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Theorem ftc1anclem7 34504
Description: Lemma for ftc1anc 34506. (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 23960 . . . . . . . . . . 11 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
21ffvelrnda 6716 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℝ)
32recnd 10515 . . . . . . . . 9 ((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℂ)
4 ax-icn 10442 . . . . . . . . . 10 i ∈ ℂ
5 i1ff 23960 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
65ffvelrnda 6716 . . . . . . . . . . 11 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℝ)
76recnd 10515 . . . . . . . . . 10 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℂ)
8 mulcl 10467 . . . . . . . . . 10 ((i ∈ ℂ ∧ (𝑔𝑥) ∈ ℂ) → (i · (𝑔𝑥)) ∈ ℂ)
94, 7, 8sylancr 587 . . . . . . . . 9 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (i · (𝑔𝑥)) ∈ ℂ)
10 addcl 10465 . . . . . . . . 9 (((𝑓𝑥) ∈ ℂ ∧ (i · (𝑔𝑥)) ∈ ℂ) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
113, 9, 10syl2an 595 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑥 ∈ ℝ)) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
1211anandirs 675 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
13 reex 10474 . . . . . . . . 9 ℝ ∈ V
1413a1i 11 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ℝ ∈ V)
152adantlr 711 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℝ)
16 ovexd 7050 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (i · (𝑔𝑥)) ∈ V)
171feqmptd 6601 . . . . . . . . 9 (𝑓 ∈ dom ∫1𝑓 = (𝑥 ∈ ℝ ↦ (𝑓𝑥)))
1817adantr 481 . . . . . . . 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 5500 . . . . . . . . . . 11 (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i)
2221a1i 11 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i))
235feqmptd 6601 . . . . . . . . . 10 (𝑔 ∈ dom ∫1𝑔 = (𝑥 ∈ ℝ ↦ (𝑔𝑥)))
2419, 20, 6, 22, 23offval2 7284 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → ((ℝ × {i}) ∘𝑓 · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔𝑥))))
2524adantl 482 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((ℝ × {i}) ∘𝑓 · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔𝑥))))
2614, 15, 16, 18, 25offval2 7284 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓𝑓 + ((ℝ × {i}) ∘𝑓 · 𝑔)) = (𝑥 ∈ ℝ ↦ ((𝑓𝑥) + (i · (𝑔𝑥)))))
27 absf 14531 . . . . . . . . 9 abs:ℂ⟶ℝ
2827a1i 11 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs:ℂ⟶ℝ)
2928feqmptd 6601 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs = (𝑡 ∈ ℂ ↦ (abs‘𝑡)))
30 fveq2 6538 . . . . . . 7 (𝑡 = ((𝑓𝑥) + (i · (𝑔𝑥))) → (abs‘𝑡) = (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))
3112, 26, 29, 30fmptco 6754 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓𝑓 + ((ℝ × {i}) ∘𝑓 · 𝑔))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))))
32 ftc1anclem3 34500 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓𝑓 + ((ℝ × {i}) ∘𝑓 · 𝑔))) ∈ dom ∫1)
3331, 32eqeltrrd 2884 . . . . 5 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))) ∈ dom ∫1)
34 ioombl 23849 . . . . 5 (𝑢(,)𝑤) ∈ dom vol
35 fveq2 6538 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑓𝑥) = (𝑓𝑡))
36 fveq2 6538 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
3736oveq2d 7032 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (i · (𝑔𝑥)) = (i · (𝑔𝑡)))
3835, 37oveq12d 7034 . . . . . . . . . . 11 (𝑥 = 𝑡 → ((𝑓𝑥) + (i · (𝑔𝑥))) = ((𝑓𝑡) + (i · (𝑔𝑡))))
3938fveq2d 6542 . . . . . . . . . 10 (𝑥 = 𝑡 → (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
40 eqid 2795 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))
41 fvex 6551 . . . . . . . . . 10 (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ V
4239, 40, 41fvmpt 6635 . . . . . . . . 9 (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
4342eqcomd 2801 . . . . . . . 8 (𝑡 ∈ ℝ → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) = ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡))
4443ifeq1d 4399 . . . . . . 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 5051 . . . . . 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 23989 . . . . 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 586 . . . 4 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1)
48 breq2 4966 . . . . . . 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 4966 . . . . . . 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 12618 . . . . . . . 8 (𝑡 ∈ (𝑢(,)𝑤) → 𝑡 ∈ ℝ)
51 eleq1w 2865 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑥 ∈ ℝ ↔ 𝑡 ∈ ℝ))
5251anbi2d 628 . . . . . . . . . . 11 (𝑥 = 𝑡 → (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) ↔ ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)))
5338eleq1d 2867 . . . . . . . . . . 11 (𝑥 = 𝑡 → (((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ ↔ ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ))
5452, 53imbi12d 346 . . . . . . . . . 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, 12chvarv 2370 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
5655absge0d 14638 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
5750, 56sylan2 592 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
58 0le0 11586 . . . . . . . 8 0 ≤ 0
5958a1i 11 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ 0)
6048, 49, 57, 59ifbothda 4418 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
6160ralrimivw 3150 . . . . 5 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
62 ax-resscn 10440 . . . . . . . 8 ℝ ⊆ ℂ
6362a1i 11 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ℝ ⊆ ℂ)
64 c0ex 10481 . . . . . . . . . 10 0 ∈ V
6541, 64ifex 4429 . . . . . . . . 9 if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V
66 eqid 2795 . . . . . . . . 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 6359 . . . . . . . 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 23957 . . . . . 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 5500 . . . . . . . 8 (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
