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Theorem ftc1anclem7 35593
Description: Lemma for ftc1anc 35595. (Contributed by Brendan Leahy, 13-May-2018.)
Hypotheses
Ref Expression
ftc1anc.g 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)
ftc1anc.a (𝜑𝐴 ∈ ℝ)
ftc1anc.b (𝜑𝐵 ∈ ℝ)
ftc1anc.le (𝜑𝐴𝐵)
ftc1anc.s (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
ftc1anc.d (𝜑𝐷 ⊆ ℝ)
ftc1anc.i (𝜑𝐹 ∈ 𝐿1)
ftc1anc.f (𝜑𝐹:𝐷⟶ℂ)
Assertion
Ref Expression
ftc1anclem7 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 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
StepHypRef Expression
1 i1ff 24573 . . . . . . . . . . 11 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
21ffvelrnda 6904 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℝ)
32recnd 10861 . . . . . . . . 9 ((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℂ)
4 ax-icn 10788 . . . . . . . . . 10 i ∈ ℂ
5 i1ff 24573 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
65ffvelrnda 6904 . . . . . . . . . . 11 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℝ)
76recnd 10861 . . . . . . . . . 10 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℂ)
8 mulcl 10813 . . . . . . . . . 10 ((i ∈ ℂ ∧ (𝑔𝑥) ∈ ℂ) → (i · (𝑔𝑥)) ∈ ℂ)
94, 7, 8sylancr 590 . . . . . . . . 9 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (i · (𝑔𝑥)) ∈ ℂ)
10 addcl 10811 . . . . . . . . 9 (((𝑓𝑥) ∈ ℂ ∧ (i · (𝑔𝑥)) ∈ ℂ) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
113, 9, 10syl2an 599 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑥 ∈ ℝ)) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
1211anandirs 679 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
13 reex 10820 . . . . . . . . 9 ℝ ∈ V
1413a1i 11 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ℝ ∈ V)
152adantlr 715 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℝ)
16 ovexd 7248 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (i · (𝑔𝑥)) ∈ V)
171feqmptd 6780 . . . . . . . . 9 (𝑓 ∈ dom ∫1𝑓 = (𝑥 ∈ ℝ ↦ (𝑓𝑥)))
1817adantr 484 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 = (𝑥 ∈ ℝ ↦ (𝑓𝑥)))
1913a1i 11 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → ℝ ∈ V)
204a1i 11 . . . . . . . . . 10 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → i ∈ ℂ)
21 fconstmpt 5611 . . . . . . . . . . 11 (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i)
2221a1i 11 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i))
235feqmptd 6780 . . . . . . . . . 10 (𝑔 ∈ dom ∫1𝑔 = (𝑥 ∈ ℝ ↦ (𝑔𝑥)))
2419, 20, 6, 22, 23offval2 7488 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → ((ℝ × {i}) ∘f · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔𝑥))))
2524adantl 485 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((ℝ × {i}) ∘f · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔𝑥))))
2614, 15, 16, 18, 25offval2 7488 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓f + ((ℝ × {i}) ∘f · 𝑔)) = (𝑥 ∈ ℝ ↦ ((𝑓𝑥) + (i · (𝑔𝑥)))))
27 absf 14901 . . . . . . . . 9 abs:ℂ⟶ℝ
2827a1i 11 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs:ℂ⟶ℝ)
2928feqmptd 6780 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs = (𝑡 ∈ ℂ ↦ (abs‘𝑡)))
30 fveq2 6717 . . . . . . 7 (𝑡 = ((𝑓𝑥) + (i · (𝑔𝑥))) → (abs‘𝑡) = (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))
3112, 26, 29, 30fmptco 6944 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))))
32 ftc1anclem3 35589 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ∈ dom ∫1)
3331, 32eqeltrrd 2839 . . . . 5 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))) ∈ dom ∫1)
34 ioombl 24462 . . . . 5 (𝑢(,)𝑤) ∈ dom vol
35 fveq2 6717 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑓𝑥) = (𝑓𝑡))
36 fveq2 6717 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
3736oveq2d 7229 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (i · (𝑔𝑥)) = (i · (𝑔𝑡)))
3835, 37oveq12d 7231 . . . . . . . . . . 11 (𝑥 = 𝑡 → ((𝑓𝑥) + (i · (𝑔𝑥))) = ((𝑓𝑡) + (i · (𝑔𝑡))))
3938fveq2d 6721 . . . . . . . . . 10 (𝑥 = 𝑡 → (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
40 eqid 2737 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))
41 fvex 6730 . . . . . . . . . 10 (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ V
4239, 40, 41fvmpt 6818 . . . . . . . . 9 (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
4342eqcomd 2743 . . . . . . . 8 (𝑡 ∈ ℝ → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) = ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡))
4443ifeq1d 4458 . . . . . . 7 (𝑡 ∈ ℝ → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = if(𝑡 ∈ (𝑢(,)𝑤), ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡), 0))
4544mpteq2ia 5146 . . . . . 6 (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡), 0))
4645i1fres 24603 . . . . 5 (((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))) ∈ dom ∫1 ∧ (𝑢(,)𝑤) ∈ dom vol) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1)
4733, 34, 46sylancl 589 . . . 4 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1)
48 breq2 5057 . . . . . . 7 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → (0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ↔ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
49 breq2 5057 . . . . . . 7 (0 = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
50 elioore 12965 . . . . . . . 8 (𝑡 ∈ (𝑢(,)𝑤) → 𝑡 ∈ ℝ)
