ℝ+
212		imassrn 5618	. . . . . . . . . . . . . . . 16 ⊢ (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ran abs
213		frn 6193	. . . . . . . . . . . . . . . . 17 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
214	27, 213	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ran abs ⊆ ℝ
215	212, 214	sstri 3761	. . . . . . . . . . . . . . 15 ⊢ (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ
216	215	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ)
217		frn 6193	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℝ⟶ℝ → ran 𝑓 ⊆ ℝ)
218	1, 217	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ dom ∫1 → ran 𝑓 ⊆ ℝ)
219	218	adantr 466	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ran 𝑓 ⊆ ℝ)
220		frn 6193	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔:ℝ⟶ℝ → ran 𝑔 ⊆ ℝ)
221	5, 220	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 ∈ dom ∫1 → ran 𝑔 ⊆ ℝ)
222	221	adantl 467	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ran 𝑔 ⊆ ℝ)
223	219, 222	unssd 3940	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℝ)
224	223, 62	syl6ss 3764	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
225		i1f0rn 23669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ dom ∫1 → 0 ∈ ran 𝑓)
226		elun1 3931	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ran 𝑓 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
227	225, 226	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ dom ∫1 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
228	227	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
229		ffn 6185	. . . . . . . . . . . . . . . . . 18 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
230	27, 229	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ abs Fn ℂ
231		fnfvima 6639	. . . . . . . . . . . . . . . . 17 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
232	230, 231	mp3an1 1559	. . . . . . . . . . . . . . . 16 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
233	224, 228, 232	syl2anc 565	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
234		ne0i 4069	. . . . . . . . . . . . . . 15 ⊢ ((abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
235	233, 234	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
236		ffun 6188	. . . . . . . . . . . . . . . . 17 ⊢ (abs:ℂ⟶ℝ → Fun abs)
237	27, 236	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ Fun abs
238		i1frn 23664	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ dom ∫1 → ran 𝑓 ∈ Fin)
239		i1frn 23664	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ dom ∫1 → ran 𝑔 ∈ Fin)
240		unfi 8383	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝑓 ∈ Fin ∧ ran 𝑔 ∈ Fin) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
241	238, 239, 240	syl2an 575	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
242		imafi 8415	. . . . . . . . . . . . . . . 16 ⊢ ((Fun abs ∧ (ran 𝑓 ∪ ran 𝑔) ∈ Fin) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
243	237, 241, 242	sylancr 567	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
244		fimaxre2 11171	. . . . . . . . . . . . . . 15 ⊢ (((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥)
245	215, 243, 244	sylancr 567	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥)
246		suprcl 11185	. . . . . . . . . . . . . 14 ⊢ (((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
247	216, 235, 245, 246	syl3anc 1476	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
248	247	adantr 466	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
249		0red 10243	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 ∈ ℝ)
250	224	sselda 3752	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → 𝑟 ∈ ℂ)
251	250	abscld 14383	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ ℝ)
252	251	adantrr 688	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ∈ ℝ)
253		absgt0 14272	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℂ → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
254	250, 253	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
255	254	biimpa 462	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) ∧ 𝑟 ≠ 0) → 0 < (abs‘𝑟))
256	255	anasss 457	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < (abs‘𝑟))
257	216, 235, 245	3jca 1122	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥))
258	257	adantr 466	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → ((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥))
259		fnfvima 6639	. . . . . . . . . . . . . . . . 17 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
260	230, 259	mp3an1 1559	. . . . . . . . . . . . . . . 16 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
261	224, 260	sylan 561	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
262		suprub 11186	. . . . . . . . . . . . . . 15 ⊢ ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥) ∧ (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
263	258, 261, 262	syl2anc 565	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
264	263	adantrr 688	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
265	249, 252, 248, 256, 264	ltletrd 10399	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
266	248, 265	elrpd 12072	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
267	266	rexlimdvaa 3180	. . . . . . . . . 10 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0 → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+))
268	267	imp 393	. . . . . . . . 9 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
269		rpmulcl 12058	. . . . . . . . 9 ⊢ ((2 ∈ ℝ+ ∧ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
270	211, 268, 269	sylancr 567	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
271	210, 270	rerpdivcld 12106	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
272	271	adantll 685	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
273	272	adantlr 686	. . . . 5 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
274	273	ad3antrrr 701	. . . 4 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
275		simp-4l 760	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → 𝜑)
