Theorem ftc1anclem7 34975

Description: Lemma for ftc1anc 34977. (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 24279	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
2	1	ffvelrnda 6853	. . . . . . . . . 10 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ∈ ℝ)
3	2	recnd 10671	. . . . . . . . 9 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ∈ ℂ)
4		ax-icn 10598	. . . . . . . . . 10 ⊢ i ∈ ℂ
5		i1ff 24279	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ dom ∫1 → 𝑔:ℝ⟶ℝ)
6	5	ffvelrnda 6853	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑔‘𝑥) ∈ ℝ)
7	6	recnd 10671	. . . . . . . . . 10 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (𝑔‘𝑥) ∈ ℂ)
8		mulcl 10623	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ (𝑔‘𝑥) ∈ ℂ) → (i · (𝑔‘𝑥)) ∈ ℂ)
9	4, 7, 8	sylancr 589	. . . . . . . . 9 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) → (i · (𝑔‘𝑥)) ∈ ℂ)
10		addcl 10621	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (i · (𝑔‘𝑥)) ∈ ℂ) → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ)
11	3, 9, 10	syl2an 597	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1 ∧ 𝑥 ∈ ℝ)) → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ)
12	11	anandirs 677	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ)
13		reex 10630	. . . . . . . . 9 ⊢ ℝ ∈ V
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ℝ ∈ V)
15	2	adantlr 713	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (𝑓‘𝑥) ∈ ℝ)
16		ovexd 7193	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (i · (𝑔‘𝑥)) ∈ V)
17	1	feqmptd 6735	. . . . . . . . 9 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑥 ∈ ℝ ↦ (𝑓‘𝑥)))
18	17	adantr 483	. . . . . . . 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 5616	. . . . . . . . . . 11 ⊢ (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i)
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑔 ∈ dom ∫1 → (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i))
23	5	feqmptd 6735	. . . . . . . . . 10 ⊢ (𝑔 ∈ dom ∫1 → 𝑔 = (𝑥 ∈ ℝ ↦ (𝑔‘𝑥)))
24	19, 20, 6, 22, 23	offval2 7428	. . . . . . . . 9 ⊢ (𝑔 ∈ dom ∫1 → ((ℝ × {i}) ∘f · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔‘𝑥))))
25	24	adantl 484	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ((ℝ × {i}) ∘f · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔‘𝑥))))
26	14, 15, 16, 18, 25	offval2 7428	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑓 ∘f + ((ℝ × {i}) ∘f · 𝑔)) = (𝑥 ∈ ℝ ↦ ((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))
27		absf 14699	. . . . . . . . 9 ⊢ abs:ℂ⟶ℝ
28	27	a1i 11	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → abs:ℂ⟶ℝ)
29	28	feqmptd 6735	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → abs = (𝑡 ∈ ℂ ↦ (abs‘𝑡)))
30		fveq2 6672	. . . . . . 7 ⊢ (𝑡 = ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) → (abs‘𝑡) = (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))
31	12, 26, 29, 30	fmptco 6893	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs ∘ (𝑓 ∘f + ((ℝ × {i}) ∘f · 𝑔))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥))))))
32		ftc1anclem3 34971	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs ∘ (𝑓 ∘f + ((ℝ × {i}) ∘f · 𝑔))) ∈ dom ∫1)
33	31, 32	eqeltrrd 2916	. . . . 5 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥))))) ∈ dom ∫1)
34		ioombl 24168	. . . . 5 ⊢ (𝑢(,)𝑤) ∈ dom vol
35		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑓‘𝑥) = (𝑓‘𝑡))
36		fveq2 6672	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
37	36	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (i · (𝑔‘𝑥)) = (i · (𝑔‘𝑡)))
38	35, 37	oveq12d 7176	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → ((𝑓‘𝑥) + (i · (𝑔‘𝑥))) = ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))
39	38	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))) = (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
40		eqid 2823	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥))))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))
41		fvex 6685	. . . . . . . . . 10 ⊢ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ V
42	39, 40, 41	fvmpt 6770	. . . . . . . . 9 ⊢ (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))‘𝑡) = (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
43	42	eqcomd 2829	. . . . . . . 8 ⊢ (𝑡 ∈ ℝ → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) = ((𝑥 ∈ ℝ ↦ (abs‘((𝑓‘𝑥) + (i · (𝑔‘𝑥)))))‘𝑡))
44	43	ifeq1d 4487	. . . . . . 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 5159	. . . . . 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 24308	. . . . 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 588	. . . 4 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)) ∈ dom ∫1)
48		breq2 5072	. . . . . . 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 5072	. . . . . . 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 12771	. . . . . . . 8 ⊢ (𝑡 ∈ (𝑢(,)𝑤) → 𝑡 ∈ ℝ)
51		eleq1w 2897	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ ℝ ↔ 𝑡 ∈ ℝ))
52	51	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) ↔ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)))
53	38	eleq1d 2899	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (((𝑓‘𝑥) + (i · (𝑔‘𝑥))) ∈ ℂ ↔ ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ))