51 eleq1w 2820 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑥 ∈ ℝ ↔ 𝑡 ∈ ℝ))
5251anbi2d 632 . . . . . . . . . . 11 (𝑥 = 𝑡 → (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) ↔ ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)))
5338eleq1d 2822 . . . . . . . . . . 11 (𝑥 = 𝑡 → (((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ ↔ ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ))
5452, 53imbi12d 348 . . . . . . . . . 10 (𝑥 = 𝑡 → ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ) ↔ (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)))
5554, 12chvarvv 2007 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
5655absge0d 15008 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
5750, 56sylan2 596 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
58 0le0 11931 . . . . . . . 8 0 ≤ 0
5958a1i 11 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ 0)
6048, 49, 57, 59ifbothda 4477 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
6160ralrimivw 3106 . . . . 5 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
62 ax-resscn 10786 . . . . . . . 8 ℝ ⊆ ℂ
6362a1i 11 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ℝ ⊆ ℂ)
64 c0ex 10827 . . . . . . . . . 10 0 ∈ V
6541, 64ifex 4489 . . . . . . . . 9 if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V
66 eqid 2737 . . . . . . . . 9 (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
6765, 66fnmpti 6521 . . . . . . . 8 (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) Fn ℝ
6867a1i 11 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) Fn ℝ)
6963, 680pledm 24570 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (0𝑝r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ↔ (ℝ × {0}) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))))
7064a1i 11 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ∈ V)
7165a1i 11 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V)
72 fconstmpt 5611 . . . . . . . 8 (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
7372a1i 11 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
74 eqidd 2738 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
7514, 70, 71, 73, 74ofrfval2 7489 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((ℝ × {0}) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
7669, 75bitrd 282 . . . . 5 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (0𝑝r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
7761, 76mpbird 260 . . . 4 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0𝑝r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
78 itg2itg1 24634 . . . . 5 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) = (∫1‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))))
79 itg1cl 24582 . . . . . 6 ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1 → (∫1‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
8079adantr 484 . . . . 5 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) → (∫1‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
8178, 80eqeltrd 2838 . . . 4 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
8247, 77, 81syl2anc 587 . . 3 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
8382ad6antlr 737 . 2 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
84 simplll 775 . . . . 5 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)))
85 ftc1anc.a . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ∈ ℝ)
8685rexrd 10883 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ*)
87 ftc1anc.b . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ ℝ)
8887rexrd 10883 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ*)
8986, 88jca 515 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
90 df-icc 12942 . . . . . . . . . . . . . . . . . . . . . 22 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑡 ∈ ℝ* ∣ (𝑥𝑡𝑡𝑦)})
9190elixx3g 12948 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑢 ∈ ℝ*) ∧ (𝐴𝑢𝑢𝐵)))
9291simprbi 500 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ (𝐴[,]𝐵) → (𝐴𝑢𝑢𝐵))
9392simpld 498 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴[,]𝐵) → 𝐴𝑢)
9490elixx3g 12948 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) ∧ (𝐴𝑤𝑤𝐵)))
9594simprbi 500 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝐴[,]𝐵) → (𝐴𝑤𝑤𝐵))
9695simprd 499 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴[,]𝐵) → 𝑤𝐵)
9793, 96anim12i 616 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴𝑢𝑤𝐵))
98 ioossioo 13029 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑢𝑤𝐵)) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
9989, 97, 98syl2an 599 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
100 ftc1anc.s . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
101100adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ 𝐷)
10299, 101sstrd 3911 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ 𝐷)
1031023adantr3 1173 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑢(,)𝑤) ⊆ 𝐷)
104103sselda 3901 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
105 ftc1anc.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝐷⟶ℂ)
106105ffvelrnda 6904 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
107106adantlr 715 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
108104, 107syldan 594 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
109108adantllr 719 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
