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Theorem ftc1anc 33623
Description: ftc1a 23845 holds for functions that obey the triangle inequality in the absence of ax-cc 9295. Theorem 565Ma of [Fremlin5] p. 220. (Contributed by Brendan Leahy, 11-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)
ftc1anc.t (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘(𝐹𝑡)), 0))))
Assertion
Ref Expression
ftc1anc (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Distinct variable groups:   𝑡,𝑠,𝑥,𝐴   𝐵,𝑠,𝑡,𝑥   𝐷,𝑠,𝑡,𝑥   𝐹,𝑠,𝑡,𝑥   𝜑,𝑠,𝑡,𝑥   𝐺,𝑠
Allowed substitution hints:   𝐺(𝑥,𝑡)

Proof of Theorem ftc1anc
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑟 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
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, 8ftc1lem2 23844 . 2 (𝜑𝐺:(𝐴[,]𝐵)⟶ℂ)
10 rphalfcl 11896 . . . . . 6 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
111, 2, 3, 4, 5, 6, 7, 8ftc1anclem6 33620 . . . . . 6 ((𝜑 ∧ (𝑦 / 2) ∈ ℝ+) → ∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2))
1210, 11sylan2 490 . . . . 5 ((𝜑𝑦 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2))
1312adantrl 752 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) → ∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2))
1410ad2antll 765 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) → (𝑦 / 2) ∈ ℝ+)
15 2rp 11875 . . . . . . . . . . . 12 2 ∈ ℝ+
16 i1ff 23488 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ dom ∫1𝑓:ℝ⟶ℝ)
17 frn 6091 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:ℝ⟶ℝ → ran 𝑓 ⊆ ℝ)
1816, 17syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ dom ∫1 → ran 𝑓 ⊆ ℝ)
1918adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑓 ⊆ ℝ)
20 i1ff 23488 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
21 frn 6091 . . . . . . . . . . . . . . . . . . . . 21 (𝑔:ℝ⟶ℝ → ran 𝑔 ⊆ ℝ)
2220, 21syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑔 ∈ dom ∫1 → ran 𝑔 ⊆ ℝ)
2322adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ran 𝑔 ⊆ ℝ)
2419, 23unssd 3822 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℝ)
25 ax-resscn 10031 . . . . . . . . . . . . . . . . . 18 ℝ ⊆ ℂ
2624, 25syl6ss 3648 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ)
27 i1f0rn 23494 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ dom ∫1 → 0 ∈ ran 𝑓)
28 elun1 3813 . . . . . . . . . . . . . . . . . . 19 (0 ∈ ran 𝑓 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
2927, 28syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ dom ∫1 → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
3029adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → 0 ∈ (ran 𝑓 ∪ ran 𝑔))
31 absf 14121 . . . . . . . . . . . . . . . . . . 19 abs:ℂ⟶ℝ
32 ffn 6083 . . . . . . . . . . . . . . . . . . 19 (abs:ℂ⟶ℝ → abs Fn ℂ)
3331, 32ax-mp 5 . . . . . . . . . . . . . . . . . 18 abs Fn ℂ
34 fnfvima 6536 . . . . . . . . . . . . . . . . . 18 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
3533, 34mp3an1 1451 . . . . . . . . . . . . . . . . 17 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 0 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
3626, 30, 35syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
37 ne0i 3954 . . . . . . . . . . . . . . . 16 ((abs‘0) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
3836, 37syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅)
39 imassrn 5512 . . . . . . . . . . . . . . . . 17 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ran abs
40 frn 6091 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
4131, 40ax-mp 5 . . . . . . . . . . . . . . . . 17 ran abs ⊆ ℝ
4239, 41sstri 3645 . . . . . . . . . . . . . . . 16 (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ
43 ffun 6086 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → Fun abs)
4431, 43ax-mp 5 . . . . . . . . . . . . . . . . 17 Fun abs
45 i1frn 23489 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ dom ∫1 → ran 𝑓 ∈ Fin)
46 i1frn 23489 . . . . . . . . . . . . . . . . . 18 (𝑔 ∈ dom ∫1 → ran 𝑔 ∈ Fin)
47 unfi 8268 . . . . . . . . . . . . . . . . . 18 ((ran 𝑓 ∈ Fin ∧ ran 𝑔 ∈ Fin) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
4845, 46, 47syl2an 493 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (ran 𝑓 ∪ ran 𝑔) ∈ Fin)
49 imafi 8300 . . . . . . . . . . . . . . . . 17 ((Fun abs ∧ (ran 𝑓 ∪ ran 𝑔) ∈ Fin) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
5044, 48, 49sylancr 696 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin)
51 fimaxre2 11007 . . . . . . . . . . . . . . . 16 (((abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ ∧ (abs “ (ran 𝑓 ∪ ran 𝑔)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥)
5242, 50, 51sylancr 696 . . . . . . . . . . . . . . 15 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥)
53 suprcl 11021 . . . . . . . . . . . . . . . 16 (((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)
5442, 53mp3an1 1451 . . . . . . . . . . . . . . 15 (((abs “ (ran 𝑓 ∪ ran 𝑔)) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (abs “ (ran 𝑓 ∪ ran 𝑔))𝑦𝑥) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
5538, 52, 54syl2anc 694 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
5655adantr 480 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
57 0red 10079 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 ∈ ℝ)
5826sselda 3636 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → 𝑟 ∈ ℂ)
5958abscld 14219 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ ℝ)
6059adantrr 753 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ∈ ℝ)
6155adantr 480 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ)
62 absgt0 14108 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ ℂ → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
6358, 62syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (𝑟 ≠ 0 ↔ 0 < (abs‘𝑟)))
6463biimpd 219 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (𝑟 ≠ 0 → 0 < (abs‘𝑟)))
6564impr 648 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < (abs‘𝑟))
6642a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (abs “ (ran 𝑓 ∪ ran 𝑔)) ⊆ ℝ)
6766, 38, 523jca 1261 . . . . . . . . . . . . . . . . . . 19 ((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))
6867adantr 480 . . . . . . . . . . . . . . . . . 18 (((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))
69 fnfvima 6536 . . . . . . . . . . . . . . . . . . . 20 ((abs Fn ℂ ∧ (ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
7033, 69mp3an1 1451 . . . . . . . . . . . . . . . . . . 19 (((ran 𝑓 ∪ ran 𝑔) ⊆ ℂ ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
7126, 70sylan 487 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ∈ (abs “ (ran 𝑓 ∪ ran 𝑔)))
72 suprub 11022 . . . . . . . . . . . . . . . . . 18 ((((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, < ))
7368, 71, 72syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
7473adantrr 753 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → (abs‘𝑟) ≤ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
7557, 60, 61, 65, 74ltletrd 10235 . . . . . . . . . . . . . . 15 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ (𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ∧ 𝑟 ≠ 0)) → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