276		iccssre 12460	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
277	85, 87, 276	syl2anc 565	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
278	277, 62	syl6ss 3764	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
279	278	sselda 3752	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℂ)
280	278	sselda 3752	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℂ)
281		subcl 10482	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑤 − 𝑢) ∈ ℂ)
282	279, 280, 281	syl2anr 576	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑤 − 𝑢) ∈ ℂ)
283	282	anandis 649	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑤 − 𝑢) ∈ ℂ)
284	283	abscld 14383	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘(𝑤 − 𝑢)) ∈ ℝ)
285	284	3adantr3 1176	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (abs‘(𝑤 − 𝑢)) ∈ ℝ)
286	275, 285	sylan 561	. . . . 5 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (abs‘(𝑤 − 𝑢)) ∈ ℝ)
287	286	adantr 466	. . . 4 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (abs‘(𝑤 − 𝑢)) ∈ ℝ)
288		rpdivcl 12059	. . . . . . . . 9 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
289	168, 270, 288	syl2anr 576	. . . . . . . 8 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
290	289	rpred 12075	. . . . . . 7 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
291	290	adantlll 689	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
292	291	adantllr 690	. . . . 5 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
293	292	ad2antrr 697	. . . 4 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
294	277	sseld 3751	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) → 𝑢 ∈ ℝ))
295	277	sseld 3751	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (𝐴[,]𝐵) → 𝑤 ∈ ℝ))
296		idd 24	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ≤ 𝑤 → 𝑢 ≤ 𝑤))
297	294, 295, 296	3anim123d 1554	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)))
298	297	ad2antrr 697	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)))
299	298	imp 393	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤))
300	55	abscld 14383	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℝ)
301	300	rexrd 10291	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℝ*)
302		elxrge0 12488	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ (0[,]+∞) ↔ ((abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
303	301, 56, 302	sylanbrc 564	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ (0[,]+∞))
304		ifcl 4269	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ∈ (0[,]+∞))
305	303, 119, 304	sylancl 566	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ∈ (0[,]+∞))
306	305, 66	fmptd 6527	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)):ℝ⟶(0[,]+∞))
307	247	recnd 10270	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℂ)
308	307	2timesd 11477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) = (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
309	247, 247	readdcld 10271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
310	309	rexrd 10291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ*)
311		abs0 14233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (abs‘0) = 0
312	311, 233	syl5eqelr 2855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
313		suprub 11186	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥) ∧ 0 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
314	257, 312, 313	syl2anc 565	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
315	247, 247, 314, 314	addge0d 10805	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
316		elxrge0 12488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞) ↔ ((sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ* ∧ 0 ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))))
317	310, 315, 316	sylanbrc 564	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
318	308, 317	eqeltrd 2850	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
319		ifcl 4269	. . . . . . . . . . . . . . . . . 18 ⊢ (((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
320	318, 119, 319	sylancl 566	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
321	320	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
322		eqid 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
323	321, 322	fmptd 6527	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)):ℝ⟶(0[,]+∞))
324	1	ffvelrnda 6502	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ℝ)
325	324	recnd 10270	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ℂ)
326	325	abscld 14383	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ∈ ℝ)
327	5	ffvelrnda 6502	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ℝ)
328	327	recnd 10270	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ℂ)
329	328	abscld 14383	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ ℝ)
330		readdcl 10221	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((abs‘(𝑓‘𝑡)) ∈ ℝ ∧ (abs‘(𝑔‘𝑡)) ∈ ℝ) → ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))) ∈ ℝ)
331	326, 329, 330	syl2an 575	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ)) → ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))) ∈ ℝ)
332	331	anandirs 650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))) ∈ ℝ)
333	309	adantr 466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
334		mulcl 10222	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((i ∈ ℂ ∧ (𝑔‘𝑡) ∈ ℂ) → (i · (𝑔‘𝑡)) ∈ ℂ)
335	4, 328, 334	sylancr 567	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (i · (𝑔‘𝑡)) ∈ ℂ)
336		abstri 14278	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓‘𝑡) ∈ ℂ ∧ (i · (𝑔‘𝑡)) ∈ ℂ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))))
337	325, 335, 336	syl2an 575	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ)) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))))
338	337	anandirs 650	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))))
339		absmul 14242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((i ∈ ℂ ∧ (𝑔‘𝑡) ∈ ℂ) → (abs‘(i · (𝑔‘𝑡))) = ((abs‘i) · (abs‘(𝑔‘𝑡))))
340	4, 328, 339	sylancr 567	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = ((abs‘i) · (abs‘(𝑔‘𝑡))))