11055adantll 714 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
11150, 110sylan2 596 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
112111adantlr 715 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
113109, 112subcld 11189 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
114113abscld 15000 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
115114rexrd 10883 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
116113absge0d 15008 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
117 elxrge0 13045 . . . . . . . . 9 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
118115, 116, 117sylanbrc 586 . . . . . . . 8 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞))
119 0e0iccpnf 13047 . . . . . . . . 9 0 ∈ (0[,]+∞)
120119a1i 11 . . . . . . . 8 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
121118, 120ifclda 4474 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,]+∞))
122121adantr 484 . . . . . 6 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,]+∞))
123122fmpttd 6932 . . . . 5 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞))
12484, 123sylan 583 . . . 4 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞))
125 rpre 12594 . . . . . 6 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
126125rehalfcld 12077 . . . . 5 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
127126ad2antlr 727 . . . 4 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑦 / 2) ∈ ℝ)
128 simpll 767 . . . . . . . 8 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) → (𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)))
129102sselda 3901 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
130129adantllr 719 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
131106adantlr 715 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
132 ftc1anc.d . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐷 ⊆ ℝ)
133132sselda 3901 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → 𝑡 ∈ ℝ)
134133adantlr 715 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 𝑡 ∈ ℝ)
135134, 110syldan 594 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
136131, 135subcld 11189 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
137136abscld 15000 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
138137rexrd 10883 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
139138adantlr 715 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
140130, 139syldan 594 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
141136absge0d 15008 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
142141adantlr 715 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
143130, 142syldan 594 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
144140, 143, 117sylanbrc 586 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞))
145119a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
146144, 145ifclda 4474 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,]+∞))
147146adantr 484 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,]+∞))
148147fmpttd 6932 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞))
149 itg2cl 24630 . . . . . . . . 9 ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ*)
150148, 149syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ*)
151128, 150sylan 583 . . . . . . 7 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ*)
152 rphalfcl 12613 . . . . . . . . 9 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
153152rpxrd 12629 . . . . . . . 8 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ*)
154153ad2antlr 727 . . . . . . 7 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑦 / 2) ∈ ℝ*)
155 0cnd 10826 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℂ)
156106, 155ifclda 4474 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ)
157 subcl 11077 . . . . . . . . . . . . . . . 16 ((if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ ∧ ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
158156, 55, 157syl2an 599 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
159158anassrs 471 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
160159abscld 15000 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
161160rexrd 10883 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
162159absge0d 15008 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
163 elxrge0 13045 . . . . . . . . . . . 12 ((abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞) ↔ ((abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ* ∧ 0 ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
164161, 162, 163sylanbrc 586 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞))
165164fmpttd 6932 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))):ℝ⟶(0[,]+∞))
166 itg2cl 24630 . . . . . . . . . 10 ((𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))):ℝ⟶(0[,]+∞) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) ∈ ℝ*)
167165, 166syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) ∈ ℝ*)
168167ad3antrrr 730 . . . . . . . 8 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) ∈ ℝ*)
169165adantr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))):ℝ⟶(0[,]+∞))
170 breq1 5056 . . . . . . . . . . . . 13 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) → ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