7675rexlimdvaa 3061 . . . . . . . . . . . . . 14 ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) → (∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0 → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))
7776imp 444 . . . . . . . . . . . . 13 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → 0 < sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))
7856, 77elrpd 11907 . . . . . . . . . . . 12 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+)
79 rpmulcl 11893 . . . . . . . . . . . 12 ((2 ∈ ℝ+ ∧ sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ) ∈ ℝ+) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
8015, 78, 79sylancr 696 . . . . . . . . . . 11 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+)
81 rpdivcl 11894 . . . . . . . . . . 11 (((𝑦 / 2) ∈ ℝ+ ∧ (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )) ∈ ℝ+) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
8214, 80, 81syl2an 493 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ ((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0)) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
8382anassrs 681 . . . . . . . . 9 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) → ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
8483adantlr 751 . . . . . . . 8 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+)
85 ancom 465 . . . . . . . . . . . 12 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+) ↔ (𝑦 ∈ ℝ+𝑢 ∈ (𝐴[,]𝐵)))
8685anbi2i 730 . . . . . . . . . . 11 (((((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) ∧ 𝑦 ∈ ℝ+)) ↔ ((((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))))
87 an32 856 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ↔ ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)))
8887anbi1i 731 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ↔ (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)))
89 an32 856 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ↔ (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)))
9088, 89bitri 264 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ↔ (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)))
9190anbi1i 731 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) ↔ ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0))
92 an32 856 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ↔ ((((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) ∧ 𝑦 ∈ ℝ+)))
9391, 92bitri 264 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) ↔ ((((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) ∧ 𝑦 ∈ ℝ+)))
94 anass 682 . . . . . . . . . . 11 ((((((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)) ↔ ((((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))))
9586, 93, 943bitr4i 292 . . . . . . . . . 10 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) ↔ (((((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)))
96 oveq12 6699 . . . . . . . . . . . . . . . 16 ((𝑏 = 𝑤𝑎 = 𝑢) → (𝑏𝑎) = (𝑤𝑢))
9796ancoms 468 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑢𝑏 = 𝑤) → (𝑏𝑎) = (𝑤𝑢))
9897fveq2d 6233 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢𝑏 = 𝑤) → (abs‘(𝑏𝑎)) = (abs‘(𝑤𝑢)))
9998breq1d 4695 . . . . . . . . . . . . 13 ((𝑎 = 𝑢𝑏 = 𝑤) → ((abs‘(𝑏𝑎)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ↔ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
100 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑤 → (𝐺𝑏) = (𝐺𝑤))
101 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑢 → (𝐺𝑎) = (𝐺𝑢))
102100, 101oveqan12rd 6710 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑢𝑏 = 𝑤) → ((𝐺𝑏) − (𝐺𝑎)) = ((𝐺𝑤) − (𝐺𝑢)))
103102fveq2d 6233 . . . . . . . . . . . . . 14 ((𝑎 = 𝑢𝑏 = 𝑤) → (abs‘((𝐺𝑏) − (𝐺𝑎))) = (abs‘((𝐺𝑤) − (𝐺𝑢))))
104103breq1d 4695 . . . . . . . . . . . . 13 ((𝑎 = 𝑢𝑏 = 𝑤) → ((abs‘((𝐺𝑏) − (𝐺𝑎))) < 𝑦 ↔ (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
10599, 104imbi12d 333 . . . . . . . . . . . 12 ((𝑎 = 𝑢𝑏 = 𝑤) → (((abs‘(𝑏𝑎)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑏) − (𝐺𝑎))) < 𝑦) ↔ ((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)))
106 oveq12 6699 . . . . . . . . . . . . . . . 16 ((𝑏 = 𝑢𝑎 = 𝑤) → (𝑏𝑎) = (𝑢𝑤))
107106ancoms 468 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑤𝑏 = 𝑢) → (𝑏𝑎) = (𝑢𝑤))
108107fveq2d 6233 . . . . . . . . . . . . . 14 ((𝑎 = 𝑤𝑏 = 𝑢) → (abs‘(𝑏𝑎)) = (abs‘(𝑢𝑤)))
109108breq1d 4695 . . . . . . . . . . . . 13 ((𝑎 = 𝑤𝑏 = 𝑢) → ((abs‘(𝑏𝑎)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ↔ (abs‘(𝑢𝑤)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
110 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑢 → (𝐺𝑏) = (𝐺𝑢))
111 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑤 → (𝐺𝑎) = (𝐺𝑤))
112110, 111oveqan12rd 6710 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑤𝑏 = 𝑢) → ((𝐺𝑏) − (𝐺𝑎)) = ((𝐺𝑢) − (𝐺𝑤)))
113112fveq2d 6233 . . . . . . . . . . . . . 14 ((𝑎 = 𝑤𝑏 = 𝑢) → (abs‘((𝐺𝑏) − (𝐺𝑎))) = (abs‘((𝐺𝑢) − (𝐺𝑤))))
114113breq1d 4695 . . . . . . . . . . . . 13 ((𝑎 = 𝑤𝑏 = 𝑢) → ((abs‘((𝐺𝑏) − (𝐺𝑎))) < 𝑦 ↔ (abs‘((𝐺𝑢) − (𝐺𝑤))) < 𝑦))
115109, 114imbi12d 333 . . . . . . . . . . . 12 ((𝑎 = 𝑤𝑏 = 𝑢) → (((abs‘(𝑏𝑎)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑏) − (𝐺𝑎))) < 𝑦) ↔ ((abs‘(𝑢𝑤)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑢) − (𝐺𝑤))) < 𝑦)))
116 iccssre 12293 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
1172, 3, 116syl2anc 694 . . . . . . . . . . . . 13 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
118117ad4antr 769 . . . . . . . . . . . 12 (((((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+) → (𝐴[,]𝐵) ⊆ ℝ)
119 simp-4l 823 . . . . . . . . . . . . 13 (((((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)
120117, 25syl6ss 3648 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
121120sselda 3636 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℂ)
122120sselda 3636 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℂ)
123 abssub 14110 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (abs‘(𝑤𝑢)) = (abs‘(𝑢𝑤)))
124121, 122, 123syl2anr 494 . . . . . . . . . . . . . . . 16 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑𝑤 ∈ (𝐴[,]𝐵))) → (abs‘(𝑤𝑢)) = (abs‘(𝑢𝑤)))
125124anandis 890 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘(𝑤𝑢)) = (abs‘(𝑢𝑤)))
126125breq1d 4695 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ↔ (abs‘(𝑢𝑤)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
1279ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝐺𝑤) ∈ ℂ)
1289ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝐺𝑢) ∈ ℂ)
129 abssub 14110 . . . . . . . . . . . . . . . . 17 (((𝐺𝑤) ∈ ℂ ∧ (𝐺𝑢) ∈ ℂ) → (abs‘((𝐺𝑤) − (𝐺𝑢))) = (abs‘((𝐺𝑢) − (𝐺𝑤))))
130127, 128, 129syl2anr 494 . . . . . . . . . . . . . . . 16 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑𝑤 ∈ (𝐴[,]𝐵))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) = (abs‘((𝐺𝑢) − (𝐺𝑤))))
131130anandis 890 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) = (abs‘((𝐺𝑢) − (𝐺𝑤))))
132131breq1d 4695 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦 ↔ (abs‘((𝐺𝑢) − (𝐺𝑤))) < 𝑦))
133126, 132imbi12d 333 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦) ↔ ((abs‘(𝑢𝑤)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑢) − (𝐺𝑤))) < 𝑦)))