54	52, 53	imbi12d 347	. . . . . . . . . 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	chvarvv 2005	. . . . . . . . 9 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
56	55	absge0d 14806	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
57	50, 56	sylan2 594	. . . . . . 7 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
58		0le0 11741	. . . . . . . 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 4506	. . . . . 6 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))
61	60	ralrimivw 3185	. . . . 5 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))
62		ax-resscn 10596	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
63	62	a1i 11	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ℝ ⊆ ℂ)
64		c0ex 10637	. . . . . . . . . 10 ⊢ 0 ∈ V
65	41, 64	ifex 4517	. . . . . . . . 9 ⊢ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ∈ V
66		eqid 2823	. . . . . . . . 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 6493	. . . . . . . 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 24276	. . . . . 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 5616	. . . . . . . 8 ⊢ (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
73	72	a1i 11	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
74		eqidd 2824	. . . . . . 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 7429	. . . . . 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 281	. . . . 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 259	. . . 4 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0𝑝 ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)))
78		itg2itg1 24339	. . . . 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 24288	. . . . . 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 483	. . . . 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 2915	. . . 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 586	. . 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 735	. 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 773	. . . . 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 10693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
87		ftc1anc.b	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ ℝ)
88	87	rexrd 10693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
89	86, 88	jca 514	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))
90		df-icc 12748	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑡 ∈ ℝ* ∣ (𝑥 ≤ 𝑡 ∧ 𝑡 ≤ 𝑦)})
91	90	elixx3g 12754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑢 ∈ ℝ*) ∧ (𝐴 ≤ 𝑢 ∧ 𝑢 ≤ 𝐵)))
92	91	simprbi 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝑢 ∧ 𝑢 ≤ 𝐵))
93	92	simpld 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (𝐴[,]𝐵) → 𝐴 ≤ 𝑢)
94	90	elixx3g 12754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐴 ≤ 𝑤 ∧ 𝑤 ≤ 𝐵)))
95	94	simprbi 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ (𝐴[,]𝐵) → (𝐴 ≤ 𝑤 ∧ 𝑤 ≤ 𝐵))
96	95	simprd 498	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (𝐴[,]𝐵) → 𝑤 ≤ 𝐵)
97	93, 96	anim12i 614	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴 ≤ 𝑢 ∧ 𝑤 ≤ 𝐵))
98		ioossioo 12832	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝑢 ∧ 𝑤 ≤ 𝐵)) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
99	89, 97, 98	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
100		ftc1anc.s	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
101	100	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ 𝐷)
102	99, 101	sstrd 3979	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ 𝐷)
103	102	3adantr3 1167	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (𝑢(,)𝑤) ⊆ 𝐷)
104	103	sselda 3969	. . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ ℂ)
107	106	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ ℂ)
108	104, 107	syldan 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹‘𝑡) ∈ ℂ)
109	108	adantllr 717	. . . . . . . . . . . 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 712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
111	50, 110	sylan2 594	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
112	111	adantlr 713	. . . . . . . . . . . 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 10999	. . . . . . . . . . 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 14798	. . . . . . . . . 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 10693	. . . . . . . . 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 14806	. . . . . . . . 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 12848	. . . . . . . . 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 585	. . . . . . . 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 12850	. . . . . . . . 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 4503	. . . . . . 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 483	. . . . . 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	122	fmpttd 6881	. . . . 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[,]+∞))
124	84, 123	sylan 582	. . . 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[,]+∞))
125		rpre 12400	. . . . . 6 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
126	125	rehalfcld 11887	. . . . 5 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
127	126	ad2antlr 725	. . . 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)