171 breq1 5056 . . . . . . . . . . . . 13 (0 = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) → (0 ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
172137leidd 11398 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
173 iftrue 4445 . . . . . . . . . . . . . . . . . . 19 (𝑡𝐷 → if(𝑡𝐷, (𝐹𝑡), 0) = (𝐹𝑡))
174173fvoveq1d 7235 . . . . . . . . . . . . . . . . . 18 (𝑡𝐷 → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
175174adantl 485 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
176172, 175breqtrrd 5081 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
177176adantlr 715 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
178130, 177syldan 594 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
179178adantlr 715 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
180162adantlr 715 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
181180adantr 484 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
182170, 171, 179, 181ifbothda 4477 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
183182ralrimiva 3105 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
18413a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℝ ∈ V)
185 fvex 6730 . . . . . . . . . . . . . . 15 (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V
186185, 64ifex 4489 . . . . . . . . . . . . . 14 if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ V
187186a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ V)
188 fvexd 6732 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V)
189 eqidd 2738 . . . . . . . . . . . . 13 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
190 eqidd 2738 . . . . . . . . . . . . 13 (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))) = (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
191184, 187, 188, 189, 190ofrfval2 7489 . . . . . . . . . . . 12 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
192191ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
193183, 192mpbird 260 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
194 itg2le 24637 . . . . . . . . . 10 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))))
195148, 169, 193, 194syl3anc 1373 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))))
196128, 195sylan 583 . . . . . . . 8 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))))
197 simpllr 776 . . . . . . . 8 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2))
198151, 168, 154, 196, 197xrlelttrd 12750 . . . . . . 7 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) < (𝑦 / 2))
199151, 154, 198xrltled 12740 . . . . . 6 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (𝑦 / 2))
200199adantllr 719 . . . . 5 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (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‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (𝑦 / 2))
2012003adantr3 1173 . . . 4 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (𝑦 / 2))
202 itg2lecl 24636 . . . 4 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑦 / 2) ∈ ℝ ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (𝑦 / 2)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ)
203124, 127, 201, 202syl3anc 1373 . . 3 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ)
204203adantr 484 . 2 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ)
205126ad3antlr 731 . 2 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (𝑦 / 2) ∈ ℝ)
20682adantr 484 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
207 2rp 12591 . . . . . . . . 9 2 ∈ ℝ+
208 imassrn 5940 . . . . . . . . . . . . . . . 16 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ran abs
209 frn 6552 . . . . . . . . . . . . . . . . 17 (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
21027, 209ax-mp 5 . . . . . . . . . . . . . . . 16 ran abs ⊆ ℝ
211208, 210sstri 3910 . . . . . . . . . . . . . . 15 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ
212211a1i 11 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ)
2131frnd 6553 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ dom ∫1 → ran 𝑓 ⊆ ℝ)
214213adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑓 ⊆ ℝ)
2155frnd 6553 . . . . . . . . . . . . . . . . . . 19 (𝑔 ∈ dom ∫1 → ran 𝑔 ⊆ ℝ)
216215adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑔 ⊆ ℝ)
217214, 216unssd 4100 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℝ)
218217, 62sstrdi 3913 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
219 i1f0rn 24579 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ dom ∫1 → 0 ∈ ran 𝑓)
220 elun1 4090 . . . . . . . . . . . . . . . . . 18 (0 ∈ ran 𝑓 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
221219, 220syl 17 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
222221adantr 484 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
223 ffn 6545 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
22427, 223ax-mp 5 . . . . . . . . . . . . . . . . 17 abs Fn ℂ
225 fnfvima 7049 . . . . . . . . . . . . . . . . 17 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
226224, 225mp3an1 1450 . . . . . . . . . . . . . . . 16 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
227218, 222, 226syl2anc 587 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
228227ne0d 4250 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
229 ffun 6548 . . . . . . . . . . . . . . . . 17 (abs:ℂ⟶ℝ → Fun abs)