134119, 133sylan 487 . . . . . . . . . . . 12 ((((((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))) → (((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦) ↔ ((abs‘(𝑢𝑤)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑢) − (𝐺𝑤))) < 𝑦)))
1352rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐴 ∈ ℝ*)
1363rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐵 ∈ ℝ*)
137135, 136jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
138 df-icc 12220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑡 ∈ ℝ* ∣ (𝑥𝑡𝑡𝑦)})
139138elixx3g 12226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑢 ∈ ℝ*) ∧ (𝐴𝑢𝑢𝐵)))
140139simprbi 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢 ∈ (𝐴[,]𝐵) → (𝐴𝑢𝑢𝐵))
141140simpld 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 ∈ (𝐴[,]𝐵) → 𝐴𝑢)
142138elixx3g 12226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 ∈ (𝐴[,]𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) ∧ (𝐴𝑤𝑤𝐵)))
143142simprbi 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ (𝐴[,]𝐵) → (𝐴𝑤𝑤𝐵))
144143simprd 478 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ (𝐴[,]𝐵) → 𝑤𝐵)
145141, 144anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴𝑢𝑤𝐵))
146 ioossioo 12303 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑢𝑤𝐵)) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
147137, 145, 146syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ (𝐴(,)𝐵))
1485adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ 𝐷)
149147, 148sstrd 3646 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ⊆ 𝐷)
150149sselda 3636 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 𝑡𝐷)
1518ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
152151abscld 14219 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑡𝐷) → (abs‘(𝐹𝑡)) ∈ ℝ)
153152rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑡𝐷) → (abs‘(𝐹𝑡)) ∈ ℝ*)
154151absge0d 14227 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑡𝐷) → 0 ≤ (abs‘(𝐹𝑡)))
155 elxrge0 12319 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((abs‘(𝐹𝑡)) ∈ (0[,]+∞) ↔ ((abs‘(𝐹𝑡)) ∈ ℝ* ∧ 0 ≤ (abs‘(𝐹𝑡))))
156153, 154, 155sylanbrc 699 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑡𝐷) → (abs‘(𝐹𝑡)) ∈ (0[,]+∞))
157156adantlr 751 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (abs‘(𝐹𝑡)) ∈ (0[,]+∞))
158150, 157syldan 486 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘(𝐹𝑡)) ∈ (0[,]+∞))
159 0e0iccpnf 12321 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ (0[,]+∞)
160159a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
161158, 160ifclda 4153 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ∈ (0[,]+∞))
162161adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ∈ (0[,]+∞))
163 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))
164162, 163fmptd 6425 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞))
165 itg2cl 23544 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
166164, 165syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
1671663adantr3 1242 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
168119, 167sylan 487 . . . . . . . . . . . . . . . 16 ((((((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)), 0))) ∈ ℝ*)
169168adantr 480 . . . . . . . . . . . . . . 15 (((((((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)), 0))) ∈ ℝ*)
170 simplll 813 . . . . . . . . . . . . . . . . 17 (((((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)))
171151adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
172150, 171syldan 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
173172adantllr 755 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
174 elioore 12243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑡 ∈ (𝑢(,)𝑤) → 𝑡 ∈ ℝ)
17516ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℝ)
176175recnd 10106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ℂ)
177 ax-icn 10033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 i ∈ ℂ
17820ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℝ)
179178recnd 10106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℂ)
180 mulcl 10058 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((i ∈ ℂ ∧ (𝑔𝑡) ∈ ℂ) → (i · (𝑔𝑡)) ∈ ℂ)
181177, 179, 180sylancr 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (i · (𝑔𝑡)) ∈ ℂ)
182 addcl 10056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑓𝑡) ∈ ℂ ∧ (i · (𝑔𝑡)) ∈ ℂ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
183176, 181, 182syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
184183anandirs 891 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
185174, 184sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
186185adantll 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
187186adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝑓𝑡) + (i · (𝑔𝑡))) ∈ ℂ)
188173, 187subcld 10430 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℂ)
189188abscld 14219 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
190185abscld 14219 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
191190adantll 750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
192191adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))) ∈ ℝ)
193189, 192readdcld 10107 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ)
194193rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ*)
195188absge0d 14227 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))))
196184absge0d 14227 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
197174, 196sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
198197adantll 750 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
199198adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))
200189, 192, 195, 199addge0d 10641 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ≤ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
201 elxrge0 12319 . . . . . . . . . . . . . . . . . . . . . . 23 (((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞) ↔ (((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ ℝ* ∧ 0 ≤ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡)))))))
202194, 200, 201sylanbrc 699 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ (0[,]+∞))
203159a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡 ∈ (𝑢(,)𝑤)) → 0 ∈ (0[,]+∞))
204202, 203ifclda 4153 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,]+∞))
205204adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ (0[,]+∞))
206 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
207205, 206fmptd 6425 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞))
208 itg2cl 23544 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ*)
209207, 208syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ*)
2102093adantr3 1242 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ*)
211170, 210sylan 487 . . . . . . . . . . . . . . . 16 ((((((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))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) ∈ ℝ*)
212211adantr 480 . . . . . . . . . . . . . . 15 (((((((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))) ∈ ℝ*)
213 rpxr 11878 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ+𝑦 ∈ ℝ*)
