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Theorem ftc1anclem7 37411
Description: Lemma for ftc1anc 37413. (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 25691 . . . . . . . . . . 11 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
21ffvelcdmda 7088 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℝ)
32recnd 11281 . . . . . . . . 9 ((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℂ)
4 ax-icn 11206 . . . . . . . . . 10 i ∈ ℂ
5 i1ff 25691 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
65ffvelcdmda 7088 . . . . . . . . . . 11 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℝ)
76recnd 11281 . . . . . . . . . 10 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℂ)
8 mulcl 11231 . . . . . . . . . 10 ((i ∈ ℂ ∧ (𝑔𝑥) ∈ ℂ) → (i · (𝑔𝑥)) ∈ ℂ)
94, 7, 8sylancr 585 . . . . . . . . 9 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (i · (𝑔𝑥)) ∈ ℂ)
10 addcl 11229 . . . . . . . . 9 (((𝑓𝑥) ∈ ℂ ∧ (i · (𝑔𝑥)) ∈ ℂ) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
113, 9, 10syl2an 594 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑥 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑥 ∈ ℝ)) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
1211anandirs 677 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → ((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ)
13 reex 11238 . . . . . . . . 9 ℝ ∈ V
1413a1i 11 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ℝ ∈ V)
152adantlr 713 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (𝑓𝑥) ∈ ℝ)
16 ovexd 7449 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑥 ∈ ℝ) → (i · (𝑔𝑥)) ∈ V)
171feqmptd 6961 . . . . . . . . 9 (𝑓 ∈ dom ∫1𝑓 = (𝑥 ∈ ℝ ↦ (𝑓𝑥)))
1817adantr 479 . . . . . . . 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 5735 . . . . . . . . . . 11 (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i)
2221a1i 11 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (ℝ × {i}) = (𝑥 ∈ ℝ ↦ i))
235feqmptd 6961 . . . . . . . . . 10 (𝑔 ∈ dom ∫1𝑔 = (𝑥 ∈ ℝ ↦ (𝑔𝑥)))
2419, 20, 6, 22, 23offval2 7700 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → ((ℝ × {i}) ∘f · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔𝑥))))
2524adantl 480 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((ℝ × {i}) ∘f · 𝑔) = (𝑥 ∈ ℝ ↦ (i · (𝑔𝑥))))
2614, 15, 16, 18, 25offval2 7700 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓f + ((ℝ × {i}) ∘f · 𝑔)) = (𝑥 ∈ ℝ ↦ ((𝑓𝑥) + (i · (𝑔𝑥)))))
27 absf 15335 . . . . . . . . 9 abs:ℂ⟶ℝ
2827a1i 11 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs:ℂ⟶ℝ)
2928feqmptd 6961 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs = (𝑡 ∈ ℂ ↦ (abs‘𝑡)))
30 fveq2 6891 . . . . . . 7 (𝑡 = ((𝑓𝑥) + (i · (𝑔𝑥))) → (abs‘𝑡) = (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))
3112, 26, 29, 30fmptco 7133 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))))
32 ftc1anclem3 37407 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ∈ dom ∫1)
3331, 32eqeltrrd 2827 . . . . 5 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))) ∈ dom ∫1)
34 ioombl 25580 . . . . 5 (𝑢(,)𝑤) ∈ dom vol
35 fveq2 6891 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑓𝑥) = (𝑓𝑡))
36 fveq2 6891 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
3736oveq2d 7430 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (i · (𝑔𝑥)) = (i · (𝑔𝑡)))
3835, 37oveq12d 7432 . . . . . . . . . . 11 (𝑥 = 𝑡 → ((𝑓𝑥) + (i · (𝑔𝑥))) = ((𝑓𝑡) + (i · (𝑔𝑡))))
3938fveq2d 6895 . . . . . . . . . 10 (𝑥 = 𝑡 → (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
40 eqid 2726 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥))))) = (𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))
41 fvex 6904 . . . . . . . . . 10 (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ V
4239, 40, 41fvmpt 6999 . . . . . . . . 9 (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
4342eqcomd 2732 . . . . . . . 8 (𝑡 ∈ ℝ → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) = ((𝑥 ∈ ℝ ↦ (abs‘((𝑓𝑥) + (i · (𝑔𝑥)))))‘𝑡))
4443ifeq1d 4543 . . . . . . 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 5247 . . . . . 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 25721 . . . . 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 584 . . . 4 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ dom ∫1)
48 breq2 5148 . . . . . . 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 5148 . . . . . . 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 13400 . . . . . . . 8 (𝑡 ∈ (𝑢(,)𝑤) → 𝑡 ∈ ℝ)
51 eleq1w 2809 . . . . . . . . . . . 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 2811 . . . . . . . . . . 11 (𝑥 = 𝑡 → (((𝑓𝑥) + (i · (𝑔𝑥))) ∈ ℂ ↔ ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ))
5452, 53imbi12d 343 . . . . . . . . . 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 1995 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
5655absge0d 15442 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
5750, 56sylan2 591 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
58 0le0 12357 . . . . . . . 8 0 ≤ 0
5958a1i 11 . . . . . . 7 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ 0)
6048, 49, 57, 59ifbothda 4562 . . . . . 6 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
6160ralrimivw 3140 . . . . 5 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ 0 ≤ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
62 ax-resscn 11204 . . . . . . . 8 ℝ ⊆ ℂ
6362a1i 11 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ℝ ⊆ ℂ)
64 c0ex 11247 . . . . . . . . . 10 0 ∈ V
6541, 64ifex 4574 . . . . . . . . 9 if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V
66 eqid 2726 . . . . . . . . 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 6694 . . . . . . . 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 25688 . . . . . 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 5735 . . . . . . . 8 (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
7372a1i 11 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