7372a1i 11 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
74 eqidd 2796 . . . . . . 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 7285 . . . . . 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 280 . . . . 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 258 . . . 4 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
78 itg2itg1 24020 . . . . 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 23969 . . . . . 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 481 . . . . 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 2883 . . . 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 584 . . 3 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
8382ad6antlr 733 . 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 771 . . . . 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 10537 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ*)
87 ftc1anc.b . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ ℝ)
8887rexrd 10537 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ*)
8986, 88jca 512 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
90 df-icc 12595 . . . . . . . . . . . . . . . . . . . . . 22 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑡 ∈ ℝ* ∣ (𝑥𝑡𝑡𝑦)})
9190elixx3g 12601 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑢 ∈ ℝ*) ∧ (𝐴𝑢𝑢𝐵)))
9291simprbi 497 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ (𝐴[,]𝐵) → (𝐴𝑢𝑢𝐵))
9392simpld 495 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴[,]𝐵) → 𝐴𝑢)
9490elixx3g 12601 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) ∧ (𝐴𝑤𝑤𝐵)))
9594simprbi 497 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝐴[,]𝐵) → (𝐴𝑤𝑤𝐵))
9695simprd 496 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴[,]𝐵) → 𝑤𝐵)
9793, 96anim12i 612 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴𝑢𝑤𝐵))
98 ioossioo 12679 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑢𝑤𝐵)) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
9989, 97, 98syl2an 595 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
100 ftc1anc.s . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
101100adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ 𝐷)
10299, 101sstrd 3899 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ 𝐷)
1031023adantr3 1164 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑢(,)𝑤) ⊆ 𝐷)
104103sselda 3889 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
105 ftc1anc.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝐷⟶ℂ)
106105ffvelrnda 6716 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
107106adantlr 711 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
108104, 107syldan 591 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
109108adantllr 715 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
11055adantll 710 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
11150, 110sylan2 592 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
112111adantlr 711 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
113109, 112subcld 10845 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
114113abscld 14630 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
115114rexrd 10537 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
116113absge0d 14638 . . . . . . . . 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 12695 . . . . . . . . 9 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
118115, 116, 117sylanbrc 583 . . . . . . . 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 12697 . . . . . . . . 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 4415 . . . . . . 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 481 . . . . . 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 6742 . . . . 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 580 . . . 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 12247 . . . . . 6 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
126125rehalfcld 11732 . . . . 5 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
127126ad2antlr 723 . . . 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 763 . . . . . . . 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 3889 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
130129adantllr 715 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
131106adantlr 711 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
132 ftc1anc.d . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐷 ⊆ ℝ)
133132sselda 3889 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → 𝑡 ∈ ℝ)
134133adantlr 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 𝑡 ∈ ℝ)
135134, 110syldan 591 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
136131, 135subcld 10845 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
137136abscld 14630 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
138137rexrd 10537 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
139138adantlr 711 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
140130, 139syldan 591 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
141136absge0d 14638 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
142141adantlr 711 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
143130, 142syldan 591 . . . . . . . . . . . . 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 583 . . . . . . . . . . . 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 4415 . . . . . . . . . . 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 481 . . . . . . . . . 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 6742 . . . . . . . . 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 24016 . . . . . . . . 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 580 . . . . . . 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 12266 . . . . . . . . 9 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