214213ad3antlr 767 . . . . . . . . . . . . . . 15 (((((((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 ∈ ℝ*)
215161adantlr 751 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ∈ (0[,]+∞))
216215adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ∈ (0[,]+∞))
217216, 163fmptd 6425 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞))
218173, 187npcand 10434 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) + ((𝑓𝑡) + (i · (𝑔𝑡)))) = (𝐹𝑡))
219218fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘(((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) + ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘(𝐹𝑡)))
220188, 187abstrid 14239 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘(((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡)))) + ((𝑓𝑡) + (i · (𝑔𝑡))))) ≤ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
221219, 220eqbrtrrd 4709 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (abs‘(𝐹𝑡)) ≤ ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
222 iftrue 4125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) = (abs‘(𝐹𝑡)))
223222adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) = (abs‘(𝐹𝑡)))
224 iftrue 4125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
225224adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))))
226221, 223, 2253brtr4d 4717 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
227226ex 449 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
228 0le0 11148 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ≤ 0
229228a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡 ∈ (𝑢(,)𝑤) → 0 ≤ 0)
230 iffalse 4128 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) = 0)
231 iffalse 4128 . . . . . . . . . . . . . . . . . . . . . . 23 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0) = 0)
232229, 230, 2313brtr4d 4717 . . . . . . . . . . . . . . . . . . . . . 22 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
233227, 232pm2.61d1 171 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
234233ralrimivw 2996 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))
235 reex 10065 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
236235a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ℝ ∈ V)
237 fvex 6239 . . . . . . . . . . . . . . . . . . . . . . . 24 (abs‘(𝐹𝑡)) ∈ V
238 c0ex 10072 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
239237, 238ifex 4189 . . . . . . . . . . . . . . . . . . . . . . 23 if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ∈ V
240239a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ∈ V)
241 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))) ∈ V
242241, 238ifex 4189 . . . . . . . . . . . . . . . . . . . . . . 23 if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ V
243242a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡 ∈ ℝ) → if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0) ∈ V)
244 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)))
245 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
246236, 240, 243, 244, 245ofrfval2 6957 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
247246ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
248234, 247mpbird 247 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)))
249 itg2le 23551 . . . . . . . . . . . . . . . . . . 19 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
250217, 207, 248, 249syl3anc 1366 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
2512503adantr3 1242 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
252170, 251sylan 487 . . . . . . . . . . . . . . . 16 ((((((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)), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
253252adantr 480 . . . . . . . . . . . . . . 15 (((((((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)), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹𝑡) − ((𝑓𝑡) + (i · (𝑔𝑡))))) + (abs‘((𝑓𝑡) + (i · (𝑔𝑡))))), 0))))
2541, 2, 3, 4, 5, 6, 7, 8ftc1anclem8 33622 . . . . . . . . . . . . . . 15 (((((((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)
255169, 212, 214, 253, 254xrlelttrd 12029 . . . . . . . . . . . . . 14 (((((((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)), 0))) < 𝑦)
256 simplll 813 . . . . . . . . . . . . . . . 16 ((((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) → 𝜑)
257 simpr2 1088 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝑤 ∈ (𝐴[,]𝐵))
258 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑤 → (𝐴(,)𝑥) = (𝐴(,)𝑤))
259 itgeq1 23584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴(,)𝑥) = (𝐴(,)𝑤) → ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡 = ∫(𝐴(,)𝑤)(𝐹𝑡) d𝑡)
260258, 259syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑤 → ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡 = ∫(𝐴(,)𝑤)(𝐹𝑡) d𝑡)
261 itgex 23582 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∫(𝐴(,)𝑤)(𝐹𝑡) d𝑡 ∈ V
262260, 1, 261fvmpt 6321 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 ∈ (𝐴[,]𝐵) → (𝐺𝑤) = ∫(𝐴(,)𝑤)(𝐹𝑡) d𝑡)
263257, 262syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝐺𝑤) = ∫(𝐴(,)𝑤)(𝐹𝑡) d𝑡)
2642adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝐴 ∈ ℝ)
265117sselda 3636 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℝ)
2662653ad2antr2 1247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝑤 ∈ ℝ)
267117sselda 3636 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℝ)
268267rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → 𝑢 ∈ ℝ*)
2692683ad2antr1 1246 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝑢 ∈ ℝ*)
270 elicc1 12257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑢 ∈ (𝐴[,]𝐵) ↔ (𝑢 ∈ ℝ*𝐴𝑢𝑢𝐵)))
271135, 136, 270syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑢 ∈ (𝐴[,]𝐵) ↔ (𝑢 ∈ ℝ*𝐴𝑢𝑢𝐵)))
272271biimpa 500 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝑢 ∈ ℝ*𝐴𝑢𝑢𝐵))
273272simp2d 1094 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → 𝐴𝑢)
2742733ad2antr1 1246 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝐴𝑢)
275 simpr3 1089 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝑢𝑤)
276135adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ*)
277265rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ ℝ*)
278 elicc1 12257 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝑢 ∈ (𝐴[,]𝑤) ↔ (𝑢 ∈ ℝ*𝐴𝑢𝑢𝑤)))
279276, 277, 278syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝑢 ∈ (𝐴[,]𝑤) ↔ (𝑢 ∈ ℝ*𝐴𝑢𝑢𝑤)))
2802793ad2antr2 1247 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑢 ∈ (𝐴[,]𝑤) ↔ (𝑢 ∈ ℝ*𝐴𝑢𝑢𝑤)))
281269, 274, 275, 280mpbir3and 1264 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝑢 ∈ (𝐴[,]𝑤))
282 iooss2 12249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵 ∈ ℝ*𝑤𝐵) → (𝐴(,)𝑤) ⊆ (𝐴(,)𝐵))
283136, 144, 282syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑤) ⊆ (𝐴(,)𝐵))
2845adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝐵) ⊆ 𝐷)
285283, 284sstrd 3646 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑤) ⊆ 𝐷)
2862853ad2antr2 1247 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝐴(,)𝑤) ⊆ 𝐷)
287286sselda 3636 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝐴(,)𝑤)) → 𝑡𝐷)