23027, 229ax-mp 5 . . . . . . . . . . . . . . . 16 Fun abs
231 i1frn 24574 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → ran 𝑓 ∈ Fin)
232 i1frn 24574 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ dom ∫1 → ran 𝑔 ∈ Fin)
233 unfi 8850 . . . . . . . . . . . . . . . . 17 ((ran 𝑓 ∈ Fin ∧ ran 𝑔 ∈ Fin) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
234231, 232, 233syl2an 599 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
235 imafi 8853 . . . . . . . . . . . . . . . 16 ((Fun abs ∧ (ran 𝑓 ∪ ran 𝑔) ∈ Fin) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
236230, 234, 235sylancr 590 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
237 fimaxre2 11777 . . . . . . . . . . . . . . 15 (((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥)
238211, 236, 237sylancr 590 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥)
239 suprcl 11792 . . . . . . . . . . . . . 14 (((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
240212, 228, 238, 239syl3anc 1373 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
241240adantr 484 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
242 0red 10836 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 ∈ ℝ)
243218sselda 3901 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → 𝑟 ∈ ℂ)
244243abscld 15000 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ ℝ)
245244adantrr 717 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ∈ ℝ)
246 absgt0 14888 . . . . . . . . . . . . . . . 16 (𝑟 ∈ ℂ → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
247243, 246syl 17 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
248247biimpa 480 . . . . . . . . . . . . . 14 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) ∧ 𝑟 ≠ 0) → 0 < (abs‘𝑟))
249248anasss 470 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < (abs‘𝑟))
250212, 228, 2383jca 1130 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥))
251250adantr 484 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ 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))𝑦𝑥))
252 fnfvima 7049 . . . . . . . . . . . . . . . . 17 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
253224, 252mp3an1 1450 . . . . . . . . . . . . . . . 16 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
254218, 253sylan 583 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
255 suprub 11793 . . . . . . . . . . . . . . 15 ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥) ∧ (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
256251, 254, 255syl2anc 587 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
257256adantrr 717 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
258242, 245, 241, 249, 257ltletrd 10992 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
259241, 258elrpd 12625 . . . . . . . . . . 11 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
260259rexlimdvaa 3204 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0 → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+))
261260imp 410 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
262 rpmulcl 12609 . . . . . . . . 9 ((2 ∈ ℝ+ ∧ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
263207, 261, 262sylancr 590 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
264206, 263rerpdivcld 12659 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
265264adantll 714 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
266265adantlr 715 . . . . 5 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
267266ad3antrrr 730 . . . 4 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
268 simp-4l 783 . . . . . 6 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → 𝜑)
269 iccssre 13017 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
27085, 87, 269syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
271270, 62sstrdi 3913 . . . . . . . . . . 11 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
272271sselda 3901 . . . . . . . . . 10 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℂ)
273271sselda 3901 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℂ)
274 subcl 11077 . . . . . . . . . 10 ((𝑤 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑤𝑢) ∈ ℂ)
275272, 273, 274syl2anr 600 . . . . . . . . 9 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑𝑤 ∈ (𝐴[,]𝐵))) → (𝑤𝑢) ∈ ℂ)
276275anandis 678 . . . . . . . 8 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑤𝑢) ∈ ℂ)
277276abscld 15000 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘(𝑤𝑢)) ∈ ℝ)
2782773adantr3 1173 . . . . . 6 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘(𝑤𝑢)) ∈ ℝ)
279268, 278sylan 583 . . . . 5 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘(𝑤𝑢)) ∈ ℝ)
280279adantr 484 . . . 4 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (abs‘(𝑤𝑢)) ∈ ℝ)
281 rpdivcl 12611 . . . . . . . . 9 (((𝑦 / 2) ∈ ℝ+ ∧ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
282152, 263, 281syl2anr 600 . . . . . . . 8 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
283282rpred 12628 . . . . . . 7 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
284283adantlll 718 . . . . . 6 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
285284adantllr 719 . . . . 5 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
286285ad2antrr 726 . . . 4 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
