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Theorem ftc1anclem8 35082
Description: Lemma for ftc1anc 35083. (Contributed by Brendan Leahy, 29-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
ftc1anclem8 (((((((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))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) < 𝑦)
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 ftc1anclem8
StepHypRef Expression
1 ftc1anc.g . . 3 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡)
2 ftc1anc.a . . 3 (𝜑𝐴 ∈ ℝ)
3 ftc1anc.b . . 3 (𝜑𝐵 ∈ ℝ)
4 ftc1anc.le . . 3 (𝜑𝐴𝐵)
5 ftc1anc.s . . 3 (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
6 ftc1anc.d . . 3 (𝜑𝐷 ⊆ ℝ)
7 ftc1anc.i . . 3 (𝜑𝐹 ∈ 𝐿1)
8 ftc1anc.f . . 3 (𝜑𝐹:𝐷⟶ℂ)
91, 2, 3, 4, 5, 6, 7, 8ftc1anclem7 35081 . 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))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))) < ((𝑦 / 2) + (𝑦 / 2)))
10 simplll 774 . . . 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+) → (𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)))
11 3simpa 1145 . . . 4 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤) → (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)))
12 ioossre 12795 . . . . . . . . 9 (𝑢(,)𝑤) ⊆ ℝ
1312a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑢(,)𝑤) ⊆ ℝ)
14 rembl 24147 . . . . . . . . 9 ℝ ∈ dom vol
1514a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ℝ ∈ dom vol)
16 fvex 6674 . . . . . . . . . 10 (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ V
17 c0ex 10633 . . . . . . . . . 10 0 ∈ V
1816, 17ifex 4498 . . . . . . . . 9 if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V
1918a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V)
20 eldifn 4090 . . . . . . . . . 10 (𝑡 ∈ (ℝ ∖ (𝑢(,)𝑤)) → ¬ 𝑡 ∈ (𝑢(,)𝑤))
2120adantl 485 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (ℝ ∖ (𝑢(,)𝑤))) → ¬ 𝑡 ∈ (𝑢(,)𝑤))
2221iffalsed 4461 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (ℝ ∖ (𝑢(,)𝑤))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
23 iftrue 4456 . . . . . . . . . . . 12 (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
2423mpteq2ia 5143 . . . . . . . . . . 11 (𝑡 ∈ (𝑢(,)𝑤) ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡 ∈ (𝑢(,)𝑤) ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
25 resmpt 5892 . . . . . . . . . . . 12 ((𝑢(,)𝑤) ⊆ ℝ → ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ↾ (𝑢(,)𝑤)) = (𝑡 ∈ (𝑢(,)𝑤) ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
2612, 25ax-mp 5 . . . . . . . . . . 11 ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ↾ (𝑢(,)𝑤)) = (𝑡 ∈ (𝑢(,)𝑤) ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
2724, 26eqtr4i 2850 . . . . . . . . . 10 (𝑡 ∈ (𝑢(,)𝑤) ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ↾ (𝑢(,)𝑤))
28 i1ff 24283 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
2928ffvelrnda 6842 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℝ)
3029recnd 10667 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℂ)
31 ax-icn 10594 . . . . . . . . . . . . . . . . 17 i ∈ ℂ
32 i1ff 24283 . . . . . . . . . . . . . . . . . . 19 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
3332ffvelrnda 6842 . . . . . . . . . . . . . . . . . 18 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℝ)
3433recnd 10667 . . . . . . . . . . . . . . . . 17 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℂ)
35 mulcl 10619 . . . . . . . . . . . . . . . . 17 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (i · (𝑔𝑡)) ∈ ℂ)
3631, 34, 35sylancr 590 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (i · (𝑔𝑡)) ∈ ℂ)
37 addcl 10617 . . . . . . . . . . . . . . . 16 (((𝑓𝑡) ∈ ℂ ∧ (i · (𝑔𝑡)) ∈ ℂ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
3830, 36, 37syl2an 598 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
3938anandirs 678 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
40 reex 10626 . . . . . . . . . . . . . . . 16 ℝ ∈ V
4140a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ℝ ∈ V)
4229adantlr 714 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℝ)
4336adantll 713 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (i · (𝑔𝑡)) ∈ ℂ)
4428feqmptd 6724 . . . . . . . . . . . . . . . 16 (𝑓 ∈ dom ∫1𝑓 = (𝑡 ∈ ℝ ↦ (𝑓𝑡)))
4544adantr 484 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓 = (𝑡 ∈ ℝ ↦ (𝑓𝑡)))
4640a1i 11 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ dom ∫1 → ℝ ∈ V)
4731a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → i ∈ ℂ)
48 fconstmpt 5601 . . . . . . . . . . . . . . . . . 18 (ℝ × {i}) = (𝑡 ∈ ℝ ↦ i)
4948a1i 11 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ dom ∫1 → (ℝ × {i}) = (𝑡 ∈ ℝ ↦ i))
5032feqmptd 6724 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ dom ∫1𝑔 = (𝑡 ∈ ℝ ↦ (𝑔𝑡)))
5146, 47, 33, 49, 50offval2 7420 . . . . . . . . . . . . . . . 16 (𝑔 ∈ dom ∫1 → ((ℝ × {i}) ∘f · 𝑔) = (𝑡 ∈ ℝ ↦ (i · (𝑔𝑡))))
5251adantl 485 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((ℝ × {i}) ∘f · 𝑔) = (𝑡 ∈ ℝ ↦ (i · (𝑔𝑡))))
5341, 42, 43, 45, 52offval2 7420 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓f + ((ℝ × {i}) ∘f · 𝑔)) = (𝑡 ∈ ℝ ↦ ((𝑓𝑡) + (i · (𝑔𝑡)))))
54 absf 14697 . . . . . . . . . . . . . . . 16 abs:ℂ⟶ℝ
5554a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs:ℂ⟶ℝ)
5655feqmptd 6724 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
57 fveq2 6661 . . . . . . . . . . . . . 14 (𝑥 = ((𝑓𝑡) + (i · (𝑔𝑡))) → (abs‘𝑥) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
5839, 53, 56, 57fmptco 6882 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) = (𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
59 ftc1anclem3 35077 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ∈ dom ∫1)
6058, 59eqeltrrd 2917 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ dom ∫1)
61 i1fmbf 24282 . . . . . . . . . . . 12 ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ MblFn)
6260, 61syl 17 . . . . . . . . . . 11 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ MblFn)
63 ioombl 24172 . . . . . . . . . . 11 (𝑢(,)𝑤) ∈ dom vol
64 mbfres 24251 . . . . . . . . . . 11 (((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ MblFn ∧ (𝑢(,)𝑤) ∈ dom vol) → ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ↾ (𝑢(,)𝑤)) ∈ MblFn)
6562, 63, 64sylancl 589 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ↾ (𝑢(,)𝑤)) ∈ MblFn)
6627, 65eqeltrid 2920 . . . . . . . . 9 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ (𝑢(,)𝑤) ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ MblFn)
6766adantl 485 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡 ∈ (𝑢(,)𝑤) ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ MblFn)
6813, 15, 19, 22, 67mbfss 24253 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ MblFn)
6968adantr 484 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ MblFn)
7039abscld 14796 . . . . . . . . . 10 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
7139absge0d 14804 . . . . . . . . . 10 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
72 elrege0 12841 . . . . . . . . . 10 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,)+∞) ↔ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ ∧ 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
7370, 71, 72sylanbrc 586 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,)+∞))
74 0e0icopnf 12845 . . . . . . . . 9 0 ∈ (0[,)+∞)
75 ifcl 4494 . . . . . . . . 9 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,)+∞) ∧ 0 ∈ (0[,)+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,)+∞))
7673, 74, 75sylancl 589 . . . . . . . 8 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,)+∞))