153152rpxrd 12282 . . . . . . . 8 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ*)
154153ad2antlr 723 . . . . . . 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 10480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℂ)
156106, 155ifclda 4415 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ)
157 subcl 10732 . . . . . . . . . . . . . . . 16 ((if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ ∧ ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
158156, 55, 157syl2an 595 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
159158anassrs 468 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
160159abscld 14630 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
161160rexrd 10537 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
162159absge0d 14638 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
163 elxrge0 12695 . . . . . . . . . . . 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 583 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞))
165164fmpttd 6742 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))):ℝ⟶(0[,]+∞))
166 itg2cl 24016 . . . . . . . . . 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 726 . . . . . . . 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 481 . . . . . . . . . 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 4965 . . . . . . . . . . . . 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 4965 . . . . . . . . . . . . 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 11054 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
173 iftrue 4387 . . . . . . . . . . . . . . . . . . 19 (𝑡𝐷 → if(𝑡𝐷, (𝐹𝑡), 0) = (𝐹𝑡))
174173fvoveq1d 7038 . . . . . . . . . . . . . . . . . 18 (𝑡𝐷 → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
175174adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
176172, 175breqtrrd 4990 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
177176adantlr 711 . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 4418 . . . . . . . . . . . 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 3149 . . . . . . . . . . 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 6551 . . . . . . . . . . . . . . 15 (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V
186185, 64ifex 4429 . . . . . . . . . . . . . 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 6553 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V)
189 eqidd 2796 . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . 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 7285 . . . . . . . . . . . 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 722 . . . . . . . . . . 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 258 . . . . . . . . . 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 24023 . . . . . . . . . 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 1364 . . . . . . . . 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 580 . . . . . . . 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 772 . . . . . . . 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 12403 . . . . . . 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 12393 . . . . . 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 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))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (𝑦 / 2))
2012003adantr3 1164 . . . 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 24022 . . . 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 1364 . . 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 481 . 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 727 . 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 481 . . . . . . . 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 12244 . . . . . . . . 9 2 ∈ ℝ+
208 imassrn 5817 . . . . . . . . . . . . . . . 16 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ran abs
209 frn 6388 . . . . . . . . . . . . . . . . 17 (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
21027, 209ax-mp 5 . . . . . . . . . . . . . . . 16 ran abs ⊆ ℝ
211208, 210sstri 3898 . . . . . . . . . . . . . . 15 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ
212211a1i 11 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ)
2131frnd 6389 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ dom ∫1 → ran 𝑓 ⊆ ℝ)
214213adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑓 ⊆ ℝ)
2155frnd 6389 . . . . . . . . . . . . . . . . . . 19 (𝑔 ∈ dom ∫1 → ran 𝑔 ⊆ ℝ)
216215adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑔 ⊆ ℝ)
217214, 216unssd 4083 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℝ)
218217, 62syl6ss 3901 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
219 i1f0rn 23966 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ dom ∫1 → 0 ∈ ran 𝑓)
220 elun1 4073 . . . . . . . . . . . . . . . . . 18 (0 ∈ ran 𝑓 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
221219, 220syl 17 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
222221adantr 481 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
223 ffn 6382 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
22427, 223ax-mp 5 . . . . . . . . . . . . . . . . 17 abs Fn ℂ
225 fnfvima 6860 . . . . . . . . . . . . . . . . 17 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
226224, 225mp3an1 1440 . . . . . . . . . . . . . . . 16 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
227218, 222, 226syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
228227ne0d 4221 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
229 ffun 6385 . . . . . . . . . . . . . . . . 17 (abs:ℂ⟶ℝ → Fun abs)
23027, 229ax-mp 5 . . . . . . . . . . . . . . . 16 Fun abs