74 eqidd 2727 . . . . . . 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 7701 . . . . . 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 278 . . . . 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 256 . . . 4 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0𝑝r ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
78 itg2itg1 25752 . . . . 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 25700 . . . . . 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 479 . . . . 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 2826 . . . 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 582 . . 3 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
8382ad6antlr 735 . 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 773 . . . . 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 11303 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ ℝ*)
87 ftc1anc.b . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ ℝ)
8887rexrd 11303 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ ℝ*)
8986, 88jca 510 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
90 df-icc 13377 . . . . . . . . . . . . . . . . . . . . . 22 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑡 ∈ ℝ* ∣ (𝑥𝑡𝑡𝑦)})
9190elixx3g 13383 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑢 ∈ ℝ*) ∧ (𝐴𝑢𝑢𝐵)))
9291simprbi 495 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ (𝐴[,]𝐵) → (𝐴𝑢𝑢𝐵))
9392simpld 493 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴[,]𝐵) → 𝐴𝑢)
9490elixx3g 13383 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) ∧ (𝐴𝑤𝑤𝐵)))
9594simprbi 495 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝐴[,]𝐵) → (𝐴𝑤𝑤𝐵))
9695simprd 494 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴[,]𝐵) → 𝑤𝐵)
9793, 96anim12i 611 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴𝑢𝑤𝐵))
98 ioossioo 13464 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑢𝑤𝐵)) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
9989, 97, 98syl2an 594 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
100 ftc1anc.s . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
101100adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ 𝐷)
10299, 101sstrd 3990 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ 𝐷)
1031023adantr3 1168 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑢(,)𝑤) ⊆ 𝐷)
104103sselda 3979 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
105 ftc1anc.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝐷⟶ℂ)
106105ffvelcdmda 7088 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
107106adantlr 713 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
108104, 107syldan 589 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
109108adantllr 717 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
11055adantll 712 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
11150, 110sylan2 591 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
112111adantlr 713 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
113109, 112subcld 11610 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
114113abscld 15434 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
115114rexrd 11303 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
116113absge0d 15442 . . . . . . . . 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 13480 . . . . . . . . 9 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
118115, 116, 117sylanbrc 581 . . . . . . . 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 13482 . . . . . . . . 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 4559 . . . . . . 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 479 . . . . . 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 7119 . . . . 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 578 . . . 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 13028 . . . . . 6 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
126125rehalfcld 12503 . . . . 5 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
127126ad2antlr 725 . . . 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 765 . . . . . . . 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 3979 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
130129adantllr 717 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
131106adantlr 713 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
132 ftc1anc.d . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐷 ⊆ ℝ)
133132sselda 3979 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → 𝑡 ∈ ℝ)
134133adantlr 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 𝑡 ∈ ℝ)
135134, 110syldan 589 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
136131, 135subcld 11610 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
137136abscld 15434 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
138137rexrd 11303 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
139138adantlr 713 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
140130, 139syldan 589 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
141136absge0d 15442 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
142141adantlr 713 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
143130, 142syldan 589 . . . . . . . . . . . . 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 581 . . . . . . . . . . . 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 4559 . . . . . . . . . . 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 479 . . . . . . . . . 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 7119 . . . . . . . . 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 25748 . . . . . . . . 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 578 . . . . . . 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 13047 . . . . . . . . 9 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
153152rpxrd 13063 . . . . . . . 8 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ*)