7776fmpttd 6870 . . . . . . 7 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,)+∞))
7877ad2antlr 726 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,)+∞))
7970rexrd 10689 . . . . . . . . . . 11 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ*)
80 elxrge0 12844 . . . . . . . . . . 11 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ↔ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
8179, 71, 80sylanbrc 586 . . . . . . . . . 10 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞))
82 0e0iccpnf 12846 . . . . . . . . . 10 0 ∈ (0[,]+∞)
83 ifcl 4494 . . . . . . . . . 10 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
8481, 82, 83sylancl 589 . . . . . . . . 9 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
8584fmpttd 6870 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞))
8685ad2antlr 726 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞))
87 ifcl 4494 . . . . . . . . . . 11 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
8881, 82, 87sylancl 589 . . . . . . . . . 10 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,]+∞))
8988fmpttd 6870 . . . . . . . . 9 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞))
90 ffn 6503 . . . . . . . . . . . . . . 15 (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
91 frn 6509 . . . . . . . . . . . . . . . 16 (𝑓:ℝ⟶ℝ → ran 𝑓 ⊆ ℝ)
92 ax-resscn 10592 . . . . . . . . . . . . . . . 16 ℝ ⊆ ℂ
9391, 92sstrdi 3965 . . . . . . . . . . . . . . 15 (𝑓:ℝ⟶ℝ → ran 𝑓 ⊆ ℂ)
94 ffn 6503 . . . . . . . . . . . . . . . . 17 (abs:ℂ⟶ℝ → abs Fn ℂ)
9554, 94ax-mp 5 . . . . . . . . . . . . . . . 16 abs Fn ℂ
96 fnco 6454 . . . . . . . . . . . . . . . 16 ((abs Fn ℂ ∧ 𝑓 Fn ℝ ∧ ran 𝑓 ⊆ ℂ) → (abs ∘ 𝑓) Fn ℝ)
9795, 96mp3an1 1445 . . . . . . . . . . . . . . 15 ((𝑓 Fn ℝ ∧ ran 𝑓 ⊆ ℂ) → (abs ∘ 𝑓) Fn ℝ)
9890, 93, 97syl2anc 587 . . . . . . . . . . . . . 14 (𝑓:ℝ⟶ℝ → (abs ∘ 𝑓) Fn ℝ)
9928, 98syl 17 . . . . . . . . . . . . 13 (𝑓 ∈ dom ∫1 → (abs ∘ 𝑓) Fn ℝ)
10099adantr 484 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ 𝑓) Fn ℝ)
101 ffn 6503 . . . . . . . . . . . . . . 15 (𝑔:ℝ⟶ℝ → 𝑔 Fn ℝ)
102 frn 6509 . . . . . . . . . . . . . . . 16 (𝑔:ℝ⟶ℝ → ran 𝑔 ⊆ ℝ)
103102, 92sstrdi 3965 . . . . . . . . . . . . . . 15 (𝑔:ℝ⟶ℝ → ran 𝑔 ⊆ ℂ)
104 fnco 6454 . . . . . . . . . . . . . . . 16 ((abs Fn ℂ ∧ 𝑔 Fn ℝ ∧ ran 𝑔 ⊆ ℂ) → (abs ∘ 𝑔) Fn ℝ)
10595, 104mp3an1 1445 . . . . . . . . . . . . . . 15 ((𝑔 Fn ℝ ∧ ran 𝑔 ⊆ ℂ) → (abs ∘ 𝑔) Fn ℝ)
106101, 103, 105syl2anc 587 . . . . . . . . . . . . . 14 (𝑔:ℝ⟶ℝ → (abs ∘ 𝑔) Fn ℝ)
10732, 106syl 17 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1 → (abs ∘ 𝑔) Fn ℝ)
108107adantl 485 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ 𝑔) Fn ℝ)
109 inidm 4180 . . . . . . . . . . . 12 (ℝ ∩ ℝ) = ℝ
11028adantr 484 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑓:ℝ⟶ℝ)
111 fvco3 6751 . . . . . . . . . . . . 13 ((𝑓:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → ((abs ∘ 𝑓)‘𝑡) = (abs‘(𝑓𝑡)))
112110, 111sylan 583 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs ∘ 𝑓)‘𝑡) = (abs‘(𝑓𝑡)))
11332adantl 485 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 𝑔:ℝ⟶ℝ)
114 fvco3 6751 . . . . . . . . . . . . 13 ((𝑔:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ) → ((abs ∘ 𝑔)‘𝑡) = (abs‘(𝑔𝑡)))
115113, 114sylan 583 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs ∘ 𝑔)‘𝑡) = (abs‘(𝑔𝑡)))
116100, 108, 41, 41, 109, 112, 115offval 7410 . . . . . . . . . . 11 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((abs ∘ 𝑓) ∘f + (abs ∘ 𝑔)) = (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))))
11730addid1d 10838 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → ((𝑓𝑡) + 0) = (𝑓𝑡))
118117mpteq2dva 5147 . . . . . . . . . . . . . . . 16 (𝑓 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ ((𝑓𝑡) + 0)) = (𝑡 ∈ ℝ ↦ (𝑓𝑡)))
11940a1i 11 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → ℝ ∈ V)
12017a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → 0 ∈ V)
12131a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → i ∈ ℂ)
12248a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ dom ∫1 → (ℝ × {i}) = (𝑡 ∈ ℝ ↦ i))
123 fconstmpt 5601 . . . . . . . . . . . . . . . . . . . 20 (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
124123a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ dom ∫1 → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
125119, 121, 120, 122, 124offval2 7420 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ dom ∫1 → ((ℝ × {i}) ∘f · (ℝ × {0})) = (𝑡 ∈ ℝ ↦ (i · 0)))
126 it0e0 11856 . . . . . . . . . . . . . . . . . . 19 (i · 0) = 0
127126mpteq2i 5144 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ ℝ ↦ (i · 0)) = (𝑡 ∈ ℝ ↦ 0)
128125, 127syl6eq 2875 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ dom ∫1 → ((ℝ × {i}) ∘f · (ℝ × {0})) = (𝑡 ∈ ℝ ↦ 0))
129119, 29, 120, 44, 128offval2 7420 . . . . . . . . . . . . . . . 16 (𝑓 ∈ dom ∫1 → (𝑓f + ((ℝ × {i}) ∘f · (ℝ × {0}))) = (𝑡 ∈ ℝ ↦ ((𝑓𝑡) + 0)))
130118, 129, 443eqtr4d 2869 . . . . . . . . . . . . . . 15 (𝑓 ∈ dom ∫1 → (𝑓f + ((ℝ × {i}) ∘f · (ℝ × {0}))) = 𝑓)
131130coeq2d 5720 . . . . . . . . . . . . . 14 (𝑓 ∈ dom ∫1 → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · (ℝ × {0})))) = (abs ∘ 𝑓))
132 i1f0 24294 . . . . . . . . . . . . . . 15 (ℝ × {0}) ∈ dom ∫1
133 ftc1anclem3 35077 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1 ∧ (ℝ × {0}) ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · (ℝ × {0})))) ∈ dom ∫1)
134132, 133mpan2 690 . . . . . . . . . . . . . 14 (𝑓 ∈ dom ∫1 → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · (ℝ × {0})))) ∈ dom ∫1)
135131, 134eqeltrrd 2917 . . . . . . . . . . . . 13 (𝑓 ∈ dom ∫1 → (abs ∘ 𝑓) ∈ dom ∫1)
136135adantr 484 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ 𝑓) ∈ dom ∫1)
137 coeq2 5716 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (abs ∘ 𝑓) = (abs ∘ 𝑔))
138137eleq1d 2900 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((abs ∘ 𝑓) ∈ dom ∫1 ↔ (abs ∘ 𝑔) ∈ dom ∫1))
139138, 135vtoclga 3560 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1 → (abs ∘ 𝑔) ∈ dom ∫1)
140139adantl 485 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ 𝑔) ∈ dom ∫1)
141136, 140i1fadd 24302 . . . . . . . . . . 11 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((abs ∘ 𝑓) ∘f + (abs ∘ 𝑔)) ∈ dom ∫1)
142116, 141eqeltrrd 2917 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))) ∈ dom ∫1)
14330abscld 14796 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ ℝ)
14430absge0d 14804 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝑓𝑡)))
145 elrege0 12841 . . . . . . . . . . . . . . . 16 ((abs‘(𝑓𝑡)) ∈ (0[,)+∞) ↔ ((abs‘(𝑓𝑡)) ∈ ℝ ∧ 0 ≤ (abs‘(𝑓𝑡))))
146143, 144, 145sylanbrc 586 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑓𝑡)) ∈ (0[,)+∞))
14734abscld 14796 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℝ)
14834absge0d 14804 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → 0 ≤ (abs‘(𝑔𝑡)))
149 elrege0 12841 . . . . . . . . . . . . . . . 16 ((abs‘(𝑔𝑡)) ∈ (0[,)+∞) ↔ ((abs‘(𝑔𝑡)) ∈ ℝ ∧ 0 ≤ (abs‘(𝑔𝑡))))
150147, 148, 149sylanbrc 586 . . . . . . . . . . . . . . 15 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ (0[,)+∞))
151 ge0addcl 12847 . . . . . . . . . . . . . . 15 (((abs‘(𝑓𝑡)) ∈ (0[,)+∞) ∧ (abs‘(𝑔𝑡)) ∈ (0[,)+∞)) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ (0[,)+∞))
152146, 150, 151syl2an 598 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ (0[,)+∞))
153152anandirs 678 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ (0[,)+∞))
154153fmpttd 6870 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶(0[,)+∞))