288151adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
289287, 288syldan 486 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) ∧ 𝑡 ∈ (𝐴(,)𝑤)) → (𝐹𝑡) ∈ ℂ)
290 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑢 → (𝑤 ∈ (𝐴[,]𝐵) ↔ 𝑢 ∈ (𝐴[,]𝐵)))
291290anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 = 𝑢 → ((𝜑𝑤 ∈ (𝐴[,]𝐵)) ↔ (𝜑𝑢 ∈ (𝐴[,]𝐵))))
292 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = 𝑢 → (𝐴(,)𝑤) = (𝐴(,)𝑢))
293292mpteq1d 4771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = 𝑢 → (𝑡 ∈ (𝐴(,)𝑤) ↦ (𝐹𝑡)) = (𝑡 ∈ (𝐴(,)𝑢) ↦ (𝐹𝑡)))
294293eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 = 𝑢 → ((𝑡 ∈ (𝐴(,)𝑤) ↦ (𝐹𝑡)) ∈ 𝐿1 ↔ (𝑡 ∈ (𝐴(,)𝑢) ↦ (𝐹𝑡)) ∈ 𝐿1))
295291, 294imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = 𝑢 → (((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑤) ↦ (𝐹𝑡)) ∈ 𝐿1) ↔ ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑢) ↦ (𝐹𝑡)) ∈ 𝐿1)))
296 ioombl 23379 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐴(,)𝑤) ∈ dom vol
297296a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑤) ∈ dom vol)
298151adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑤 ∈ (𝐴[,]𝐵)) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
2998feqmptd 6288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐹 = (𝑡𝐷 ↦ (𝐹𝑡)))
300299, 7eqeltrrd 2731 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ 𝐿1)
301300adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ 𝐿1)
302285, 297, 298, 301iblss 23616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑤 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑤) ↦ (𝐹𝑡)) ∈ 𝐿1)
303295, 302chvarv 2299 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑢) ↦ (𝐹𝑡)) ∈ 𝐿1)
3043033ad2antr1 1246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑡 ∈ (𝐴(,)𝑢) ↦ (𝐹𝑡)) ∈ 𝐿1)
305 ioombl 23379 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢(,)𝑤) ∈ dom vol
306305a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑢(,)𝑤) ∈ dom vol)
307 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → (𝐹𝑡) ∈ V)
308300adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ 𝐿1)
309149, 306, 307, 308iblss 23616 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ (𝑢(,)𝑤) ↦ (𝐹𝑡)) ∈ 𝐿1)
3103093adantr3 1242 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝑡 ∈ (𝑢(,)𝑤) ↦ (𝐹𝑡)) ∈ 𝐿1)
311264, 266, 281, 289, 304, 310itgsplitioo 23649 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ∫(𝐴(,)𝑤)(𝐹𝑡) d𝑡 = (∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 + ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡))
312263, 311eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝐺𝑤) = (∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 + ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡))
313 simpr1 1087 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝑢 ∈ (𝐴[,]𝐵))
314 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑢 → (𝐴(,)𝑥) = (𝐴(,)𝑢))
315 itgeq1 23584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴(,)𝑥) = (𝐴(,)𝑢) → ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡 = ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡)
316314, 315syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑢 → ∫(𝐴(,)𝑥)(𝐹𝑡) d𝑡 = ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡)
317 itgex 23582 . . . . . . . . . . . . . . . . . . . . . . . 24 ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 ∈ V
318316, 1, 317fvmpt 6321 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 ∈ (𝐴[,]𝐵) → (𝐺𝑢) = ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡)
319313, 318syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (𝐺𝑢) = ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡)
320312, 319oveq12d 6708 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ((𝐺𝑤) − (𝐺𝑢)) = ((∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 + ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) − ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡))
321 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑢)) → (𝐹𝑡) ∈ V)
322321, 303itgcl 23595 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑢 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 ∈ ℂ)
323322adantrr 753 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 ∈ ℂ)
324 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝑢(,)𝑤)) → (𝐹𝑡) ∈ V)
325324, 309itgcl 23595 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡 ∈ ℂ)
326323, 325pncan2d 10432 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 + ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) − ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡) = ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡)
3273263adantr3 1242 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ((∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡 + ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) − ∫(𝐴(,)𝑢)(𝐹𝑡) d𝑡) = ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡)
328320, 327eqtrd 2685 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ((𝐺𝑤) − (𝐺𝑢)) = ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡)
329328fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) = (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡))
330 ftc1anc.t . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘(𝐹𝑡)), 0))))
331 df-ov 6693 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢(,)𝑤) = ((,)‘⟨𝑢, 𝑤⟩)
332 opelxpi 5182 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ⟨𝑢, 𝑤⟩ ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))
333 ioof 12309 . . . . . . . . . . . . . . . . . . . . . . . . 25 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
334 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
335333, 334ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 (,) Fn (ℝ* × ℝ*)
336 iccssxr 12294 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴[,]𝐵) ⊆ ℝ*
337 xpss12 5158 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐴[,]𝐵) ⊆ ℝ* ∧ (𝐴[,]𝐵) ⊆ ℝ*) → ((𝐴[,]𝐵) × (𝐴[,]𝐵)) ⊆ (ℝ* × ℝ*))
338336, 336, 337mp2an 708 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴[,]𝐵) × (𝐴[,]𝐵)) ⊆ (ℝ* × ℝ*)
339 fnfvima 6536 . . . . . . . . . . . . . . . . . . . . . . . 24 (((,) Fn (ℝ* × ℝ*) ∧ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) ⊆ (ℝ* × ℝ*) ∧ ⟨𝑢, 𝑤⟩ ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((,)‘⟨𝑢, 𝑤⟩) ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
340335, 338, 339mp3an12 1454 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑢, 𝑤⟩ ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘⟨𝑢, 𝑤⟩) ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
341332, 340syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ((,)‘⟨𝑢, 𝑤⟩) ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
342331, 341syl5eqel 2734 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝑢(,)𝑤) ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
343 itgeq1 23584 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 = (𝑢(,)𝑤) → ∫𝑠(𝐹𝑡) d𝑡 = ∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡)
344343fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = (𝑢(,)𝑤) → (abs‘∫𝑠(𝐹𝑡) d𝑡) = (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡))
345 eleq2 2719 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠 = (𝑢(,)𝑤) → (𝑡𝑠𝑡 ∈ (𝑢(,)𝑤)))
