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Theorem ftc1anclem5 33795
Description: Lemma for ftc1anc 33799, the existence of a simple function the integral of whose pointwise difference from the function is less than a given positive real. (Contributed by Brendan Leahy, 17-Jun-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
ftc1anclem5 ((𝜑𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌)
Distinct variable groups:   𝑡,𝑓,𝑥,𝐴   𝐵,𝑓,𝑡,𝑥   𝐷,𝑓,𝑡,𝑥   𝑓,𝐹,𝑡,𝑥   𝜑,𝑓,𝑡,𝑥   𝑓,𝐺   𝑓,𝑌,𝑡,𝑥
Allowed substitution hints:   𝐺(𝑥,𝑡)

Proof of Theorem ftc1anclem5
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 iftrue 4279 . . . . . . . . 9 (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
21mpteq2ia 4927 . . . . . . . 8 (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
32fveq2i 6405 . . . . . . 7 (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
4 ftc1anc.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝐷⟶ℂ)
54ffvelrnda 6575 . . . . . . . . . . . . 13 ((𝜑𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
6 0cnd 10312 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℂ)
75, 6ifclda 4307 . . . . . . . . . . . 12 (𝜑 → if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ)
87recld 14151 . . . . . . . . . . 11 (𝜑 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
98adantr 468 . . . . . . . . . 10 ((𝜑𝑡 ∈ ℝ) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
10 ftc1anc.d . . . . . . . . . . 11 (𝜑𝐷 ⊆ ℝ)
11 rembl 23515 . . . . . . . . . . . 12 ℝ ∈ dom vol
1211a1i 11 . . . . . . . . . . 11 (𝜑 → ℝ ∈ dom vol)
138adantr 468 . . . . . . . . . . 11 ((𝜑𝑡𝐷) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
14 eldifn 3926 . . . . . . . . . . . . 13 (𝑡 ∈ (ℝ ∖ 𝐷) → ¬ 𝑡𝐷)
1514adantl 469 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ (ℝ ∖ 𝐷)) → ¬ 𝑡𝐷)
16 iffalse 4282 . . . . . . . . . . . . . 14 𝑡𝐷 → if(𝑡𝐷, (𝐹𝑡), 0) = 0)
1716fveq2d 6406 . . . . . . . . . . . . 13 𝑡𝐷 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = (ℜ‘0))
18 re0 14109 . . . . . . . . . . . . 13 (ℜ‘0) = 0
1917, 18syl6eq 2852 . . . . . . . . . . . 12 𝑡𝐷 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = 0)
2015, 19syl 17 . . . . . . . . . . 11 ((𝜑𝑡 ∈ (ℝ ∖ 𝐷)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = 0)
21 iftrue 4279 . . . . . . . . . . . . . 14 (𝑡𝐷 → if(𝑡𝐷, (𝐹𝑡), 0) = (𝐹𝑡))
2221fveq2d 6406 . . . . . . . . . . . . 13 (𝑡𝐷 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = (ℜ‘(𝐹𝑡)))
2322mpteq2ia 4927 . . . . . . . . . . . 12 (𝑡𝐷 ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡)))
244feqmptd 6464 . . . . . . . . . . . . . . 15 (𝜑𝐹 = (𝑡𝐷 ↦ (𝐹𝑡)))
25 ftc1anc.i . . . . . . . . . . . . . . 15 (𝜑𝐹 ∈ 𝐿1)
2624, 25eqeltrrd 2882 . . . . . . . . . . . . . 14 (𝜑 → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ 𝐿1)
275iblcn 23773 . . . . . . . . . . . . . 14 (𝜑 → ((𝑡𝐷 ↦ (𝐹𝑡)) ∈ 𝐿1 ↔ ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)))
2826, 27mpbid 223 . . . . . . . . . . . . 13 (𝜑 → ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
2928simpld 484 . . . . . . . . . . . 12 (𝜑 → (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1)
3023, 29syl5eqel 2885 . . . . . . . . . . 11 (𝜑 → (𝑡𝐷 ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ 𝐿1)
3110, 12, 13, 20, 30iblss2 23780 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ 𝐿1)
328recnd 10347 . . . . . . . . . . . . 13 (𝜑 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ)
3332adantr 468 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ℝ) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ)
34 eqidd 2803 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
35 absf 14294 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
3635a1i 11 . . . . . . . . . . . . 13 (𝜑 → abs:ℂ⟶ℝ)
3736feqmptd 6464 . . . . . . . . . . . 12 (𝜑 → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
38 fveq2 6402 . . . . . . . . . . . 12 (𝑥 = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (abs‘𝑥) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3933, 34, 37, 38fmptco 6613 . . . . . . . . . . 11 (𝜑 → (abs ∘ (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) = (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
409fmpttd 6601 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))):ℝ⟶ℝ)
41 iblmbf 23742 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
4225, 41syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 ∈ MblFn)
4324, 42eqeltrrd 2882 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ MblFn)
445ismbfcn2 23613 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑡𝐷 ↦ (𝐹𝑡)) ∈ MblFn ↔ ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)))
4543, 44mpbid 223 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
4645simpld 484 . . . . . . . . . . . . . 14 (𝜑 → (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn)
4723, 46syl5eqel 2885 . . . . . . . . . . . . 13 (𝜑 → (𝑡𝐷 ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ MblFn)
4810, 12, 13, 20, 47mbfss 23621 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ MblFn)