155 0plef 24279 . . . . . . . . . . . 12 ((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶(0[,)+∞) ↔ ((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶ℝ ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))))
156154, 155sylib 221 . . . . . . . . . . 11 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶ℝ ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))))
157156simprd 499 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0𝑝r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))))
158 itg2itg1 24343 . . . . . . . . . . 11 (((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) → (∫2‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) = (∫1‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))))
159 itg1cl 24292 . . . . . . . . . . . 12 ((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))) ∈ dom ∫1 → (∫1‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) ∈ ℝ)
160159adantr 484 . . . . . . . . . . 11 (((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) → (∫1‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) ∈ ℝ)
161158, 160eqeltrd 2916 . . . . . . . . . 10 (((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))) ∈ dom ∫1 ∧ 0𝑝r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) → (∫2‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) ∈ ℝ)
162142, 157, 161syl2anc 587 . . . . . . . . 9 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) ∈ ℝ)
163 icossicc 12823 . . . . . . . . . . 11 (0[,)+∞) ⊆ (0[,]+∞)
164 fss 6517 . . . . . . . . . . 11 (((𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶(0[,]+∞))
165154, 163, 164sylancl 589 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶(0[,]+∞))
166 0re 10641 . . . . . . . . . . . . . 14 0 ∈ ℝ
167 ifcl 4494 . . . . . . . . . . . . . 14 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ ℝ)
16870, 166, 167sylancl 589 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ ℝ)
169 readdcl 10618 . . . . . . . . . . . . . . 15 (((abs‘(𝑓𝑡)) ∈ ℝ ∧ (abs‘(𝑔𝑡)) ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
170143, 147, 169syl2an 598 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
171170anandirs 678 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))) ∈ ℝ)
17270leidd 11204 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
173 breq1 5055 . . . . . . . . . . . . . . 15 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) = if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ↔ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
174 breq1 5055 . . . . . . . . . . . . . . 15 (0 = if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → (0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ↔ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
175173, 174ifboth 4488 . . . . . . . . . . . . . 14 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∧ 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
176172, 71, 175syl2anc 587 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
177 abstri 14690 . . . . . . . . . . . . . . . 16 (((𝑓𝑡) ∈ ℂ ∧ (i · (𝑔𝑡)) ∈ ℂ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
17830, 36, 177syl2an 598 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
179178anandirs 678 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))))
180 absmul 14654 . . . . . . . . . . . . . . . . . . 19 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
18131, 34, 180sylancr 590 . . . . . . . . . . . . . . . . . 18 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = ((abs‘i) · (abs‘(𝑔𝑡))))
182 absi 14646 . . . . . . . . . . . . . . . . . . 19 (abs‘i) = 1
183182oveq1i 7159 . . . . . . . . . . . . . . . . . 18 ((abs‘i) · (abs‘(𝑔𝑡))) = (1 · (abs‘(𝑔𝑡)))
184181, 183syl6eq 2875 . . . . . . . . . . . . . . . . 17 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (1 · (abs‘(𝑔𝑡))))
185147recnd 10667 . . . . . . . . . . . . . . . . . 18 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(𝑔𝑡)) ∈ ℂ)
186185mulid2d 10657 . . . . . . . . . . . . . . . . 17 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (1 · (abs‘(𝑔𝑡))) = (abs‘(𝑔𝑡)))
187184, 186eqtrd 2859 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
188187adantll 713 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘(i · (𝑔𝑡))) = (abs‘(𝑔𝑡)))
189188oveq2d 7165 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((abs‘(𝑓𝑡)) + (abs‘(i · (𝑔𝑡)))) = ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
190179, 189breqtrd 5078 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
191168, 70, 171, 176, 190letrd 10795 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
192191ralrimiva 3177 . . . . . . . . . . 11 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∀𝑡 ∈ ℝ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))
193 eqidd 2825 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
194 eqidd 2825 . . . . . . . . . . . 12 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))) = (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))))
19541, 168, 171, 193, 194ofrfval2 7421 . . . . . . . . . . 11 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))) ↔ ∀𝑡 ∈ ℝ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))))
196192, 195mpbird 260 . . . . . . . . . 10 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))))
197 itg2le 24346 . . . . . . . . . 10 (((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))))
19889, 165, 196, 197syl3anc 1368 . . . . . . . . 9 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))))
199 itg2lecl 24345 . . . . . . . . 9 (((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞) ∧ (∫2‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡))))) ∈ ℝ ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ ((abs‘(𝑓𝑡)) + (abs‘(𝑔𝑡)))))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
20089, 162, 198, 199syl3anc 1368 . . . . . . . 8 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
201200ad2antlr 726 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
20289ad2antlr 726 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞))
203 breq1 5055 . . . . . . . . . . 11 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
204 breq1 5055 . . . . . . . . . . 11 (0 = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → (0 ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
205 elioore 12765 . . . . . . . . . . . . . . 15 (𝑡 ∈ (𝑢(,)𝑤) → 𝑡 ∈ ℝ)
206205, 172sylan2 595 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
207206adantll 713 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
208207adantlr 714 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
2092rexrd 10689 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ ℝ*)
2103rexrd 10689 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ ℝ*)
211209, 210jca 515 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
212 df-icc 12742 . . . . . . . . . . . . . . . . . . . . 21 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑡 ∈ ℝ* ∣ (𝑥𝑡𝑡𝑦)})
213212elixx3g 12748 . . . . . . . . . . . . . . . . . . . 20 (𝑢 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑢 ∈ ℝ*) ∧ (𝐴𝑢𝑢𝐵)))
214213simprbi 500 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ (𝐴[,]𝐵) → (𝐴𝑢𝑢𝐵))
215214simpld 498 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (𝐴[,]𝐵) → 𝐴𝑢)
216212elixx3g 12748 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) ∧ (𝐴𝑤𝑤𝐵)))
217216simprbi 500 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝐴[,]𝐵) → (𝐴𝑤𝑤𝐵))
218217simprd 499 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝐴[,]𝐵) → 𝑤𝐵)
219215, 218anim12i 615 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴𝑢𝑤𝐵))