346345ifbid 4141 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠 = (𝑢(,)𝑤) → if(𝑡𝑠, (abs‘(𝐹𝑡)), 0) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))
347346mpteq2dv 4778 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 = (𝑢(,)𝑤) → (𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘(𝐹𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)))
348347fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = (𝑢(,)𝑤) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘(𝐹𝑡)), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))))
349344, 348breq12d 4698 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑢(,)𝑤) → ((abs‘∫𝑠(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘(𝐹𝑡)), 0))) ↔ (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)))))
350349rspccva 3339 . . . . . . . . . . . . . . . . . . . . 21 ((∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘(𝐹𝑡)), 0))) ∧ (𝑢(,)𝑤) ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))))
351330, 342, 350syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))))
3523513adantr3 1242 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))))
353329, 352eqbrtrd 4707 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))))
354353adantlr 751 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))))
355 subcl 10318 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐺𝑤) ∈ ℂ ∧ (𝐺𝑢) ∈ ℂ) → ((𝐺𝑤) − (𝐺𝑢)) ∈ ℂ)
356127, 128, 355syl2anr 494 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ (𝜑𝑤 ∈ (𝐴[,]𝐵))) → ((𝐺𝑤) − (𝐺𝑢)) ∈ ℂ)
357356anandis 890 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝐺𝑤) − (𝐺𝑢)) ∈ ℂ)
358357abscld 14219 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ)
359358rexrd 10127 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ*)
3603593adantr3 1242 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ*)
361360adantlr 751 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ*)
362167adantlr 751 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
363213ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → 𝑦 ∈ ℝ*)
364 xrlelttr 12025 . . . . . . . . . . . . . . . . . 18 (((abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ* ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ∈ ℝ*𝑦 ∈ ℝ*) → (((abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) < 𝑦) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
365361, 362, 363, 364syl3anc 1366 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (((abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) < 𝑦) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
366354, 365mpand 711 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) < 𝑦 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
367256, 366sylanl1 683 . . . . . . . . . . . . . . 15 ((((((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)), 0))) < 𝑦 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
368367adantr 480 . . . . . . . . . . . . . 14 (((((((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)), 0))) < 𝑦 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
369255, 368mpd 15 . . . . . . . . . . . . 13 (((((((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‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)
370369ex 449 . . . . . . . . . . . 12 ((((((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‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
371105, 115, 118, 134, 370wlogle 10599 . . . . . . . . . . 11 ((((((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))) → ((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
372371anassrs 681 . . . . . . . . . 10 (((((((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)) → ((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
37395, 372sylanb 488 . . . . . . . . 9 ((((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
374373ralrimiva 2995 . . . . . . . 8 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) → ∀𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
375 breq2 4689 . . . . . . . . . . 11 (𝑧 = ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → ((abs‘(𝑤𝑢)) < 𝑧 ↔ (abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))))
376375imbi1d 330 . . . . . . . . . 10 (𝑧 = ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦) ↔ ((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)))
377376ralbidv 3015 . . . . . . . . 9 (𝑧 = ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (∀𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦) ↔ ∀𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)))
378377rspcev 3340 . . . . . . . 8 ((((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) ∈ ℝ+ ∧ ∀𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < ))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
37984, 374, 378syl2anc 694 . . . . . . 7 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
380 ralnex 3021 . . . . . . . . 9 (∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ¬ 𝑟 ≠ 0 ↔ ¬ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0)
381 nne 2827 . . . . . . . . . 10 𝑟 ≠ 0 ↔ 𝑟 = 0)
382381ralbii 3009 . . . . . . . . 9 (∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔) ¬ 𝑟 ≠ 0 ↔ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0)
383380, 382bitr3i 266 . . . . . . . 8 (¬ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0 ↔ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0)
384 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
38516, 384syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 ∈ dom ∫1𝑓 Fn ℝ)
386 fnfvelrn 6396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑓 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
387385, 386sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ ran 𝑓)
388 elun1 3813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑓𝑡) ∈ ran 𝑓 → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
389387, 388syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
390 eqeq1 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑟 = (𝑓𝑡) → (𝑟 = 0 ↔ (𝑓𝑡) = 0))
391390rspcva 3338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑓𝑡) ∈ (ran 𝑓 ∪ ran 𝑔) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (𝑓𝑡) = 0)
392389, 391sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑓 ∈ dom ∫1𝑡 ∈ ℝ) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (𝑓𝑡) = 0)
393392adantllr 755 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (𝑓𝑡) = 0)
394 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑔:ℝ⟶ℝ → 𝑔 Fn ℝ)
39520, 394syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
396 fnfvelrn 6396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑔 Fn ℝ ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
397395, 396sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ran 𝑔)
398 elun2 3814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑔𝑡) ∈ ran 𝑔 → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
399397, 398syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔))
400 eqeq1 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑟 = (𝑔𝑡) → (𝑟 = 0 ↔ (𝑔𝑡) = 0))
401400rspcva 3338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (𝑔𝑡) = 0)
402401oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (i · (𝑔𝑡)) = (i · 0))