128		simpll 765	. . . . . . . 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+) → (𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)))
129	102	sselda 3969	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡 ∈ 𝐷)
130	129	adantllr 717	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡 ∈ 𝐷)
131	106	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ ℂ)
132		ftc1anc.d	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
133	132	sselda 3969	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ ℝ)
134	133	adantlr 713	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → 𝑡 ∈ ℝ)
135	134, 110	syldan 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → ((𝑓‘𝑡) + (i · (𝑔‘𝑡))) ∈ ℂ)
136	131, 135	subcld 10999	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → ((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
137	136	abscld 14798	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ)
138	137	rexrd 10693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
139	138	adantlr 713	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ 𝐷) → (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
140	130, 139	syldan 593	. . . . . . . . . . . . 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))))) ∈ ℝ*)
141	136	absge0d 14806	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ 𝐷) → 0 ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
142	141	adantlr 713	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ 𝐷) → 0 ≤ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
143	130, 142	syldan 593	. . . . . . . . . . . . 13 ⊢ ((((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))))))
144	140, 143, 117	sylanbrc 585	. . . . . . . . . . . 12 ⊢ ((((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[,]+∞))
145	119	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
146	144, 145	ifclda 4503	. . . . . . . . . . 11 ⊢ (((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[,]+∞))
147	146	adantr 483	. . . . . . . . . 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) ∈ (0[,]+∞))
148	147	fmpttd 6881	. . . . . . . . 9 ⊢ (((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[,]+∞))
149		itg2cl 24335	. . . . . . . . 9 ⊢ ((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*)
150	148, 149	syl 17	. . . . . . . 8 ⊢ (((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*)
151	128, 150	sylan 582	. . . . . . 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))) ∈ ℝ*)
152		rphalfcl 12419	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
153	152	rpxrd 12435	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ*)
154	153	ad2antlr 725	. . . . . . 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))) → (𝑦 / 2) ∈ ℝ*)
155		0cnd 10636	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝑡 ∈ 𝐷) → 0 ∈ ℂ)
156	106, 155	ifclda 4503	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) ∈ ℂ)
157		subcl 10887	. . . . . . . . . . . . . . . 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)
158	156, 55, 157	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)) → (if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
159	158	anassrs 470	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℂ)
160	159	abscld 14798	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ)
161	160	rexrd 10693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ ℝ*)
162	159	absge0d 14806	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
163		elxrge0 12848	. . . . . . . . . . . 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)))))))
164	161, 162, 163	sylanbrc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ (0[,]+∞))
165	164	fmpttd 6881	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))):ℝ⟶(0[,]+∞))
166		itg2cl 24335	. . . . . . . . . 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*)
167	165, 166	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*)
168	167	ad3antrrr 728	. . . . . . . 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*)
169	165	adantr 483	. . . . . . . . . 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[,]+∞))
170		breq1 5071	. . . . . . . . . . . . 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)))))))
171		breq1 5071	. . . . . . . . . . . . 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)))))))
172	137	leidd 11208	. . . . . . . . . . . . . . . . 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))))))
173		iftrue 4475	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ 𝐷 → if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) = (𝐹‘𝑡))
174	173	fvoveq1d 7180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ 𝐷 → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) = (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
175	174	adantl 484	. . . . . . . . . . . . . . . . 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))))))
176	172, 175	breqtrrd 5096	. . . . . . . . . . . . . . . 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))))))
177	176	adantlr 713	. . . . . . . . . . . . . . 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))))))
178	130, 177	syldan 593	. . . . . . . . . . . . . 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))))))