220 ioossioo 12828 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑢𝑤𝐵)) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
221211, 219, 220syl2an 598 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
2225adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ 𝐷)
223221, 222sstrd 3963 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ 𝐷)
224223sselda 3953 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
225 iftrue 4456 . . . . . . . . . . . . . 14 (𝑡𝐷 → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
226224, 225syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
227226adantllr 718 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
228208, 227breqtrrd 5080 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
229 breq2 5056 . . . . . . . . . . . . 13 ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) = if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → (0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ↔ 0 ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
230 breq2 5056 . . . . . . . . . . . . 13 (0 = if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
2316sselda 3953 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝐷) → 𝑡 ∈ ℝ)
232231adantlr 714 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 𝑡 ∈ ℝ)
23371adantll 713 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
234232, 233syldan 594 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
235 0le0 11735 . . . . . . . . . . . . . 14 0 ≤ 0
236235a1i 11 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ¬ 𝑡𝐷) → 0 ≤ 0)
237229, 230, 234, 236ifbothda 4487 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → 0 ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
238237ad2antrr 725 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
239203, 204, 228, 238ifbothda 4487 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
240239ralrimivw 3178 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))
24140a1i 11 . . . . . . . . . . 11 (𝜑 → ℝ ∈ V)
24218a1i 11 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V)
24316, 17ifex 4498 . . . . . . . . . . . 12 if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V
244243a1i 11 . . . . . . . . . . 11 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V)
245 eqidd 2825 . . . . . . . . . . 11 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
246 eqidd 2825 . . . . . . . . . . 11 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
247241, 242, 244, 245, 246ofrfval2 7421 . . . . . . . . . 10 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
248247ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ≤ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
249240, 248mpbird 260 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
250 itg2le 24346 . . . . . . . 8 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘r ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))))
25186, 202, 249, 250syl3anc 1368 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))))
252 itg2lecl 24345 . . . . . . 7 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,]+∞) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
25386, 201, 251, 252syl3anc 1368 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
2548ffvelrnda 6842 . . . . . . . . . . . . . . 15 ((𝜑𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
255254adantlr 714 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
256224, 255syldan 594 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
257256adantllr 718 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
258205, 39sylan2 595 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
259258adantll 713 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
260259adantlr 714 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
261257, 260subcld 10995 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
262261abscld 14796 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
263261absge0d 14804 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
264 elrege0 12841 . . . . . . . . . 10 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,)+∞) ↔ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ ∧ 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
265262, 263, 264sylanbrc 586 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,)+∞))
26674a1i 11 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,)+∞))
267265, 266ifclda 4484 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,)+∞))
268267adantr 484 . . . . . . 7 ((((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[,)+∞))
269268fmpttd 6870 . . . . . 6 (((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[,)+∞))
270262rexrd 10689 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
271 elxrge0 12844 . . . . . . . . . . 11 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ* ∧ 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
272270, 263, 271sylanbrc 586 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞))
27382a1i 11 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
274272, 273ifclda 4484 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,]+∞))
275274adantr 484 . . . . . . . 8 ((((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[,]+∞))
276275fmpttd 6870 . . . . . . 7 (((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[,]+∞))
277 recncf 23510 . . . . . . . . . . . . 13 ℜ ∈ (ℂ–cn→ℝ)
278 prid1g 4681 . . . . . . . . . . . . 13 (ℜ ∈ (ℂ–cn→ℝ) → ℜ ∈ {ℜ, ℑ})
279277, 278ax-mp 5 . . . . . . . . . . . 12 ℜ ∈ {ℜ, ℑ}
280 ftc1anclem2 35076 . . . . . . . . . . . 12 ((𝐹:𝐷⟶ℂ ∧ 𝐹 ∈ 𝐿1 ∧ ℜ ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
281279, 280mp3an3 1447 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
2828, 7, 281syl2anc 587 . . . . . . . . . 10 (𝜑 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) ∈ ℝ)
283 imcncf 23511 . . . . . . . . . . . . 13 ℑ ∈ (ℂ–cn→ℝ)
284 prid2g 4682 . . . . . . . . . . . . 13 (ℑ ∈ (ℂ–cn→ℝ) → ℑ ∈ {ℜ, ℑ})
285283, 284ax-mp 5 . . . . . . . . . . . 12 ℑ ∈ {ℜ, ℑ}
286 ftc1anclem2 35076 . . . . . . . . . . . 12 ((𝐹:𝐷⟶ℂ ∧ 𝐹 ∈ 𝐿1 ∧ ℑ ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
287285, 286mp3an3 1447 . . . . . . . . . . 11 ((𝐹:𝐷⟶ℂ ∧ 𝐹 ∈ 𝐿1) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
2888, 7, 287syl2anc 587 . . . . . . . . . 10 (𝜑 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
289282, 288readdcld 10668 . . . . . . . . 9 (𝜑 → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) ∈ ℝ)
290289ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) ∈ ℝ)
291201, 290readdcld 10668 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))) ∈ ℝ)
292 ge0addcl 12847 . . . . . . . . . . . . 13 ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞))
293292adantl 485 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,)+∞))
294 ifcl 4494 . . . . . . . . . . . . . . 15 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ (0[,)+∞) ∧ 0 ∈ (0[,)+∞)) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,)+∞))
29573, 74, 294sylancl 589 . . . . . . . . . . . . . 14 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ (0[,)+∞))
296295fmpttd 6870 . . . . . . . . . . . . 13 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,)+∞))
297296adantl 485 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)):ℝ⟶(0[,)+∞))
298292adantl 485 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,)+∞))
299254recld 14553 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → (ℜ‘(𝐹𝑡)) ∈ ℝ)
300299recnd 10667 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑡𝐷) → (ℜ‘(𝐹𝑡)) ∈ ℂ)
301300abscld 14796 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝐷) → (abs‘(ℜ‘(𝐹𝑡))) ∈ ℝ)