287270sseld 3900 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) → 𝑢 ∈ ℝ))
288270sseld 3900 . . . . . . . . . . 11 (𝜑 → (𝑤 ∈ (𝐴[,]𝐵) → 𝑤 ∈ ℝ))
289 idd 24 . . . . . . . . . . 11 (𝜑 → (𝑢𝑤𝑢𝑤))
290287, 288, 2893anim123d 1445 . . . . . . . . . 10 (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)))
291290ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)))
292291imp 410 . . . . . . . 8 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤))
29355abscld 15000 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
294293rexrd 10883 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ*)
295 elxrge0 13045 . . . . . . . . . . . . . . . . . 18 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ↔ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
296294, 56, 295sylanbrc 586 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞))
297 ifcl 4484 . . . . . . . . . . . . . . . . 17 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
298296, 119, 297sylancl 589 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
299298fmpttd 6932 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞))
300240recnd 10861 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℂ)
3013002timesd 12073 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) = (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
302240, 240readdcld 10862 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
303302rexrd 10883 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ*)
304 abs0 14849 . . . . . . . . . . . . . . . . . . . . . . 23 (abs‘0) = 0
305304, 227eqeltrrid 2843 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
306 suprub 11793 . . . . . . . . . . . . . . . . . . . . . 22 ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥) ∧ 0 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
307250, 305, 306syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
308240, 240, 307, 307addge0d 11408 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
309 elxrge0 13045 . . . . . . . . . . . . . . . . . . . 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, < ))))
310303, 308, 309sylanbrc 586 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
311301, 310eqeltrd 2838 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
312 ifcl 4484 . . . . . . . . . . . . . . . . . 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[,]+∞))
313311, 119, 312sylancl 589 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
314313adantr 484 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
315314fmpttd 6932 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)):ℝ⟶(0[,]+∞))
3161ffvelrnda 6904 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℝ)
317316recnd 10861 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℂ)
318317abscld 15000 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ ℝ)
3195ffvelrnda 6904 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℝ)
320319recnd 10861 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℂ)
321320abscld 15000 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℝ)
322 readdcl 10812 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((abs‘(𝑓𝑡)) ∈ ℝ ∧ (abs‘(𝑔𝑡)) ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
323318, 321, 322syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
324323anandirs 679 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
325302adantr 484 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
326 mulcl 10813 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (i · (𝑔𝑡)) ∈ ℂ)
3274, 320, 326sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (i · (𝑔𝑡)) ∈ ℂ)
328 abstri 14894 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑡) ∈ ℂ ∧ (i · (𝑔𝑡)) ∈ ℂ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
329317, 327, 328syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
330329anandirs 679 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
331 absmul 14858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
3324, 320, 331sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
333 absi 14850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (abs‘i) = 1
334333oveq1i 7223 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((abs‘i) · (abs‘(𝑔𝑡))) = (1 · (abs‘(𝑔𝑡)))
335332, 334eqtrdi 2794 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (1 · (abs‘(𝑔𝑡))))
336321recnd 10861 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℂ)
337336mulid2d 10851 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (1 · (abs‘(𝑔𝑡))) = (abs‘(𝑔𝑡)))
338335, 337eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
339338adantll 714 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
340339oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))) = ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
341330, 340breqtrd 5079 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
342318adantlr 715 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ ℝ)
343321adantll 714 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℝ)
344240adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
345250adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥))
346218adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
3471ffnd 6546 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
348 fnfvelrn 6901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