179	178	adantlr 713	. . . . . . . . . . . . 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))))))
180	162	adantlr 713	. . . . . . . . . . . . . 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))))))
181	180	adantr 483	. . . . . . . . . . . . 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))))))
182	170, 171, 179, 181	ifbothda 4506	. . . . . . . . . . . 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))))))
183	182	ralrimiva 3184	. . . . . . . . . . 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))))))
184	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
185		fvex 6685	. . . . . . . . . . . . . . 15 ⊢ (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ V
186	185, 64	ifex 4517	. . . . . . . . . . . . . 14 ⊢ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ V
187	186	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0) ∈ V)
188		fvexd 6687	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) ∈ V)
189		eqidd 2824	. . . . . . . . . . . . 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)))
190		eqidd 2824	. . . . . . . . . . . . 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)))))))
191	184, 187, 188, 189, 190	ofrfval2 7429	. . . . . . . . . . . 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)))))))
192	191	ad2antrr 724	. . . . . . . . . . 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)))))))
193	183, 192	mpbird 259	. . . . . . . . . 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)))))))
194		itg2le 24342	. . . . . . . . . 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))))))))
195	148, 169, 193, 194	syl3anc 1367	. . . . . . . . 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))))))))
196	128, 195	sylan 582	. . . . . . . 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))))))))
197		simpllr 774	. . . . . . . 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))
198	151, 168, 154, 196, 197	xrlelttrd 12556	. . . . . . 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))
199	151, 154, 198	xrltled 12546	. . . . . 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))
200	199	adantllr 717	. . . . 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))
201	200	3adantr3 1167	. . . 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))
202		itg2lecl 24341	. . . 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)
203	124, 127, 201, 202	syl3anc 1367	. . 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)
204	203	adantr 483	. 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)
205	126	ad3antlr 729	. 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)
206	82	adantr 483	. . . . . . . 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)
207		2rp 12397	. . . . . . . . 9 ⊢ 2 ∈ ℝ+
208		imassrn 5942	. . . . . . . . . . . . . . . 16 ⊢ (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ran abs
209		frn 6522	. . . . . . . . . . . . . . . . 17 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
210	27, 209	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ran abs ⊆ ℝ
211	208, 210	sstri 3978	. . . . . . . . . . . . . . 15 ⊢ (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ
212	211	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ)
213	1	frnd 6523	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ dom ∫1 → ran 𝑓 ⊆ ℝ)
214	213	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ran 𝑓 ⊆ ℝ)
215	5	frnd 6523	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 ∈ dom ∫1 → ran 𝑔 ⊆ ℝ)
216	215	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ran 𝑔 ⊆ ℝ)
217	214, 216	unssd 4164	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℝ)
218	217, 62	sstrdi 3981	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
219		i1f0rn 24285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ dom ∫1 → 0 ∈ ran 𝑓)
220		elun1 4154	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ran 𝑓 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
221	219, 220	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ dom ∫1 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
222	221	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
223		ffn 6516	. . . . . . . . . . . . . . . . . 18 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
224	27, 223	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ abs Fn ℂ
225		fnfvima 6997	. . . . . . . . . . . . . . . . 17 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
226	224, 225	mp3an1 1444	. . . . . . . . . . . . . . . 16 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
227	218, 222, 226	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
228	227	ne0d 4303	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
229		ffun 6519	. . . . . . . . . . . . . . . . 17 ⊢ (abs:ℂ⟶ℝ → Fun abs)
230	27, 229	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ Fun abs
231		i1frn 24280	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ dom ∫1 → ran 𝑓 ∈ Fin)
232		i1frn 24280	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ dom ∫1 → ran 𝑔 ∈ Fin)
233		unfi 8787	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝑓 ∈ Fin ∧ ran 𝑔 ∈ Fin) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
234	231, 232, 233	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