403 it0e0 11292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (i · 0) = 0
404402, 403syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑔𝑡) ∈ (ran 𝑓 ∪ ran 𝑔) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (i · (𝑔𝑡)) = 0)
405399, 404sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (i · (𝑔𝑡)) = 0)
406405adantlll 754 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (i · (𝑔𝑡)) = 0)
407393, 406oveq12d 6708 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → ((𝑓𝑡) + (i · (𝑔𝑡))) = (0 + 0))
408407an32s 863 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) = (0 + 0))
409 00id 10249 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 + 0) = 0
410408, 409syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) = 0)
411410adantlll 754 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → ((𝑓𝑡) + (i · (𝑔𝑡))) = 0)
412411oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) = (if(𝑡𝐷, (𝐹𝑡), 0) − 0))
413 0cnd 10071 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℂ)
414151, 413ifclda 4153 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ)
415414subid1d 10419 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (if(𝑡𝐷, (𝐹𝑡), 0) − 0) = if(𝑡𝐷, (𝐹𝑡), 0))
416415ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → (if(𝑡𝐷, (𝐹𝑡), 0) − 0) = if(𝑡𝐷, (𝐹𝑡), 0))
417412, 416eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → (if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))) = if(𝑡𝐷, (𝐹𝑡), 0))
418417fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = (abs‘if(𝑡𝐷, (𝐹𝑡), 0)))
419 fvif 6242 . . . . . . . . . . . . . . . . . . . . . 22 (abs‘if(𝑡𝐷, (𝐹𝑡), 0)) = if(𝑡𝐷, (abs‘(𝐹𝑡)), (abs‘0))
420 abs0 14069 . . . . . . . . . . . . . . . . . . . . . . 23 (abs‘0) = 0
421 ifeq2 4124 . . . . . . . . . . . . . . . . . . . . . . 23 ((abs‘0) = 0 → if(𝑡𝐷, (abs‘(𝐹𝑡)), (abs‘0)) = if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))
422420, 421ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 if(𝑡𝐷, (abs‘(𝐹𝑡)), (abs‘0)) = if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)
423419, 422eqtri 2673 . . . . . . . . . . . . . . . . . . . . 21 (abs‘if(𝑡𝐷, (𝐹𝑡), 0)) = if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)
424418, 423syl6eq 2701 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) ∧ 𝑡 ∈ ℝ) → (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))) = if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))
425424mpteq2dva 4777 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡)))))) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))
426425fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
427426breq1d 4695 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2) ↔ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)))
428427biimpd 219 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ ∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)))
429428ex 449 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → (∀𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 = 0 → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2))))
430429com23 86 . . . . . . . . . . . . . 14 ((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(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2))))
431430imp32 448 . . . . . . . . . . . . 13 (((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(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2))
432431anasss 680 . . . . . . . . . . . 12 ((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(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2))
433432adantlr 751 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ ((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(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2))
434 1rp 11874 . . . . . . . . . . . . 13 1 ∈ ℝ+
435434ne0ii 3956 . . . . . . . . . . . 12 + ≠ ∅
436357anassrs 681 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ((𝐺𝑤) − (𝐺𝑢)) ∈ ℂ)
437436abscld 14219 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ)
438437adantlrr 757 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ)
439438adantlr 751 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ)
440 rpre 11877 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
441440rehalfcld 11317 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
442441adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+) → (𝑦 / 2) ∈ ℝ)
443442ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝑦 / 2) ∈ ℝ)
444440adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
445444ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ)
446439rexrd 10127 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ∈ ℝ*)
447159a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ (0[,]+∞))
448156, 447ifclda 4153 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → if(𝑡𝐷, (abs‘(𝐹𝑡)), 0) ∈ (0[,]+∞))
449448adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘(𝐹𝑡)), 0) ∈ (0[,]+∞))
450 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))
451449, 450fmptd 6425 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞))
452 itg2cl 23544 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
453451, 452syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
454453ad3antrrr 766 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
455443rexrd 10127 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝑦 / 2) ∈ ℝ*)
456111, 110oveqan12rd 6710 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 = 𝑢𝑎 = 𝑤) → ((𝐺𝑎) − (𝐺𝑏)) = ((𝐺𝑤) − (𝐺𝑢)))
457456fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 = 𝑢𝑎 = 𝑤) → (abs‘((𝐺𝑎) − (𝐺𝑏))) = (abs‘((𝐺𝑤) − (𝐺𝑢))))
458457breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑢𝑎 = 𝑤) → ((abs‘((𝐺𝑎) − (𝐺𝑏))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) ↔ (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))))
459101, 100oveqan12rd 6710 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 = 𝑤𝑎 = 𝑢) → ((𝐺𝑎) − (𝐺𝑏)) = ((𝐺𝑢) − (𝐺𝑤)))
460459fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑏 = 𝑤𝑎 = 𝑢) → (abs‘((𝐺𝑎) − (𝐺𝑏))) = (abs‘((𝐺𝑢) − (𝐺𝑤))))
461460breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑤𝑎 = 𝑢) → ((abs‘((𝐺𝑎) − (𝐺𝑏))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) ↔ (abs‘((𝐺𝑢) − (𝐺𝑤))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))))
462131breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) ↔ (abs‘((𝐺𝑢) − (𝐺𝑤))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))))
463325abscld 14219 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ∈ ℝ)
464463rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ∈ ℝ*)
465453adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) ∈ ℝ*)
466451adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞))
467 breq2 4689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((abs‘(𝐹𝑡)) = if(𝑡𝐷, (abs‘(𝐹𝑡)), 0) → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ (abs‘(𝐹𝑡)) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))
468 breq2 4689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 = if(𝑡𝐷, (abs‘(𝐹𝑡)), 0) → (if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ 0 ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))
469152leidd 10632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑡𝐷) → (abs‘(𝐹𝑡)) ≤ (abs‘(𝐹𝑡)))