341		absi 14234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (abs‘i) = 1
342	341	oveq1i 6803	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((abs‘i) · (abs‘(𝑔‘𝑡))) = (1 · (abs‘(𝑔‘𝑡)))
343	340, 342	syl6eq 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = (1 · (abs‘(𝑔‘𝑡))))
344	329	recnd 10270	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ ℂ)
345	344	mulid2d 10260	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (1 · (abs‘(𝑔‘𝑡))) = (abs‘(𝑔‘𝑡)))
346	343, 345	eqtrd 2805	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = (abs‘(𝑔‘𝑡)))
347	346	adantll 685	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = (abs‘(𝑔‘𝑡)))
348	347	oveq2d 6809	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))) = ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))))
349	338, 348	breqtrd 4812	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))))
350	326	adantlr 686	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ∈ ℝ)
351	329	adantll 685	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ ℝ)
352	247	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
353	257	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥))
354	224	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
355		ffn 6185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
356	1, 355	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
357		fnfvelrn 6499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ran 𝑓)
358	356, 357	sylan 561	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ran 𝑓)
359		elun1 3931	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓‘𝑡) ∈ ran 𝑓 → (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
360	358, 359	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
361	360	adantlr 686	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
362		fnfvima 6639	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
363	230, 362	mp3an1 1559	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
364	354, 361, 363	syl2anc 565	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
365		suprub 11186	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥) ∧ (abs‘(𝑓‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → (abs‘(𝑓‘𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
366	353, 364, 365	syl2anc 565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
367		ffn 6185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑔:ℝ⟶ℝ → 𝑔 Fn ℝ)
368	5, 367	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑔 ∈ dom ∫1 → 𝑔 Fn ℝ)
369		fnfvelrn 6499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ran 𝑔)
370	368, 369	sylan 561	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ran 𝑔)
371		elun2 3932	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔‘𝑡) ∈ ran 𝑔 → (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
372	370, 371	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
373	372	adantll 685	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
374		fnfvima 6639	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
375	230, 374	mp3an1 1559	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
376	354, 373, 375	syl2anc 565	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
377		suprub 11186	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥) ∧ (abs‘(𝑔‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → (abs‘(𝑔‘𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
378	353, 376, 377	syl2anc 565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
379	350, 351, 352, 352, 366, 378	le2addd 10848	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))) ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
380	300, 332, 333, 349, 379	letrd 10396	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
381	308	adantr 466	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) = (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
382	380, 381	breqtrrd 4814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
383	50, 382	sylan2 572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
384		iftrue 4231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) = (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
385	384	adantl 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) = (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
386		iftrue 4231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) = (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
387	386	adantl 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) = (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
388	383, 385, 387	3brtr4d 4818	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
389	388	ex 397	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)))
390	58	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑡 ∈ (𝑢(,)𝑤) → 0 ≤ 0)
391		iffalse 4234	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) = 0)
392		iffalse 4234	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) = 0)
393	390, 391, 392	3brtr4d 4818	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
394	389, 393	pm2.61d1 172	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
395	394	ralrimivw 3116	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
396		ovex 6823	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ V
397	396, 64	ifex 4295	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ V
398	397	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ V)
399		eqidd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)))
400	14, 71, 398, 74, 399	ofrfval2 7062	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)))
401	395, 400	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)))
402		itg2le 23726	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))))
403	306, 323, 401, 402	syl3anc 1476	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))))
404	403	adantr 466	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))))
405		mblvol 23518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢(,)𝑤) ∈ dom vol → (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤)))