302300absge0d 14804 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝐷) → 0 ≤ (abs‘(ℜ‘(𝐹𝑡))))
303 elrege0 12841 . . . . . . . . . . . . . . . . . 18 ((abs‘(ℜ‘(𝐹𝑡))) ∈ (0[,)+∞) ↔ ((abs‘(ℜ‘(𝐹𝑡))) ∈ ℝ ∧ 0 ≤ (abs‘(ℜ‘(𝐹𝑡)))))
304301, 302, 303sylanbrc 586 . . . . . . . . . . . . . . . . 17 ((𝜑𝑡𝐷) → (abs‘(ℜ‘(𝐹𝑡))) ∈ (0[,)+∞))
30574a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ (0[,)+∞))
306304, 305ifclda 4484 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) ∈ (0[,)+∞))
307306adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) ∈ (0[,)+∞))
308307fmpttd 6870 . . . . . . . . . . . . . 14 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)):ℝ⟶(0[,)+∞))
309254imcld 14554 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → (ℑ‘(𝐹𝑡)) ∈ ℝ)
310309recnd 10667 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑡𝐷) → (ℑ‘(𝐹𝑡)) ∈ ℂ)
311310abscld 14796 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝐷) → (abs‘(ℑ‘(𝐹𝑡))) ∈ ℝ)
312310absge0d 14804 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝐷) → 0 ≤ (abs‘(ℑ‘(𝐹𝑡))))
313 elrege0 12841 . . . . . . . . . . . . . . . . . 18 ((abs‘(ℑ‘(𝐹𝑡))) ∈ (0[,)+∞) ↔ ((abs‘(ℑ‘(𝐹𝑡))) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘(𝐹𝑡)))))
314311, 312, 313sylanbrc 586 . . . . . . . . . . . . . . . . 17 ((𝜑𝑡𝐷) → (abs‘(ℑ‘(𝐹𝑡))) ∈ (0[,)+∞))
315314, 305ifclda 4484 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0) ∈ (0[,)+∞))
316315adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0) ∈ (0[,)+∞))
317316fmpttd 6870 . . . . . . . . . . . . . 14 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)):ℝ⟶(0[,)+∞))
318298, 308, 317, 241, 241, 109off 7418 . . . . . . . . . . . . 13 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))):ℝ⟶(0[,)+∞))
319318adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))):ℝ⟶(0[,)+∞))
32040a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ℝ ∈ V)
321293, 297, 319, 320, 320, 109off 7418 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))):ℝ⟶(0[,)+∞))
322 fss 6517 . . . . . . . . . . 11 ((((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))):ℝ⟶(0[,]+∞))
323321, 163, 322sylancl 589 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))):ℝ⟶(0[,]+∞))
324323adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))):ℝ⟶(0[,]+∞))
325 0xr 10686 . . . . . . . . . . . . . 14 0 ∈ ℝ*
326325a1i 11 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ ℝ*)
327270, 326ifclda 4484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ ℝ*)
328254adantlr 714 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
32939adantll 713 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
330232, 329syldan 594 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
331328, 330subcld 10995 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
332331abscld 14796 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
333332rexrd 10689 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
334325a1i 11 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ¬ 𝑡𝐷) → 0 ∈ ℝ*)
335333, 334ifclda 4484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ ℝ*)
336335adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ ℝ*)
337330abscld 14796 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
338 0red 10642 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ¬ 𝑡𝐷) → 0 ∈ ℝ)
339337, 338ifclda 4484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ ℝ)
340 0red 10642 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℝ)
341301, 340ifclda 4484 . . . . . . . . . . . . . . . . 17 (𝜑 → if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) ∈ ℝ)
342311, 340ifclda 4484 . . . . . . . . . . . . . . . . 17 (𝜑 → if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0) ∈ ℝ)
343341, 342readdcld 10668 . . . . . . . . . . . . . . . 16 (𝜑 → (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)) ∈ ℝ)
344343adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)) ∈ ℝ)
345339, 344readdcld 10668 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ)
346345rexrd 10689 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ*)
347346adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ ℝ*)
348 breq1 5055 . . . . . . . . . . . . 13 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) → ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
349 breq1 5055 . . . . . . . . . . . . 13 (0 = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) → (0 ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
350224adantllr 718 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
351332leidd 11204 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
352351adantlr 714 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
353 iftrue 4456 . . . . . . . . . . . . . . . 16 (𝑡𝐷 → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
354353adantl 485 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
355352, 354breqtrrd 5080 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
356350, 355syldan 594 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
357 breq2 5056 . . . . . . . . . . . . . . 15 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) → (0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ↔ 0 ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
358 breq2 5056 . . . . . . . . . . . . . . 15 (0 = if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) → (0 ≤ 0 ↔ 0 ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
359331absge0d 14804 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
360357, 358, 359, 236ifbothda 4487 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → 0 ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
361360ad2antrr 725 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
362348, 349, 356, 361ifbothda 4487 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
363254negcld 10982 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝐷) → -(𝐹𝑡) ∈ ℂ)
364363adantlr 714 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → -(𝐹𝑡) ∈ ℂ)
365330, 364addcld 10658 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (((𝑓𝑡) + (i · (𝑔𝑡))) + -(𝐹𝑡)) ∈ ℂ)
366365abscld 14796 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘(((𝑓𝑡) + (i · (𝑔𝑡))) + -(𝐹𝑡))) ∈ ℝ)
367363abscld 14796 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → (abs‘-(𝐹𝑡)) ∈ ℝ)
368367adantlr 714 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘-(𝐹𝑡)) ∈ ℝ)
369337, 368readdcld 10668 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘-(𝐹𝑡))) ∈ ℝ)
370301, 311readdcld 10668 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡)))) ∈ ℝ)
371370adantlr 714 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡)))) ∈ ℝ)
372337, 371readdcld 10668 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))) ∈ ℝ)
373330, 364abstrid 14816 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘(((𝑓𝑡) + (i · (𝑔𝑡))) + -(𝐹𝑡))) ≤ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘-(𝐹𝑡))))
374 mulcl 10619 . . . . . . . . . . . . . . . . . . . . . . 23 ((i ∈ ℂ ∧ (ℑ‘(𝐹𝑡)) ∈ ℂ) → (i · (ℑ‘(𝐹𝑡))) ∈ ℂ)
37531, 310, 374sylancr 590 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝐷) → (i · (ℑ‘(𝐹𝑡))) ∈ ℂ)
376300, 375abstrid 14816 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝐷) → (abs‘((ℜ‘(𝐹𝑡)) + (i · (ℑ‘(𝐹𝑡))))) ≤ ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(i · (ℑ‘(𝐹𝑡))))))
377254absnegd 14809 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝐷) → (abs‘-(𝐹𝑡)) = (abs‘(𝐹𝑡)))
378254replimd 14556 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑡𝐷) → (𝐹𝑡) = ((ℜ‘(𝐹𝑡)) + (i · (ℑ‘(𝐹𝑡)))))