235		imafi 8819	. . . . . . . . . . . . . . . 16 ⊢ ((Fun abs ∧ (ran 𝑓 ∪ ran 𝑔) ∈ Fin) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
236	230, 234, 235	sylancr 589	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
237		fimaxre2 11588	. . . . . . . . . . . . . . 15 ⊢ (((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥)
238	211, 236, 237	sylancr 589	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦 ≤ 𝑥)
239		suprcl 11603	. . . . . . . . . . . . . 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)
240	212, 228, 238, 239	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
241	240	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
242		0red 10646	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 ∈ ℝ)
243	218	sselda 3969	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → 𝑟 ∈ ℂ)
244	243	abscld 14798	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ ℝ)
245	244	adantrr 715	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ∈ ℝ)
246		absgt0 14686	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℂ → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
247	243, 246	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
248	247	biimpa 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) ∧ 𝑟 ≠ 0) → 0 < (abs‘𝑟))
249	248	anasss 469	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < (abs‘𝑟))
250	212, 228, 238	3jca 1124	. . . . . . . . . . . . . . . 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))
251	250	adantr 483	. . . . . . . . . . . . . . 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))
252		fnfvima 6997	. . . . . . . . . . . . . . . . 17 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
253	224, 252	mp3an1 1444	. . . . . . . . . . . . . . . 16 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
254	218, 253	sylan 582	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
255		suprub 11604	. . . . . . . . . . . . . . 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, < ))
256	251, 254, 255	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
257	256	adantrr 715	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
258	242, 245, 241, 249, 257	ltletrd 10802	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
259	241, 258	elrpd 12431	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
260	259	rexlimdvaa 3287	. . . . . . . . . 10 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0 → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+))
261	260	imp 409	. . . . . . . . 9 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
262		rpmulcl 12415	. . . . . . . . 9 ⊢ ((2 ∈ ℝ+ ∧ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
263	207, 261, 262	sylancr 589	. . . . . . . 8 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
264	206, 263	rerpdivcld 12465	. . . . . . 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)
265	264	adantll 712	. . . . . 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)
266	265	adantlr 713	. . . . 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)
267	266	ad3antrrr 728	. . . 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)
268		simp-4l 781	. . . . . 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)
269		iccssre 12821	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
270	85, 87, 269	syl2anc 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
271	270, 62	sstrdi 3981	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
272	271	sselda 3969	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℂ)
273	271	sselda 3969	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℂ)
274		subcl 10887	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑤 − 𝑢) ∈ ℂ)
275	272, 273, 274	syl2anr 598	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑤 − 𝑢) ∈ ℂ)
276	275	anandis 676	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑤 − 𝑢) ∈ ℂ)
277	276	abscld 14798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘(𝑤 − 𝑢)) ∈ ℝ)
278	277	3adantr3 1167	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) → (abs‘(𝑤 − 𝑢)) ∈ ℝ)
279	268, 278	sylan 582	. . . . 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)
280	279	adantr 483	. . . 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)
281		rpdivcl 12417	. . . . . . . . 9 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
282	152, 263, 281	syl2anr 598	. . . . . . . 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+)
283	282	rpred 12434	. . . . . . 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)
284	283	adantlll 716	. . . . . 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)
285	284	adantllr 717	. . . . 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)
286	285	ad2antrr 724	. . . 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)
287	270	sseld 3968	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) → 𝑢 ∈ ℝ))
288	270	sseld 3968	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (𝐴[,]𝐵) → 𝑤 ∈ ℝ))
289		idd 24	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ≤ 𝑤 → 𝑢 ≤ 𝑤))
290	287, 288, 289	3anim123d 1439	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤)))
291	290	ad2antrr 724	. . . . . . . . 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)))
292	291	imp 409	. . . . . . . 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))