154153ad2antlr 725 . . . . . . 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 11246 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℂ)
156106, 155ifclda 4559 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ)
157 subcl 11498 . . . . . . . . . . . . . . . 16 ((if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ ∧ ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
158156, 55, 157syl2an 594 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ)) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
159158anassrs 466 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
160159abscld 15434 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
161160rexrd 11303 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
162159absge0d 15442 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
163 elxrge0 13480 . . . . . . . . . . . 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 581 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞))
165164fmpttd 7119 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))):ℝ⟶(0[,]+∞))
166 itg2cl 25748 . . . . . . . . . 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 728 . . . . . . . 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 479 . . . . . . . . . 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 5147 . . . . . . . . . . . . 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 5147 . . . . . . . . . . . . 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 11819 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
173 iftrue 4530 . . . . . . . . . . . . . . . . . . 19 (𝑡𝐷 → if(𝑡𝐷, (𝐹𝑡), 0) = (𝐹𝑡))
174173fvoveq1d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡𝐷 → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
175174adantl 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
176172, 175breqtrrd 5172 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
177176adantlr 713 . . . . . . . . . . . . . . 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 589 . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 4562 . . . . . . . . . . . 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 3136 . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . 15 (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V
186185, 64ifex 4574 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V)
189 eqidd 2727 . . . . . . . . . . . . 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 2727 . . . . . . . . . . . . 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 7701 . . . . . . . . . . . 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 724 . . . . . . . . . . 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 256 . . . . . . . . . 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 25755 . . . . . . . . . 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 1368 . . . . . . . . 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 578 . . . . . . . 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 774 . . . . . . . 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 13185 . . . . . . 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 13175 . . . . . 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 717 . . . . 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 1168 . . . 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 25754 . . . 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 1368 . . 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 479 . 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 729 . 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 479 . . . . . . . 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 13025 . . . . . . . . 9 2 ∈ ℝ+
208 imassrn 6071 . . . . . . . . . . . . . . . 16 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ran abs
209 frn 6725 . . . . . . . . . . . . . . . . 17 (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
21027, 209ax-mp 5 . . . . . . . . . . . . . . . 16 ran abs ⊆ ℝ
211208, 210sstri 3989 . . . . . . . . . . . . . . 15 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ
212211a1i 11 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ)
2131frnd 6726 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ dom ∫1 → ran 𝑓 ⊆ ℝ)
214213adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑓 ⊆ ℝ)
2155frnd 6726 . . . . . . . . . . . . . . . . . . 19 (𝑔 ∈ dom ∫1 → ran 𝑔 ⊆ ℝ)
216215adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑔 ⊆ ℝ)
217214, 216unssd 4185 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℝ)
218217, 62sstrdi 3992 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
219 i1f0rn 25697 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ dom ∫1 → 0 ∈ ran 𝑓)
220 elun1 4175 . . . . . . . . . . . . . . . . . 18 (0 ∈ ran 𝑓 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
221219, 220syl 17 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
222221adantr 479 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
223 ffn 6718 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
22427, 223ax-mp 5 . . . . . . . . . . . . . . . . 17 abs Fn ℂ
225 fnfvima 7240 . . . . . . . . . . . . . . . . 17 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
226224, 225mp3an1 1445 . . . . . . . . . . . . . . . 16 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
227218, 222, 226syl2anc 582 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
228227ne0d 4336 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
229 ffun 6721 . . . . . . . . . . . . . . . . 17 (abs:ℂ⟶ℝ → Fun abs)
23027, 229ax-mp 5 . . . . . . . . . . . . . . . 16 Fun abs
231 i1frn 25692 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → ran 𝑓 ∈ Fin)
232 i1frn 25692 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ dom ∫1 → ran 𝑔 ∈ Fin)
233 unfi 9200 . . . . . . . . . . . . . . . . 17 ((ran 𝑓 ∈ Fin ∧ ran 𝑔 ∈ Fin) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
234231, 232, 233syl2an 594 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