379378fveq2d 6665 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝐷) → (abs‘(𝐹𝑡)) = (abs‘((ℜ‘(𝐹𝑡)) + (i · (ℑ‘(𝐹𝑡))))))
380377, 379eqtrd 2859 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝐷) → (abs‘-(𝐹𝑡)) = (abs‘((ℜ‘(𝐹𝑡)) + (i · (ℑ‘(𝐹𝑡))))))
381 absmul 14654 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((i ∈ ℂ ∧ (ℑ‘(𝐹𝑡)) ∈ ℂ) → (abs‘(i · (ℑ‘(𝐹𝑡)))) = ((abs‘i) · (abs‘(ℑ‘(𝐹𝑡)))))
38231, 310, 381sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑡𝐷) → (abs‘(i · (ℑ‘(𝐹𝑡)))) = ((abs‘i) · (abs‘(ℑ‘(𝐹𝑡)))))
383182oveq1i 7159 . . . . . . . . . . . . . . . . . . . . . . . 24 ((abs‘i) · (abs‘(ℑ‘(𝐹𝑡)))) = (1 · (abs‘(ℑ‘(𝐹𝑡))))
384382, 383syl6eq 2875 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑡𝐷) → (abs‘(i · (ℑ‘(𝐹𝑡)))) = (1 · (abs‘(ℑ‘(𝐹𝑡)))))
385311recnd 10667 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑡𝐷) → (abs‘(ℑ‘(𝐹𝑡))) ∈ ℂ)
386385mulid2d 10657 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑡𝐷) → (1 · (abs‘(ℑ‘(𝐹𝑡)))) = (abs‘(ℑ‘(𝐹𝑡))))
387384, 386eqtr2d 2860 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡𝐷) → (abs‘(ℑ‘(𝐹𝑡))) = (abs‘(i · (ℑ‘(𝐹𝑡)))))
388387oveq2d 7165 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑡𝐷) → ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡)))) = ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(i · (ℑ‘(𝐹𝑡))))))
389376, 380, 3883brtr4d 5084 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡𝐷) → (abs‘-(𝐹𝑡)) ≤ ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡)))))
390389adantlr 714 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘-(𝐹𝑡)) ≤ ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡)))))
391368, 371, 337, 390leadd2dd 11253 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘-(𝐹𝑡))) ≤ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))))
392366, 369, 372, 373, 391letrd 10795 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘(((𝑓𝑡) + (i · (𝑔𝑡))) + -(𝐹𝑡))) ≤ ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))))
393328, 330abssubd 14813 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘(((𝑓𝑡) + (i · (𝑔𝑡))) − (𝐹𝑡))))
394353adantl 485 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
395330, 328negsubd 11001 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (((𝑓𝑡) + (i · (𝑔𝑡))) + -(𝐹𝑡)) = (((𝑓𝑡) + (i · (𝑔𝑡))) − (𝐹𝑡)))
396395fveq2d 6665 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → (abs‘(((𝑓𝑡) + (i · (𝑔𝑡))) + -(𝐹𝑡))) = (abs‘(((𝑓𝑡) + (i · (𝑔𝑡))) − (𝐹𝑡))))
397393, 394, 3963eqtr4d 2869 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = (abs‘(((𝑓𝑡) + (i · (𝑔𝑡))) + -(𝐹𝑡))))
398 iftrue 4456 . . . . . . . . . . . . . . . . . 18 (𝑡𝐷 → if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0) = ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))))
399398adantl 485 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0) = ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))))
400392, 397, 3993brtr4d 5084 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0))
401400ex 416 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡𝐷 → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0)))
402235a1i 11 . . . . . . . . . . . . . . . 16 𝑡𝐷 → 0 ≤ 0)
403 iffalse 4459 . . . . . . . . . . . . . . . 16 𝑡𝐷 → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = 0)
404 iffalse 4459 . . . . . . . . . . . . . . . 16 𝑡𝐷 → if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0) = 0)
405402, 403, 4043brtr4d 5084 . . . . . . . . . . . . . . 15 𝑡𝐷 → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0))
406401, 405pm2.61d1 183 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0))
407 iftrue 4456 . . . . . . . . . . . . . . . . . 18 (𝑡𝐷 → if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) = (abs‘(ℜ‘(𝐹𝑡))))
408 iftrue 4456 . . . . . . . . . . . . . . . . . 18 (𝑡𝐷 → if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0) = (abs‘(ℑ‘(𝐹𝑡))))
409407, 408oveq12d 7167 . . . . . . . . . . . . . . . . 17 (𝑡𝐷 → (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)) = ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡)))))
410225, 409oveq12d 7167 . . . . . . . . . . . . . . . 16 (𝑡𝐷 → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) = ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))))
411410, 398eqtr4d 2862 . . . . . . . . . . . . . . 15 (𝑡𝐷 → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) = if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0))
412 00id 10813 . . . . . . . . . . . . . . . . . 18 (0 + 0) = 0
413412oveq2i 7160 . . . . . . . . . . . . . . . . 17 (0 + (0 + 0)) = (0 + 0)
414413, 412eqtri 2847 . . . . . . . . . . . . . . . 16 (0 + (0 + 0)) = 0
415 iffalse 4459 . . . . . . . . . . . . . . . . 17 𝑡𝐷 → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
416 iffalse 4459 . . . . . . . . . . . . . . . . . 18 𝑡𝐷 → if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) = 0)
417 iffalse 4459 . . . . . . . . . . . . . . . . . 18 𝑡𝐷 → if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0) = 0)
418416, 417oveq12d 7167 . . . . . . . . . . . . . . . . 17 𝑡𝐷 → (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)) = (0 + 0))
419415, 418oveq12d 7167 . . . . . . . . . . . . . . . 16 𝑡𝐷 → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) = (0 + (0 + 0)))
420414, 419, 4043eqtr4a 2885 . . . . . . . . . . . . . . 15 𝑡𝐷 → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) = if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0))
421411, 420pm2.61i 185 . . . . . . . . . . . . . 14 (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) = if(𝑡𝐷, ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + ((abs‘(ℜ‘(𝐹𝑡))) + (abs‘(ℑ‘(𝐹𝑡))))), 0)
422406, 421breqtrrdi 5094 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))
423422adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡𝐷, (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))
424327, 336, 347, 362, 423xrletrd 12552 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))
425424ralrimivw 3178 . . . . . . . . . 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) ≤ (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))
426 fvex 6674 . . . . . . . . . . . . . 14 (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V
427426, 17ifex 4498 . . . . . . . . . . . . 13 if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ V
428427a1i 11 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ V)
429 ovexd 7184 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ℝ) → (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) ∈ V)
430 eqidd 2825 . . . . . . . . . . . 12 (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)))
431 ovexd 7184 . . . . . . . . . . . . 13 ((𝜑𝑡 ∈ ℝ) → (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)) ∈ V)
432341adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) ∈ ℝ)
433342adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0) ∈ ℝ)
434 eqidd 2825 . . . . . . . . . . . . . 14 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)))
435 eqidd 2825 . . . . . . . . . . . . . 14 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))
436241, 432, 433, 434, 435offval2 7420 . . . . . . . . . . . . 13 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))) = (𝑡 ∈ ℝ ↦ (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))
437241, 244, 431, 246, 436offval2 7420 . . . . . . . . . . . 12 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) = (𝑡 ∈ ℝ ↦ (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))))
438241, 428, 429, 430, 437ofrfval2 7421 . . . . . . . . . . 11 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ∘r ≤ ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))))