293	55	abscld 14798	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℝ)
294	293	rexrd 10693	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℝ*)
295		elxrge0 12848	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ (0[,]+∞) ↔ ((abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))
296	294, 56, 295	sylanbrc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ (0[,]+∞))
297		ifcl 4513	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ∈ (0[,]+∞))
298	296, 119, 297	sylancl 588	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) ∈ (0[,]+∞))
299	298	fmpttd 6881	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0)):ℝ⟶(0[,]+∞))
300	240	recnd 10671	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℂ)
301	300	2timesd 11883	. . . . . . . . . . . . . . . . . . 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, < )))
302	240, 240	readdcld 10672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
303	302	rexrd 10693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ*)
304		abs0 14647	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (abs‘0) = 0
305	304, 227	eqeltrrid 2920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
306		suprub 11604	. . . . . . . . . . . . . . . . . . . . . 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, < ))
307	250, 305, 306	syl2anc 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
308	240, 240, 307, 307	addge0d 11218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → 0 ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
309		elxrge0 12848	. . . . . . . . . . . . . . . . . . . 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, < ))))
310	303, 308, 309	sylanbrc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
311	301, 310	eqeltrd 2915	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
312		ifcl 4513	. . . . . . . . . . . . . . . . . 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[,]+∞))
313	311, 119, 312	sylancl 588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
314	313	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
315	314	fmpttd 6881	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)):ℝ⟶(0[,]+∞))
316	1	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ℝ)
317	316	recnd 10671	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ℂ)
318	317	abscld 14798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ∈ ℝ)
319	5	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ℝ)
320	319	recnd 10671	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ℂ)
321	320	abscld 14798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ ℝ)
322		readdcl 10622	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((abs‘(𝑓‘𝑡)) ∈ ℝ ∧ (abs‘(𝑔‘𝑡)) ∈ ℝ) → ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))) ∈ ℝ)
323	318, 321, 322	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ)) → ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))) ∈ ℝ)
324	323	anandirs 677	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))) ∈ ℝ)
325	302	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
326		mulcl 10623	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((i ∈ ℂ ∧ (𝑔‘𝑡) ∈ ℂ) → (i · (𝑔‘𝑡)) ∈ ℂ)
327	4, 320, 326	sylancr 589	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (i · (𝑔‘𝑡)) ∈ ℂ)
328		abstri 14692	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓‘𝑡) ∈ ℂ ∧ (i · (𝑔‘𝑡)) ∈ ℂ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))))
329	317, 327, 328	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ)) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))))
330	329	anandirs 677	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))))
331		absmul 14656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((i ∈ ℂ ∧ (𝑔‘𝑡) ∈ ℂ) → (abs‘(i · (𝑔‘𝑡))) = ((abs‘i) · (abs‘(𝑔‘𝑡))))
332	4, 320, 331	sylancr 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = ((abs‘i) · (abs‘(𝑔‘𝑡))))
333		absi 14648	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (abs‘i) = 1
334	333	oveq1i 7168	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((abs‘i) · (abs‘(𝑔‘𝑡))) = (1 · (abs‘(𝑔‘𝑡)))
335	332, 334	syl6eq 2874	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = (1 · (abs‘(𝑔‘𝑡))))
336	321	recnd 10671	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ ℂ)
337	336	mulid2d 10661	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (1 · (abs‘(𝑔‘𝑡))) = (abs‘(𝑔‘𝑡)))
338	335, 337	eqtrd 2858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = (abs‘(𝑔‘𝑡)))
339	338	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔‘𝑡))) = (abs‘(𝑔‘𝑡)))
340	339	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓‘𝑡)) + (abs‘(i · (𝑔‘𝑡)))) = ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))))
341	330, 340	breqtrd 5094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ ((abs‘(𝑓‘𝑡)) + (abs‘(𝑔‘𝑡))))
342	318	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ∈ ℝ)
343	321	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ ℝ)
344	240	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
345	250	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . 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))
346	218	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
347	1	ffnd 6517	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