349347, 348sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
350 elun1 4090 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓𝑡) ∈ ran 𝑓 → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
351349, 350syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
352351adantlr 715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
353 fnfvima 7049 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
354224, 353mp3an1 1450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
355346, 352, 354syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
356 suprub 11793 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥) ∧ (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → (abs‘(𝑓𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
357345, 355, 356syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
3585ffnd 6546 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
359 fnfvelrn 6901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑔 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
360358, 359sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
361 elun2 4091 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔𝑡) ∈ ran 𝑔 → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
362360, 361syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
363362adantll 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
364 fnfvima 7049 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
365224, 364mp3an1 1450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
366346, 363, 365syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
367 suprub 11793 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥) ∧ (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))) → (abs‘(𝑔𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
368345, 366, 367syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
369342, 343, 344, 344, 357, 368le2addd 11451 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
370293, 324, 325, 341, 369letrd 10989 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
371301adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 ∈ dom ∫1𝑔 ∈ 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, < )))
372370, 371breqtrrd 5081 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
37350, 372sylan2 596 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
374 iftrue 4445 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
375374adantl 485 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
376 iftrue 4445 . . . . . . . . . . . . . . . . . . . . 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, < )))
377376adantl 485 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑔 ∈ 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, < )))
378373, 375, 3773brtr4d 5085 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
379378ex 416 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)))
38058a1i 11 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ (𝑢(,)𝑤) → 0 ≤ 0)
381 iffalse 4448 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
382 iffalse 4448 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) = 0)
383380, 381, 3823brtr4d 5085 . . . . . . . . . . . . . . . . . 18 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
384379, 383pm2.61d1 183 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
385384ralrimivw 3106 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))
386 ovex 7246 . . . . . . . . . . . . . . . . . . 19 (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ V
387386, 64ifex 4489 . . . . . . . . . . . . . . . . . 18 if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ V
388387a1i 11 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ V)
389 eqidd 2738 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ 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)))
39014, 71, 388, 74, 389ofrfval2 7489 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)))
391385, 390mpbird 260 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)))
392 itg2le 24637 . . . . . . . . . . . . . . 15 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 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‘((𝑓𝑡) + (i · (𝑔𝑡)))), 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‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))))
393299, 315, 391, 392syl3anc 1373 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))))
394393adantr 484 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0))))
395 mblvol 24427 . . . . . . . . . . . . . . . . 17 ((𝑢(,)𝑤) ∈ dom vol → (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤)))
39634, 395ax-mp 5 . . . . . . . . . . . . . . . 16 (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤))
397 ovolioo 24465 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol*‘(𝑢(,)𝑤)) = (𝑤𝑢))
398396, 397syl5eq 2790 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) = (𝑤𝑢))
399 resubcl 11142 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑤𝑢) ∈ ℝ)
400399ancoms 462 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑤𝑢) ∈ ℝ)
4014003adant3 1134 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (𝑤𝑢) ∈ ℝ)
402398, 401eqeltrd 2838 . . . . . . . . . . . . . 14 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
403 elrege0 13042 . . . . . . . . . . . . . . . . 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, < )))