439438ad2antrr 725 . . . . . . . . . 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 ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ≤ (if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + (if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0) + if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))))
440425, 439mpbird 260 . . . . . . . . 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 ≤ ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))))
441 itg2le 24346 . . . . . . . . 9 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞) ∧ ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ∘r ≤ ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))))
442276, 324, 440, 441syl3anc 1368 . . . . . . . 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))) ≤ (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))))
4436adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → 𝐷 ⊆ ℝ)
444243a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡𝐷) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) ∈ V)
445 eldifn 4090 . . . . . . . . . . . . . 14 (𝑡 ∈ (ℝ ∖ 𝐷) → ¬ 𝑡𝐷)
446445iffalsed 4461 . . . . . . . . . . . . 13 (𝑡 ∈ (ℝ ∖ 𝐷) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
447446adantl 485 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (ℝ ∖ 𝐷)) → if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
448 ovexd 7184 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (i · (𝑔𝑡)) ∈ V)
44941, 42, 448, 45, 52offval2 7420 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (𝑓f + ((ℝ × {i}) ∘f · 𝑔)) = (𝑡 ∈ ℝ ↦ ((𝑓𝑡) + (i · (𝑔𝑡)))))
45039, 449, 56, 57fmptco 6882 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) = (𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
451450reseq1d 5839 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ((abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ↾ 𝐷) = ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ↾ 𝐷))
4526resmptd 5895 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑡 ∈ ℝ ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ↾ 𝐷) = (𝑡𝐷 ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
453451, 452sylan9eqr 2881 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ↾ 𝐷) = (𝑡𝐷 ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
454225mpteq2ia 5143 . . . . . . . . . . . . . 14 (𝑡𝐷 ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) = (𝑡𝐷 ↦ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
455453, 454syl6eqr 2877 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ↾ 𝐷) = (𝑡𝐷 ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)))
456 i1fmbf 24282 . . . . . . . . . . . . . . 15 ((abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ∈ dom ∫1 → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ∈ MblFn)
45759, 456syl 17 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ∈ MblFn)
4588fdmd 6513 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐹 = 𝐷)
459 iblmbf 24374 . . . . . . . . . . . . . . . 16 (𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
460 mbfdm 24233 . . . . . . . . . . . . . . . 16 (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
4617, 459, 4603syl 18 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐹 ∈ dom vol)
462458, 461eqeltrrd 2917 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ dom vol)
463 mbfres 24251 . . . . . . . . . . . . . 14 (((abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ∈ MblFn ∧ 𝐷 ∈ dom vol) → ((abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ↾ 𝐷) ∈ MblFn)
464457, 462, 463syl2anr 599 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((abs ∘ (𝑓f + ((ℝ × {i}) ∘f · 𝑔))) ↾ 𝐷) ∈ MblFn)
465455, 464eqeltrrd 2917 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡𝐷 ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ MblFn)
466443, 15, 444, 447, 465mbfss 24253 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∈ MblFn)
467200adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) ∈ ℝ)
468 0cnd 10632 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℂ)
469300, 468ifclda 4484 . . . . . . . . . . . . . . . . 17 (𝜑 → if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) ∈ ℂ)
470469adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) ∈ ℂ)
471 eqidd 2825 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)))
47254a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → abs:ℂ⟶ℝ)
473472feqmptd 6724 . . . . . . . . . . . . . . . 16 (𝜑 → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
474 fveq2 6661 . . . . . . . . . . . . . . . . 17 (𝑥 = if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) → (abs‘𝑥) = (abs‘if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)))
475 fvif 6677 . . . . . . . . . . . . . . . . . 18 (abs‘if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)) = if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), (abs‘0))
476 abs0 14645 . . . . . . . . . . . . . . . . . . 19 (abs‘0) = 0
477 ifeq2 4455 . . . . . . . . . . . . . . . . . . 19 ((abs‘0) = 0 → if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), (abs‘0)) = if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))
478476, 477ax-mp 5 . . . . . . . . . . . . . . . . . 18 if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), (abs‘0)) = if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)
479475, 478eqtri 2847 . . . . . . . . . . . . . . . . 17 (abs‘if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)) = if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)
480474, 479syl6eq 2875 . . . . . . . . . . . . . . . 16 (𝑥 = if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) → (abs‘𝑥) = if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))
481470, 471, 473, 480fmptco 6882 . . . . . . . . . . . . . . 15 (𝜑 → (abs ∘ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0))) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)))
482299, 340ifclda 4484 . . . . . . . . . . . . . . . . . 18 (𝜑 → if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) ∈ ℝ)
483482adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) ∈ ℝ)
484483fmpttd 6870 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)):ℝ⟶ℝ)
48514a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℝ ∈ dom vol)
486482adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑡𝐷) → if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) ∈ ℝ)
487445adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (ℝ ∖ 𝐷)) → ¬ 𝑡𝐷)
488487iffalsed 4461 . . . . . . . . . . . . . . . . 17 ((𝜑𝑡 ∈ (ℝ ∖ 𝐷)) → if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) = 0)
489 iftrue 4456 . . . . . . . . . . . . . . . . . . 19 (𝑡𝐷 → if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0) = (ℜ‘(𝐹𝑡)))
490489mpteq2ia 5143 . . . . . . . . . . . . . . . . . 18 (𝑡𝐷 ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)) = (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡)))
4918feqmptd 6724 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐹 = (𝑡𝐷 ↦ (𝐹𝑡)))
4927, 459syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐹 ∈ MblFn)
493491, 492eqeltrrd 2917 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ MblFn)
494254ismbfcn2 24245 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑡𝐷 ↦ (𝐹𝑡)) ∈ MblFn ↔ ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)))
495493, 494mpbid 235 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
496495simpld 498 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn)
497490, 496eqeltrid 2920 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑡𝐷 ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)) ∈ MblFn)