406	34, 405	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤))
407		ovolioo 23556	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol*‘(𝑢(,)𝑤)) = (𝑤 − 𝑢))
408	406, 407	syl5eq 2817	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol‘(𝑢(,)𝑤)) = (𝑤 − 𝑢))
409		resubcl 10547	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑤 − 𝑢) ∈ ℝ)
410	409	ancoms 455	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑤 − 𝑢) ∈ ℝ)
411	410	3adant3 1126	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (𝑤 − 𝑢) ∈ ℝ)
412	408, 411	eqeltrd 2850	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
413		elrege0 12485	. . . . . . . . . . . . . . . . 17 ⊢ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞) ↔ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ ∧ 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
414	247, 314, 413	sylanbrc 564	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞))
415		ge0addcl 12491	. . . . . . . . . . . . . . . 16 ⊢ ((sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞) ∧ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞)) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
416	414, 414, 415	syl2anc 565	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
417	308, 416	eqeltrd 2850	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
418		itg2const 23727	. . . . . . . . . . . . . . 15 ⊢ (((𝑢(,)𝑤) ∈ dom vol ∧ (vol‘(𝑢(,)𝑤)) ∈ ℝ ∧ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))) = ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
419	34, 418	mp3an1 1559	. . . . . . . . . . . . . 14 ⊢ (((vol‘(𝑢(,)𝑤)) ∈ ℝ ∧ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))) = ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
420	412, 417, 419	syl2anr 576	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))) = ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
421	404, 420	breqtrd 4812	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
422	421	adantll 685	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
423	422	adantlr 686	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
424	82	ad3antlr 702	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
425	412	adantl 467	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
426	270	adantll 685	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
427	426	adantr 466	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
428	424, 425, 427	ledivmuld 12128	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (vol‘(𝑢(,)𝑤)) ↔ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤)))))
429	423, 428	mpbird 247	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (vol‘(𝑢(,)𝑤)))
430		abssubge0 14275	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (abs‘(𝑤 − 𝑢)) = (𝑤 − 𝑢))
431	407, 430	eqtr4d 2808	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol*‘(𝑢(,)𝑤)) = (abs‘(𝑤 − 𝑢)))
432	406, 431	syl5eq 2817	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol‘(𝑢(,)𝑤)) = (abs‘(𝑤 − 𝑢)))
433	432	adantl 467	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → (vol‘(𝑢(,)𝑤)) = (abs‘(𝑤 − 𝑢)))
434	429, 433	breqtrd 4812	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤 − 𝑢)))
435	299, 434	syldan 571	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤 − 𝑢)))
436	435	adantllr 690	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤 − 𝑢)))
437	436	adantlr 686	. . . . 5 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤 − 𝑢)))
438	437	adantr 466	. . . 4 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤 − 𝑢)))
439		simpr 471	. . . 4 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))))
440	274, 287, 293, 438, 439	lelttrd 10397	. . 3 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))))
441	82	adantl 467	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
442	441	ad3antrrr 701	. . . . 5 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
443	127	adantl 467	. . . . 5 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (𝑦 / 2) ∈ ℝ)
444	426	adantlr 686	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
445	444	adantr 466	. . . . 5 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
446	442, 443, 445	ltdiv1d 12120	. . . 4 ⊢ (((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) < (𝑦 / 2) ↔ ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
447	446	ad2antrr 697	. . 3 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) < (𝑦 / 2) ↔ ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
448	440, 447	mpbird 247	. 2 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) < (𝑦 / 2))
449	200	adantllr 690	. . . 4 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < (𝑦 / 2))
450	449	3adantr3 1176	. . 3 ⊢ ((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < (𝑦 / 2))
451	450	adantr 466	. 2 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < (𝑦 / 2))
452	83, 208, 209, 209, 448, 451	lt2addd 10852	1 ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)))) < ((𝑦 / 2) + (𝑦 / 2)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∪ cun 3721   ⊆ wss 3723  ∅c0 4063  ifcif 4225  {csn 4316   class class class wbr 4786   ↦ cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250   “ cima 5252   ∘ ccom 5253  Fun wfun 6025   Fn wfn 6026  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793   ∘_𝑓 cof 7042   ∘_𝑟 cofr 7043  Fincfn 8109  supcsup 8502  ℂcc 10136  ℝcr 10137  0cc0 10138  1c1 10139  ici 10140   + caddc 10141   · cmul 10143  +∞cpnf 10273  ℝ^*cxr 10275   < clt 10276   ≤ cle 10277   − cmin 10468   / cdiv 10886  2c2 11272  ℝ⁺crp 12035  (,)cioo 12380  [,)cico 12382  [,]cicc 12383  abscabs 14182  vol*covol 23450  volcvol 23451  ∫₁citg1 23603  ∫₂citg2 23604  𝐿¹cibl 23605  ∫citg 23606  0_𝑝c0p 23656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-ofr 7045  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-rest 16291  df-topgen 16312  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-mopn 19957  df-top 20919  df-topon 20936  df-bases 20971  df-cmp 21411  df-ovol 23452  df-vol 23453  df-mbf 23607  df-itg1 23608  df-itg2 23609  df-0p 23657

This theorem is referenced by:  ftc1anclem8  33824