348		fnfvelrn 6850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ran 𝑓)
349	347, 348	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ ran 𝑓)
350		elun1 4154	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓‘𝑡) ∈ ran 𝑓 → (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
351	349, 350	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
352	351	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
353		fnfvima 6997	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
354	224, 353	mp3an1 1444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
355	346, 352, 354	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
356		suprub 11604	. . . . . . . . . . . . . . . . . . . . . . . . 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, < ))
357	345, 355, 356	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓‘𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
358	5	ffnd 6517	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑔 ∈ dom ∫1 → 𝑔 Fn ℝ)
359		fnfvelrn 6850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ran 𝑔)
360	358, 359	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ ran 𝑔)
361		elun2 4155	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔‘𝑡) ∈ ran 𝑔 → (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
362	360, 361	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑔 ∈ dom ∫1 ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
363	362	adantll 712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
364		fnfvima 6997	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
365	224, 364	mp3an1 1444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔‘𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
366	346, 363, 365	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
367		suprub 11604	. . . . . . . . . . . . . . . . . . . . . . . . 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, < ))
368	345, 366, 367	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔‘𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
369	342, 343, 344, 344, 357, 368	le2addd 11261	. . . . . . . . . . . . . . . . . . . . . . 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, < )))
370	293, 324, 325, 341, 369	letrd 10799	. . . . . . . . . . . . . . . . . . . . . 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, < )))
371	301	adantr 483	. . . . . . . . . . . . . . . . . . . . . 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, < )))
372	370, 371	breqtrrd 5096	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
373	50, 372	sylan2 594	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
374		iftrue 4475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) = (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))))
375	374	adantl 484	. . . . . . . . . . . . . . . . . . . 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)))))
376		iftrue 4475	. . . . . . . . . . . . . . . . . . . . 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, < )))
377	376	adantl 484	. . . . . . . . . . . . . . . . . . . 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, < )))
378	373, 375, 377	3brtr4d 5100	. . . . . . . . . . . . . . . . . . 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))
379	378	ex 415	. . . . . . . . . . . . . . . . . 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)))
380	58	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑡 ∈ (𝑢(,)𝑤) → 0 ≤ 0)
381		iffalse 4478	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0) = 0)
382		iffalse 4478	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) = 0)
383	380, 381, 382	3brtr4d 5100	. . . . . . . . . . . . . . . . . 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))
384	379, 383	pm2.61d1 182	. . . . . . . . . . . . . . . . 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))
385	384	ralrimivw 3185	. . . . . . . . . . . . . . . 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))
386		ovex 7191	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ V
387	386, 64	ifex 4517	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ V
388	387	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)
389		eqidd 2824	. . . . . . . . . . . . . . . . 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)))
390	14, 71, 388, 74, 389	ofrfval2 7429	. . . . . . . . . . . . . . . 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)))
391	385, 390	mpbird 259	. . . . . . . . . . . . . . 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)))
392		itg2le 24342	. . . . . . . . . . . . . . 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))))
393	299, 315, 391, 392	syl3anc 1367	. . . . . . . . . . . . . 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))))
394	393	adantr 483	. . . . . . . . . . . . 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))))
395		mblvol 24133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢(,)𝑤) ∈ dom vol → (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤)))
396	34, 395	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤))
397		ovolioo 24171	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol*‘(𝑢(,)𝑤)) = (𝑤 − 𝑢))
398	396, 397	syl5eq 2870	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol‘(𝑢(,)𝑤)) = (𝑤 − 𝑢))
399		resubcl 10952	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑤 − 𝑢) ∈ ℝ)