231 i1frn 23961 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → ran 𝑓 ∈ Fin)
232 i1frn 23961 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ dom ∫1 → ran 𝑔 ∈ Fin)
233 unfi 8631 . . . . . . . . . . . . . . . . 17 ((ran 𝑓 ∈ Fin ∧ ran 𝑔 ∈ Fin) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
234231, 232, 233syl2an 595 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
235 imafi 8663 . . . . . . . . . . . . . . . 16 ((Fun abs ∧ (ran 𝑓 ∪ ran 𝑔) ∈ Fin) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
236230, 234, 235sylancr 587 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
237 fimaxre2 11434 . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥)
239 suprcl 11449 . . . . . . . . . . . . . 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 1364 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
241240adantr 481 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
242 0red 10490 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 ∈ ℝ)
243218sselda 3889 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → 𝑟 ∈ ℂ)
244243abscld 14630 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ ℝ)
245244adantrr 713 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ∈ ℝ)
246 absgt0 14518 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . 14 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) ∧ 𝑟 ≠ 0) → 0 < (abs‘𝑟))
249248anasss 467 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < (abs‘𝑟))
250212, 228, 2383jca 1121 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 6860 . . . . . . . . . . . . . . . . 17 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
253224, 252mp3an1 1440 . . . . . . . . . . . . . . . 16 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
254218, 253sylan 580 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
255 suprub 11450 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
257256adantrr 713 . . . . . . . . . . . . 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 10647 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
259241, 258elrpd 12278 . . . . . . . . . . 11 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
260259rexlimdvaa 3248 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0 → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+))
261260imp 407 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
262 rpmulcl 12262 . . . . . . . . 9 ((2 ∈ ℝ+ ∧ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
263207, 261, 262sylancr 587 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
264206, 263rerpdivcld 12312 . . . . . . 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 710 . . . . . 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 711 . . . . 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 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, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
268 simp-4l 779 . . . . . 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 12668 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
27085, 87, 269syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
271270, 62syl6ss 3901 . . . . . . . . . . 11 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
272271sselda 3889 . . . . . . . . . 10 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℂ)
273271sselda 3889 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℂ)
274 subcl 10732 . . . . . . . . . 10 ((𝑤 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑤𝑢) ∈ ℂ)
275272, 273, 274syl2anr 596 . . . . . . . . 9 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑𝑤 ∈ (𝐴[,]𝐵))) → (𝑤𝑢) ∈ ℂ)
276275anandis 674 . . . . . . . 8 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑤𝑢) ∈ ℂ)
277276abscld 14630 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘(𝑤𝑢)) ∈ ℝ)
2782773adantr3 1164 . . . . . 6 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘(𝑤𝑢)) ∈ ℝ)
279268, 278sylan 580 . . . . 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 481 . . . 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 12264 . . . . . . . . 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 596 . . . . . . . 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 12281 . . . . . . 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 714 . . . . . 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 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+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ)
286285ad2antrr 722 . . . 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 3888 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) → 𝑢 ∈ ℝ))
288270sseld 3888 . . . . . . . . . . 11 (𝜑 → (𝑤 ∈ (𝐴[,]𝐵) → 𝑤 ∈ ℝ))
289 idd 24 . . . . . . . . . . 11 (𝜑 → (𝑢𝑤𝑢𝑤))
290287, 288, 2893anim123d 1435 . . . . . . . . . 10 (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)))
291290ad2antrr 722 . . . . . . . . 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 407 . . . . . . . 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 14630 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
294293rexrd 10537 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ*)
295 elxrge0 12695 . . . . . . . . . . . . . . . . . 18 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ↔ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
296294, 56, 295sylanbrc 583 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞))
297 ifcl 4425 . . . . . . . . . . . . . . . . 17 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
298296, 119, 297sylancl 586 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
299298fmpttd 6742 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞))
300240recnd 10515 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℂ)
3013002timesd 11728 . . . . . . . . . . . . . . . . . . 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 10516 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