470 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((abs‘(𝐹𝑡)) = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) → ((abs‘(𝐹𝑡)) ≤ (abs‘(𝐹𝑡)) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ (abs‘(𝐹𝑡))))
471 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0 = if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) → (0 ≤ (abs‘(𝐹𝑡)) ↔ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ (abs‘(𝐹𝑡))))
472470, 471ifboth 4157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((abs‘(𝐹𝑡)) ≤ (abs‘(𝐹𝑡)) ∧ 0 ≤ (abs‘(𝐹𝑡))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ (abs‘(𝐹𝑡)))
473469, 154, 472syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑡𝐷) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ (abs‘(𝐹𝑡)))
474473adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ 𝑡𝐷) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ (abs‘(𝐹𝑡)))
475149ssneld 3638 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (¬ 𝑡𝐷 → ¬ 𝑡 ∈ (𝑢(,)𝑤)))
476475imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡𝐷) → ¬ 𝑡 ∈ (𝑢(,)𝑤))
477230, 228syl6eqbr 4724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑡 ∈ (𝑢(,)𝑤) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ 0)
478476, 477syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) ∧ ¬ 𝑡𝐷) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ 0)
479467, 468, 474, 478ifbothda 4156 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))
480479ralrimivw 2996 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))
481237, 238ifex 4189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 if(𝑡𝐷, (abs‘(𝐹𝑡)), 0) ∈ V
482481a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑡 ∈ ℝ) → if(𝑡𝐷, (abs‘(𝐹𝑡)), 0) ∈ V)
483 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))
484236, 240, 482, 244, 483ofrfval2 6957 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))
485484adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → ((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)) ↔ ∀𝑡 ∈ ℝ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0) ≤ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))
486480, 485mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)))
487 itg2le 23551 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0)):ℝ⟶(0[,]+∞) ∧ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0)) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
488164, 466, 486, 487syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘(𝐹𝑡)), 0))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
489464, 166, 465, 351, 488xrletrd 12031 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
4904893adantr3 1242 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘∫(𝑢(,)𝑤)(𝐹𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
491329, 490eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢𝑤)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
492458, 461, 117, 462, 491wlogle 10599 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵))) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
493492anassrs 681 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑢 ∈ (𝐴[,]𝐵)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
494493adantlrr 757 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
495494adantlr 751 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))))
496 simplr 807 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2))
497446, 454, 455, 495, 496xrlelttrd 12029 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < (𝑦 / 2))
498 rphalflt 11898 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
499498adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+) → (𝑦 / 2) < 𝑦)
500499ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝑦 / 2) < 𝑦)
501439, 443, 445, 497, 500lttrd 10236 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)
502501a1d 25 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
503502ralrimiva 2995 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) → ∀𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
504503ralrimivw 2996 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) → ∀𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
505 r19.2z 4093 . . . . . . . . . . . 12 ((ℝ+ ≠ ∅ ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
506435, 504, 505sylancr 696 . . . . . . . . . . 11 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝐷, (abs‘(𝐹𝑡)), 0))) < (𝑦 / 2)) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
507433, 506syldan 486 . . . . . . . . . 10 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ ((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))) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
508507anassrs 681 . . . . . . . . 9 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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)) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
509508anassrs 681 . . . . . . . 8 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
510383, 509sylan2b 491 . . . . . . 7 (((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (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) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
511379, 510pm2.61dan 849 . . . . . 6 ((((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2)) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
512511ex 449 . . . . 5 (((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) ∧ (𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1)) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)))
513512rexlimdvva 3067 . . . 4 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) → (∃𝑓 ∈ dom ∫1𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡𝐷, (𝐹𝑡), 0) − ((𝑓𝑡) + (i · (𝑔𝑡))))))) < (𝑦 / 2) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦)))
51413, 513mpd 15 . . 3 ((𝜑 ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ ℝ+)) → ∃𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
515514ralrimivva 3000 . 2 (𝜑 → ∀𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))
516 ssid 3657 . . 3 ℂ ⊆ ℂ
517 elcncf2 22740 . . 3 (((𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ ∀𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))))
518120, 516, 517sylancl 695 . 2 (𝜑 → (𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ) ↔ (𝐺:(𝐴[,]𝐵)⟶ℂ ∧ ∀𝑢 ∈ (𝐴[,]𝐵)∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ (𝐴[,]𝐵)((abs‘(𝑤𝑢)) < 𝑧 → (abs‘((𝐺𝑤) − (𝐺𝑢))) < 𝑦))))
5199, 515, 518mpbir2and 977 1 (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cun 3605  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑟 cofr 6938  Fincfn 7997  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975  ici 9976   + caddc 9977   · cmul 9979  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  2c2 11108  +crp 11870  (,)cioo 12213  [,]cicc 12216  abscabs 14018  cnccncf 22726  volcvol 23278  1citg1 23429  2citg2 23430  𝐿1cibl 23431  citg 23432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-cncf 22728  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435  df-ibl 23436  df-itg 23437  df-0p 23482
This theorem is referenced by:  ftc2nc  33624
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