4986, 485, 486, 488, 497mbfss 24253 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)) ∈ MblFn)
499 ftc1anclem1 35075 . . . . . . . . . . . . . . . 16 (((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)):ℝ⟶ℝ ∧ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0)) ∈ MblFn) → (abs ∘ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0))) ∈ MblFn)
500484, 498, 499syl2anc 587 . . . . . . . . . . . . . . 15 (𝜑 → (abs ∘ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (ℜ‘(𝐹𝑡)), 0))) ∈ MblFn)
501481, 500eqeltrrd 2917 . . . . . . . . . . . . . 14 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∈ MblFn)
502501, 308, 282, 317, 288itg2addnc 35056 . . . . . . . . . . . . 13 (𝜑 → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) = ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))))
503502, 289eqeltrd 2916 . . . . . . . . . . . 12 (𝜑 → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) ∈ ℝ)
504503adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) ∈ ℝ)
505466, 297, 467, 319, 504itg2addnc 35056 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))) = ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))))
506502adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))) = ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))))
507506oveq2d 7165 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))) = ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))))
508505, 507eqtrd 2859 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))) = ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))))
509508adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))) = ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))))
510442, 509breqtrd 5078 . . . . . . 7 (((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 ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))))
511 itg2lecl 24345 . . . . . . 7 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞) ∧ ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0))))) ∈ ℝ ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℜ‘(𝐹𝑡))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(ℑ‘(𝐹𝑡))), 0)))))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ)
512276, 291, 510, 511syl3anc 1368 . . . . . 6 (((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)
51369, 78, 253, 269, 512itg2addnc 35056 . . . . 5 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))) = ((∫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)))))
514241, 242, 428, 245, 430offval2 7420 . . . . . . . . 9 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) = (𝑡 ∈ ℝ ↦ (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
515 eqeq2 2836 . . . . . . . . . . 11 (((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))) = if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0) → ((if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))) ↔ (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0)))
516 eqeq2 2836 . . . . . . . . . . 11 (0 = if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0) → ((if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = 0 ↔ (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0)))
517 iftrue 4456 . . . . . . . . . . . . 13 (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
51823, 517oveq12d 7167 . . . . . . . . . . . 12 (𝑡 ∈ (𝑢(,)𝑤) → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
519518adantl 485 . . . . . . . . . . 11 ((𝜑𝑡 ∈ (𝑢(,)𝑤)) → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))))
520 iffalse 4459 . . . . . . . . . . . . . 14 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) = 0)
521 iffalse 4459 . . . . . . . . . . . . . 14 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = 0)
522520, 521oveq12d 7167 . . . . . . . . . . . . 13 𝑡 ∈ (𝑢(,)𝑤) → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = (0 + 0))
523522, 412syl6eq 2875 . . . . . . . . . . . 12 𝑡 ∈ (𝑢(,)𝑤) → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = 0)
524523adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = 0)
525515, 516, 519, 524ifbothda 4487 . . . . . . . . . 10 (𝜑 → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0))
526525mpteq2dv 5148 . . . . . . . . 9 (𝜑 → (𝑡 ∈ ℝ ↦ (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0) + if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0)))
527514, 526eqtrd 2859 . . . . . . . 8 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0)))
528527ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0)))
529 simplr 768 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1))
530258abscld 14796 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
531530recnd 10667 . . . . . . . . . . 11 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
532529, 531sylan 583 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
533262recnd 10667 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℂ)
534532, 533addcomd 10840 . . . . . . . . 9 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))) = ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
535534ifeq1da 4480 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0) = if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
536535mpteq2dv 5148 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) + (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
537528, 536eqtrd 2859 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
538537fveq2d 6665 . . . . 5 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0)) ∘f + (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
539513, 538eqtr3d 2861 . . . 4 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((∫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)))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
54010, 11, 539syl2an 598 . . 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) + (i · (𝑔𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
541540adantr 484 . 2 (((((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
542 rpcn 12396 . . . 4 (𝑦 ∈ ℝ+𝑦 ∈ ℂ)
5435422halvesd 11880 . . 3 (𝑦 ∈ ℝ+ → ((𝑦 / 2) + (𝑦 / 2)) = 𝑦)
544543ad3antlr 730 . 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) + (𝑦 / 2)) = 𝑦)
5459, 541, 5443brtr3d 5083 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) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) < 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  Vcvv 3480  cdif 3916  cun 3917  wss 3919  ifcif 4450  {csn 4550  {cpr 4552   class class class wbr 5052  cmpt 5132   × cxp 5540  dom cdm 5542  ran crn 5543  cres 5544  cima 5545  ccom 5546   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  f cof 7401  r cofr 7402  supcsup 8901  cc 10533  cr 10534  0cc0 10535  1c1 10536  ici 10537   + caddc 10538   · cmul 10540  +∞cpnf 10670  *cxr 10672   < clt 10673  cle 10674  cmin 10868  -cneg 10869   / cdiv 11295  2c2 11689  +crp 12386  (,)cioo 12735  [,)cico 12737  [,]cicc 12738  cre 14456  cim 14457  abscabs 14593  cnccncf 23484  volcvol 24070  MblFncmbf 24221  1citg1 24222  2citg2 24223  𝐿1cibl 24224  citg 24225  0𝑝c0p 24276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-addf 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-disj 5018  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fi 8872  df-sup 8903  df-inf 8904  df-oi 8971  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ioo 12739  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-rest 16696  df-topgen 16717  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-cncf 23486  df-ovol 24071  df-vol 24072  df-mbf 24226  df-itg1 24227  df-itg2 24228  df-ibl 24229  df-0p 24277
This theorem is referenced by:  ftc1anc  35083
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