303302rexrd 10537 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ*)
304 abs0 14479 . . . . . . . . . . . . . . . . . . . . . . 23 (abs‘0) = 0
305304, 227syl5eqelr 2888 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
306 suprub 11450 . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
308240, 240, 307, 307addge0d 11064 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
309 elxrge0 12695 . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
311301, 310eqeltrd 2883 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
312 ifcl 4425 . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
314313adantr 481 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
315314fmpttd 6742 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)):ℝ⟶(0[,]+∞))
3161ffvelrnda 6716 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℝ)
317316recnd 10515 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℂ)
318317abscld 14630 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ ℝ)
3195ffvelrnda 6716 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℝ)
320319recnd 10515 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℂ)
321320abscld 14630 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℝ)
322 readdcl 10466 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((abs‘(𝑓𝑡)) ∈ ℝ ∧ (abs‘(𝑔𝑡)) ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
323318, 321, 322syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
324323anandirs 675 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
325302adantr 481 . . . . . . . . . . . . . . . . . . . . . . 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 10467 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (i · (𝑔𝑡)) ∈ ℂ)
3274, 320, 326sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (i · (𝑔𝑡)) ∈ ℂ)
328 abstri 14524 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑡) ∈ ℂ ∧ (i · (𝑔𝑡)) ∈ ℂ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
329317, 327, 328syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
330329anandirs 675 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
331 absmul 14488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
3324, 320, 331sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
333 absi 14480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (abs‘i) = 1
334333oveq1i 7026 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((abs‘i) · (abs‘(𝑔𝑡))) = (1 · (abs‘(𝑔𝑡)))
335332, 334syl6eq 2847 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (1 · (abs‘(𝑔𝑡))))
336321recnd 10515 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℂ)
337336mulid2d 10505 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (1 · (abs‘(𝑔𝑡))) = (abs‘(𝑔𝑡)))
338335, 337eqtrd 2831 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
339338adantll 710 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
340339oveq2d 7032 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))) = ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
341330, 340breqtrd 4988 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
342318adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ ℝ)
343321adantll 710 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℝ)
344240adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
345250adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
3471ffnd 6383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
348 fnfvelrn 6713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
349347, 348sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
350 elun1 4073 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓𝑡) ∈ ran 𝑓 → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
351349, 350syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
352351adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
353 fnfvima 6860 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
354224, 353mp3an1 1440 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
355346, 352, 354syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
356 suprub 11450 . . . . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
3585ffnd 6383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
359 fnfvelrn 6713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑔 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
360358, 359sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
361 elun2 4074 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔𝑡) ∈ ran 𝑔 → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
362360, 361syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
363362adantll 710 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
364 fnfvima 6860 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
365224, 364mp3an1 1440 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
366346, 363, 365syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
367 suprub 11450 . . . . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
369342, 343, 344, 344, 357, 368le2addd 11107 . . . . . . . . . . . . . . . . . . . . . . 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 10644 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 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 4990 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
37350, 372sylan2 592 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
374 iftrue 4387 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
375374adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
376 iftrue 4387 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . 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 4994 . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . 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 4390 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
382 iffalse 4390 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) = 0)