404240, 307, 403sylanbrc 586 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞))
405 ge0addcl 13048 . . . . . . . . . . . . . . . 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[,)+∞))
406404, 404, 405syl2anc 587 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
407301, 406eqeltrd 2838 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
408 itg2const 24638 . . . . . . . . . . . . . . 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))))
40934, 408mp3an1 1450 . . . . . . . . . . . . . 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))))
410402, 407, 409syl2anr 600 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (∫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))))
411394, 410breqtrd 5079 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
412411adantll 714 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
413412adantlr 715 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤))))
41482ad3antlr 731 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
415402adantl 485 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
416263adantll 714 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
417416adantr 484 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
418414, 415, 417ledivmuld 12681 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (vol‘(𝑢(,)𝑤)) ↔ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ ((2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) · (vol‘(𝑢(,)𝑤)))))
419413, 418mpbird 260 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (vol‘(𝑢(,)𝑤)))
420 abssubge0 14891 . . . . . . . . . . . 12 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (abs‘(𝑤𝑢)) = (𝑤𝑢))
421397, 420eqtr4d 2780 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol*‘(𝑢(,)𝑤)) = (abs‘(𝑤𝑢)))
422396, 421syl5eq 2790 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) = (abs‘(𝑤𝑢)))
423422adantl 485 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (vol‘(𝑢(,)𝑤)) = (abs‘(𝑤𝑢)))
424419, 423breqtrd 5079 . . . . . . . 8 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤𝑢)))
425292, 424syldan 594 . . . . . . 7 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤𝑢)))
426425adantllr 719 . . . . . 6 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤𝑢)))
427426adantlr 715 . . . . 5 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤𝑢)))
428427adantr 484 . . . 4 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ≤ (abs‘(𝑤𝑢)))
429 simpr 488 . . . 4 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))))
430267, 280, 286, 428, 429lelttrd 10990 . . 3 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))))
43182adantl 485 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
432431ad3antrrr 730 . . . . 5 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
433126adantl 485 . . . . 5 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (𝑦 / 2) ∈ ℝ)
434416adantlr 715 . . . . . 6 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
435434adantr 484 . . . . 5 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
436432, 433, 435ltdiv1d 12673 . . . 4 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) < (𝑦 / 2) ↔ ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
437436ad2antrr 726 . . 3 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) < (𝑦 / 2) ↔ ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
438430, 437mpbird 260 . 2 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) < (𝑦 / 2))
439198adantllr 719 . . . 4 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (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‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) < (𝑦 / 2))
4404393adantr3 1173 . . 3 ((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) < (𝑦 / 2))
441440adantr 484 . 2 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) < (𝑦 / 2))
44283, 204, 205, 205, 438, 441lt2addd 11455 1 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))) < ((𝑦 / 2) + (𝑦 / 2)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  Vcvv 3408  cun 3864  wss 3866  c0 4237  ifcif 4439  {csn 4541   class class class wbr 5053  cmpt 5135   × cxp 5549  dom cdm 5551  ran crn 5552  cima 5554  ccom 5555  Fun wfun 6374   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  f cof 7467  r cofr 7468  Fincfn 8626  supcsup 9056  cc 10727  cr 10728  0cc0 10729  1c1 10730  ici 10731   + caddc 10732   · cmul 10734  +∞cpnf 10864  *cxr 10866   < clt 10867  cle 10868  cmin 11062   / cdiv 11489  2c2 11885  +crp 12586  (,)cioo 12935  [,)cico 12937  [,]cicc 12938  abscabs 14797  vol*covol 24359  volcvol 24360  1citg1 24512  2citg2 24513  𝐿1cibl 24514  citg 24515  0𝑝c0p 24566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-addf 10808
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-disj 5019  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-ofr 7470  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ioo 12939  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-rlim 15050  df-sum 15250  df-rest 16927  df-topgen 16948  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-top 21791  df-topon 21808  df-bases 21843  df-cmp 22284  df-ovol 24361  df-vol 24362  df-mbf 24516  df-itg1 24517  df-itg2 24518  df-0p 24567
This theorem is referenced by:  ftc1anclem8  35594
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