235 imafi 9346 . . . . . . . . . . . . . . . 16 ((Fun abs ∧ (ran 𝑓 ∪ ran 𝑔) ∈ Fin) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
236230, 234, 235sylancr 585 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
237 fimaxre2 12203 . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥)
239 suprcl 12218 . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
241240adantr 479 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
242 0red 11256 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 ∈ ℝ)
243218sselda 3979 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → 𝑟 ∈ ℂ)
244243abscld 15434 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ ℝ)
245244adantrr 715 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ∈ ℝ)
246 absgt0 15322 . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . 14 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) ∧ 𝑟 ≠ 0) → 0 < (abs‘𝑟))
249248anasss 465 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < (abs‘𝑟))
250212, 228, 2383jca 1125 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 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 7240 . . . . . . . . . . . . . . . . 17 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
253224, 252mp3an1 1445 . . . . . . . . . . . . . . . 16 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
254218, 253sylan 578 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
255 suprub 12219 . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
257256adantrr 715 . . . . . . . . . . . . 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 11413 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
259241, 258elrpd 13059 . . . . . . . . . . 11 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
260259rexlimdvaa 3146 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0 → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+))
261260imp 405 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
262 rpmulcl 13043 . . . . . . . . 9 ((2 ∈ ℝ+ ∧ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
263207, 261, 262sylancr 585 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
264206, 263rerpdivcld 13093 . . . . . . 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 712 . . . . . 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 713 . . . . 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 728 . . . 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 781 . . . . . 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 13452 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
27085, 87, 269syl2anc 582 . . . . . . . . . . . 12 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
271270, 62sstrdi 3992 . . . . . . . . . . 11 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
272271sselda 3979 . . . . . . . . . 10 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℂ)
273271sselda 3979 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℂ)
274 subcl 11498 . . . . . . . . . 10 ((𝑤 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑤𝑢) ∈ ℂ)
275272, 273, 274syl2anr 595 . . . . . . . . 9 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑𝑤 ∈ (𝐴[,]𝐵))) → (𝑤𝑢) ∈ ℂ)
276275anandis 676 . . . . . . . 8 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑤𝑢) ∈ ℂ)
277276abscld 15434 . . . . . . 7 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘(𝑤𝑢)) ∈ ℝ)
2782773adantr3 1168 . . . . . 6 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘(𝑤𝑢)) ∈ ℝ)
279268, 278sylan 578 . . . . 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 479 . . . 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 13045 . . . . . . . . 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 595 . . . . . . . 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 13062 . . . . . . 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 716 . . . . . 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 717 . . . . 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 724 . . . 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 3978 . . . . . . . . . . 11 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) → 𝑢 ∈ ℝ))
288270sseld 3978 . . . . . . . . . . 11 (𝜑 → (𝑤 ∈ (𝐴[,]𝐵) → 𝑤 ∈ ℝ))
289 idd 24 . . . . . . . . . . 11 (𝜑 → (𝑢𝑤𝑢𝑤))
290287, 288, 2893anim123d 1440 . . . . . . . . . 10 (𝜑 → ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤) → (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)))
291290ad2antrr 724 . . . . . . . . 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 405 . . . . . . . 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 15434 . . . . . . . . . . . . . . . . . . 19 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
294293rexrd 11303 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ*)
295 elxrge0 13480 . . . . . . . . . . . . . . . . . 18 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ↔ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
296294, 56, 295sylanbrc 581 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞))
297 ifcl 4569 . . . . . . . . . . . . . . . . 17 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
298296, 119, 297sylancl 584 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
299298fmpttd 7119 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞))
300240recnd 11281 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℂ)
3013002timesd 12499 . . . . . . . . . . . . . . . . . . 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 11282 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ)
303302rexrd 11303 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ*)
304 abs0 15283 . . . . . . . . . . . . . . . . . . . . . . 23 (abs‘0) = 0
305304, 227eqeltrrid 2831 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
306 suprub 12219 . . . . . . . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
308240, 240, 307, 307addge0d 11829 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ≤ (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