400	399	ancoms 461	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑤 − 𝑢) ∈ ℝ)
401	400	3adant3 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (𝑤 − 𝑢) ∈ ℝ)
402	398, 401	eqeltrd 2915	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
403		elrege0 12845	. . . . . . . . . . . . . . . . 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, < )))
404	240, 307, 403	sylanbrc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞))
405		ge0addcl 12851	. . . . . . . . . . . . . . . 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[,)+∞))
406	404, 404, 405	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
407	301, 406	eqeltrd 2915	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
408		itg2const 24343	. . . . . . . . . . . . . . 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))))
409	34, 408	mp3an1 1444	. . . . . . . . . . . . . 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))))
410	402, 407, 409	syl2anr 598	. . . . . . . . . . . . 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))))
411	394, 410	breqtrd 5094	. . . . . . . . . . . 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))))
412	411	adantll 712	. . . . . . . . . . 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))))
413	412	adantlr 713	. . . . . . . . . 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))))
414	82	ad3antlr 729	. . . . . . . . . . 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)
415	402	adantl 484	. . . . . . . . . . 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)
416	263	adantll 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
417	416	adantr 483	. . . . . . . . . . 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+)
418	414, 415, 417	ledivmuld 12487	. . . . . . . . . 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)))))
419	413, 418	mpbird 259	. . . . . . . . 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)))
420		abssubge0 14689	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (abs‘(𝑤 − 𝑢)) = (𝑤 − 𝑢))
421	397, 420	eqtr4d 2861	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol*‘(𝑢(,)𝑤)) = (abs‘(𝑤 − 𝑢)))
422	396, 421	syl5eq 2870	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢 ≤ 𝑤) → (vol‘(𝑢(,)𝑤)) = (abs‘(𝑤 − 𝑢)))
423	422	adantl 484	. . . . . . . . 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)))
424	419, 423	breqtrd 5094	. . . . . . . 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)))
425	292, 424	syldan 593	. . . . . . 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)))
426	425	adantllr 717	. . . . . 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)))
427	426	adantlr 713	. . . . 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)))
428	427	adantr 483	. . . 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)))
429		simpr 487	. . . 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, < ))))
430	267, 280, 286, 428, 429	lelttrd 10800	. . 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, < ))))
431	82	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) ∈ ℝ)
432	431	ad3antrrr 728	. . . . 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)
433	126	adantl 484	. . . . 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)
434	416	adantlr 713	. . . . . 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+)
435	434	adantr 483	. . . . 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+)
436	432, 433, 435	ltdiv1d 12479	. . . 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, < )))))
437	436	ad2antrr 724	. . 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, < )))))
438	430, 437	mpbird 259	. 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))
439	198	adantllr 717	. . . 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))
440	439	3adantr3 1167	. . 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))
441	440	adantr 483	. 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))
442	83, 204, 205, 205, 438, 441	lt2addd 11265	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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293  ifcif 4469  {csn 4569   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  dom cdm 5557  ran crn 5558   “ cima 5560   ∘ ccom 5561  Fun wfun 6351   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∘_f cof 7409   ∘_r cofr 7410  Fincfn 8511  supcsup 8906  ℂcc 10537  ℝcr 10538  0cc0 10539  1c1 10540  ici 10541   + caddc 10542   · cmul 10544  +∞cpnf 10674  ℝ^*cxr 10676   < clt 10677   ≤ cle 10678   − cmin 10872   / cdiv 11299  2c2 11695  ℝ⁺crp 12392  (,)cioo 12741  [,)cico 12743  [,]cicc 12744  abscabs 14595  vol*covol 24065  volcvol 24066  ∫₁citg1 24218  ∫₂citg2 24219  𝐿¹cibl 24220  ∫citg 24221  0_𝑝c0p 24272

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-addf 10618

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-disj 5034  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-rlim 14848  df-sum 15045  df-rest 16698  df-topgen 16719  df-psmet 20539  df-xmet 20540  df-met 20541  df-bl 20542  df-mopn 20543  df-top 21504  df-topon 21521  df-bases 21556  df-cmp 21997  df-ovol 24067  df-vol 24068  df-mbf 24222  df-itg1 24223  df-itg2 24224  df-0p 24273

This theorem is referenced by:  ftc1anclem8  34976