383380, 381, 3823brtr4d 4994 . . . . . . . . . . . . . . . . . 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 181 . . . . . . . . . . . . . . . . 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 3150 . . . . . . . . . . . . . . . 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 7048 . . . . . . . . . . . . . . . . . . 19 (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ V
387386, 64ifex 4429 . . . . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . . . . 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 7285 . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . 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 24023 . . . . . . . . . . . . . . 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 1364 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 23814 . . . . . . . . . . . . . . . . 17 ((𝑢(,)𝑤) ∈ dom vol → (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤)))
39634, 395ax-mp 5 . . . . . . . . . . . . . . . 16 (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤))
397 ovolioo 23852 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol*‘(𝑢(,)𝑤)) = (𝑤𝑢))
398396, 397syl5eq 2843 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) = (𝑤𝑢))
399 resubcl 10798 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑤𝑢) ∈ ℝ)
400399ancoms 459 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑤𝑢) ∈ ℝ)
4014003adant3 1125 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (𝑤𝑢) ∈ ℝ)
402398, 401eqeltrd 2883 . . . . . . . . . . . . . 14 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
403 elrege0 12692 . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞))
405 ge0addcl 12698 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
407301, 406eqeltrd 2883 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
408 itg2const 24024 . . . . . . . . . . . . . . 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 1440 . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . 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 4988 . . . . . . . . . . . 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 710 . . . . . . . . . . 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 711 . . . . . . . . . 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 727 . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
416263adantll 710 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
417416adantr 481 . . . . . . . . . . 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 12334 . . . . . . . . . 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 258 . . . . . . . . 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 14521 . . . . . . . . . . . 12 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (abs‘(𝑤𝑢)) = (𝑤𝑢))
421397, 420eqtr4d 2834 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol*‘(𝑢(,)𝑤)) = (abs‘(𝑤𝑢)))
422396, 421syl5eq 2843 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) = (abs‘(𝑤𝑢)))
423422adantl 482 . . . . . . . . 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 4988 . . . . . . . 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 591 . . . . . . 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 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) ∧ (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 711 . . . . 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 481 . . . 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 485 . . . 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 10645 . . 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 482 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
432431ad3antrrr 726 . . . . 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 482 . . . . 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 711 . . . . . 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 481 . . . . 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 12326 . . . 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 722 . . 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 258 . 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 715 . . . 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 1164 . . 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 481 . 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 11111 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  Vcvv 3437  cun 3857  wss 3859  c0 4211  ifcif 4381  {csn 4472   class class class wbr 4962  cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444  cima 5446  ccom 5447  Fun wfun 6219   Fn wfn 6220  wf 6221  cfv 6225  (class class class)co 7016  𝑓 cof 7265  𝑟 cofr 7266  Fincfn 8357  supcsup 8750  cc 10381  cr 10382  0cc0 10383  1c1 10384  ici 10385   + caddc 10386   · cmul 10388  +∞cpnf 10518  *cxr 10520   < clt 10521  cle 10522  cmin 10717   / cdiv 11145  2c2 11540  +crp 12239  (,)cioo 12588  [,)cico 12590  [,]cicc 12591  abscabs 14427  vol*covol 23746  volcvol 23747  1citg1 23899  2citg2 23900  𝐿1cibl 23901  citg 23902  0𝑝c0p 23953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-addf 10462
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-disj 4931  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-ofr 7268  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fi 8721  df-sup 8752  df-inf 8753  df-oi 8820  df-dju 9176  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ioo 12592  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-fl 13012  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-sum 14877  df-rest 16525  df-topgen 16546  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-top 21186  df-topon 21203  df-bases 21238  df-cmp 21679  df-ovol 23748  df-vol 23749  df-mbf 23903  df-itg1 23904  df-itg2 23905  df-0p 23954
This theorem is referenced by:  ftc1anclem8  34505
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