309 elxrge0 13480 . . . . . . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
311301, 310eqeltrd 2826 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,]+∞))
312 ifcl 4569 . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
314313adantr 479 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) ∈ (0[,]+∞))
315314fmpttd 7119 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0)):ℝ⟶(0[,]+∞))
3161ffvelcdmda 7088 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℝ)
317316recnd 11281 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℂ)
318317abscld 15434 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ ℝ)
3195ffvelcdmda 7088 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℝ)
320319recnd 11281 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℂ)
321320abscld 15434 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℝ)
322 readdcl 11230 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((abs‘(𝑓𝑡)) ∈ ℝ ∧ (abs‘(𝑔𝑡)) ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
323318, 321, 322syl2an 594 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
324323anandirs 677 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
325302adantr 479 . . . . . . . . . . . . . . . . . . . . . . 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 11231 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (i · (𝑔𝑡)) ∈ ℂ)
3274, 320, 326sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (i · (𝑔𝑡)) ∈ ℂ)
328 abstri 15328 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓𝑡) ∈ ℂ ∧ (i · (𝑔𝑡)) ∈ ℂ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
329317, 327, 328syl2an 594 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
330329anandirs 677 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
331 absmul 15292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
3324, 320, 331sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
333 absi 15284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (abs‘i) = 1
334333oveq1i 7424 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((abs‘i) · (abs‘(𝑔𝑡))) = (1 · (abs‘(𝑔𝑡)))
335332, 334eqtrdi 2782 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (1 · (abs‘(𝑔𝑡))))
336321recnd 11281 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℂ)
337336mullidd 11271 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (1 · (abs‘(𝑔𝑡))) = (abs‘(𝑔𝑡)))
338335, 337eqtrd 2766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
339338adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
340339oveq2d 7430 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))) = ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
341330, 340breqtrd 5170 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
342318adantlr 713 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ ℝ)
343321adantll 712 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℝ)
344240adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
345250adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
3471ffnd 6719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
348 fnfvelrn 7084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
349347, 348sylan 578 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
350 elun1 4175 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓𝑡) ∈ ran 𝑓 → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
351349, 350syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
352351adantlr 713 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
353 fnfvima 7240 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
354224, 353mp3an1 1445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
355346, 352, 354syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
356 suprub 12219 . . . . . . . . . . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
3585ffnd 6719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
359 fnfvelrn 7084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑔 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
360358, 359sylan 578 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
361 elun2 4176 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑔𝑡) ∈ ran 𝑔 → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
362360, 361syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
363362adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
364 fnfvima 7240 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
365224, 364mp3an1 1445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
366346, 363, 365syl2anc 582 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
367 suprub 12219 . . . . . . . . . . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
369342, 343, 344, 344, 357, 368le2addd 11872 . . . . . . . . . . . . . . . . . . . . . . 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 11410 . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . 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 5172 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
37350, 372sylan2 591 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
374 iftrue 4530 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
375374adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
376 iftrue 4530 . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 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 5176 . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . 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 4533 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
382 iffalse 4533 . . . . . . . . . . . . . . . . . . 19 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )), 0) = 0)
383380, 381, 3823brtr4d 5176 . . . . . . . . . . . . . . . . . 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 180 . . . . . . . . . . . . . . . . 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 3140 . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . . . 19 (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ V
387386, 64ifex 4574 . . . . . . . . . . . . . . . . . 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 2727 . . . . . . . . . . . . . . . . 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 7701 . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . 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 25755 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 25545 . . . . . . . . . . . . . . . . 17 ((𝑢(,)𝑤) ∈ dom vol → (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤)))
39634, 395ax-mp 5 . . . . . . . . . . . . . . . 16 (vol‘(𝑢(,)𝑤)) = (vol*‘(𝑢(,)𝑤))
397 ovolioo 25583 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol*‘(𝑢(,)𝑤)) = (𝑤𝑢))
398396, 397eqtrid 2778 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) = (𝑤𝑢))
399 resubcl 11563 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑤𝑢) ∈ ℝ)
400399ancoms 457 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (𝑤𝑢) ∈ ℝ)
4014003adant3 1129 . . . . . . . . . . . . . . 15 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (𝑤𝑢) ∈ ℝ)
402398, 401eqeltrd 2826 . . . . . . . . . . . . . 14 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
403 elrege0 13477 . . . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ (0[,)+∞))
405 ge0addcl 13483 . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) + sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
407301, 406eqeltrd 2826 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ (0[,)+∞))
408 itg2const 25756 . . . . . . . . . . . . . . 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 1445 . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . 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 5170 . . . . . . . . . . . 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 712 . . . . . . . . . . 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 713 . . . . . . . . . 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 729 . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ (𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤)) → (vol‘(𝑢(,)𝑤)) ∈ ℝ)
416263adantll 712 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
417416adantr 479 . . . . . . . . . . 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 13115 . . . . . . . . . 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 256 . . . . . . . . 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 15325 . . . . . . . . . . . 12 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (abs‘(𝑤𝑢)) = (𝑤𝑢))
421397, 420eqtr4d 2769 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol*‘(𝑢(,)𝑤)) = (abs‘(𝑤𝑢)))
422396, 421eqtrid 2778 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ 𝑢𝑤) → (vol‘(𝑢(,)𝑤)) = (abs‘(𝑤𝑢)))
423422adantl 480 . . . . . . . . 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 5170 . . . . . . . 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 589 . . . . . . 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 717 . . . . . 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 713 . . . . 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 479 . . . 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 483 . . . 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 11411 . . 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 480 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
432431ad3antrrr 728 . . . . 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 480 . . . . 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 713 . . . . . 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 479 . . . . 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 13107 . . . 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 724 . . 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 256 . 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 717 . . . 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 1168 . . 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 479 . 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 11876 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 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3463  cun 3945  wss 3947  c0 4323  ifcif 4524  {csn 4624   class class class wbr 5144  cmpt 5227   × cxp 5671  dom cdm 5673  ran crn 5674  cima 5676  ccom 5677  Fun wfun 6538   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7414  f cof 7678  r cofr 7679  Fincfn 8964  supcsup 9474  cc 11145  cr 11146  0cc0 11147  1c1 11148  ici 11149   + caddc 11150   · cmul 11152  +∞cpnf 11284  *cxr 11286   < clt 11287  cle 11288  cmin 11483   / cdiv 11910  2c2 12311  +crp 13020  (,)cioo 13370  [,)cico 13372  [,]cicc 13373  abscabs 15232  vol*covol 25477  volcvol 25478  1citg1 25630  2citg2 25631  𝐿1cibl 25632  citg 25633  0𝑝c0p 25684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-inf2 9675  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224  ax-pre-sup 11225  ax-addf 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-disj 5112  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7680  df-ofr 7681  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-er 8724  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9445  df-sup 9476  df-inf 9477  df-oi 9544  df-dju 9935  df-card 9973  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-div 11911  df-nn 12257  df-2 12319  df-3 12320  df-n0 12517  df-z 12603  df-uz 12867  df-q 12977  df-rp 13021  df-xneg 13138  df-xadd 13139  df-xmul 13140  df-ioo 13374  df-ico 13376  df-icc 13377  df-fz 13531  df-fzo 13674  df-fl 13804  df-seq 14014  df-exp 14074  df-hash 14341  df-cj 15097  df-re 15098  df-im 15099  df-sqrt 15233  df-abs 15234  df-clim 15483  df-rlim 15484  df-sum 15684  df-rest 17430  df-topgen 17451  df-psmet 21329  df-xmet 21330  df-met 21331  df-bl 21332  df-mopn 21333  df-top 22882  df-topon 22899  df-bases 22935  df-cmp 23377  df-ovol 25479  df-vol 25480  df-mbf 25634  df-itg1 25635  df-itg2 25636  df-0p 25685
This theorem is referenced by:  ftc1anclem8  37412
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