49 ftc1anclem1 33791 . . . . . . . . . . . 12 (((𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))):ℝ⟶ℝ ∧ (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ MblFn) → (abs ∘ (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ MblFn)
5040, 48, 49syl2anc 575 . . . . . . . . . . 11 (𝜑 → (abs ∘ (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ MblFn)
5139, 50eqeltrrd 2882 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ MblFn)
529, 31, 51iblabsnc 33780 . . . . . . . . 9 (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ 𝐿1)
5332abscld 14392 . . . . . . . . . . 11 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ)
5453adantr 468 . . . . . . . . . 10 ((𝜑𝑡 ∈ ℝ) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ)
5532absge0d 14400 . . . . . . . . . . 11 (𝜑 → 0 ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
5655adantr 468 . . . . . . . . . 10 ((𝜑𝑡 ∈ ℝ) → 0 ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
5754, 56iblpos 23767 . . . . . . . . 9 (𝜑 → ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ 𝐿1 ↔ ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0))) ∈ ℝ)))
5852, 57mpbid 223 . . . . . . . 8 (𝜑 → ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ MblFn ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0))) ∈ ℝ))
5958simprd 485 . . . . . . 7 (𝜑 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0))) ∈ ℝ)
603, 59syl5eqelr 2886 . . . . . 6 (𝜑 → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ)
61 ltsubrp 12074 . . . . . 6 (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ ∧ 𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))))
6260, 61sylan 571 . . . . 5 ((𝜑𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))))
63 rpre 12047 . . . . . . 7 (𝑌 ∈ ℝ+𝑌 ∈ ℝ)
64 resubcl 10624 . . . . . . 7 (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ ∧ 𝑌 ∈ ℝ) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ)
6560, 63, 64syl2an 585 . . . . . 6 ((𝜑𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ)
6660adantr 468 . . . . . 6 ((𝜑𝑌 ∈ ℝ+) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ)
6765, 66ltnled 10463 . . . . 5 ((𝜑𝑌 ∈ ℝ+) → (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ↔ ¬ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)))
6862, 67mpbid 223 . . . 4 ((𝜑𝑌 ∈ ℝ+) → ¬ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌))
6953rexrd 10368 . . . . . . . . 9 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ*)
70 elxrge0 12495 . . . . . . . . 9 ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ (0[,]+∞) ↔ ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ* ∧ 0 ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
7169, 55, 70sylanbrc 574 . . . . . . . 8 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ (0[,]+∞))
7271adantr 468 . . . . . . 7 ((𝜑𝑡 ∈ ℝ) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ (0[,]+∞))
7372fmpttd 6601 . . . . . 6 (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))):ℝ⟶(0[,]+∞))
7473adantr 468 . . . . 5 ((𝜑𝑌 ∈ ℝ+) → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))):ℝ⟶(0[,]+∞))
7565rexrd 10368 . . . . 5 ((𝜑𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ*)
76 itg2leub 23709 . . . . 5 (((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))):ℝ⟶(0[,]+∞) ∧ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ*) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ↔ ∀𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌))))
7774, 75, 76syl2anc 575 . . . 4 ((𝜑𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ↔ ∀𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌))))
7868, 77mtbid 315 . . 3 ((𝜑𝑌 ∈ ℝ+) → ¬ ∀𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)))
79 rexanali 3181 . . 3 (∃𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) ↔ ¬ ∀𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)))
8078, 79sylibr 225 . 2 ((𝜑𝑌 ∈ ℝ+) → ∃𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)))
8165ad2antrr 708 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ)
82 itg1cl 23660 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → (∫1𝑔) ∈ ℝ)
8382ad2antlr 709 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → (∫1𝑔) ∈ ℝ)
84 eqid 2802 . . . . . . . . . . . 12 (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
8584i1fpos 23681 . . . . . . . . . . 11 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1)
86 0re 10321 . . . . . . . . . . . . . 14 0 ∈ ℝ
87 i1ff 23651 . . . . . . . . . . . . . . 15 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
8887ffvelrnda 6575 . . . . . . . . . . . . . 14 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℝ)
89 max1 12228 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝑔𝑡) ∈ ℝ) → 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
9086, 88, 89sylancr 577 . . . . . . . . . . . . 13 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
9190ralrimiva 3150 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → ∀𝑡 ∈ ℝ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
92 ax-resscn 10272 . . . . . . . . . . . . . . 15 ℝ ⊆ ℂ
9392a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → ℝ ⊆ ℂ)
94 fvex 6415 . . . . . . . . . . . . . . . . 17 (𝑔𝑡) ∈ V
95 c0ex 10313 . . . . . . . . . . . . . . . . 17 0 ∈ V
9694, 95ifex 4321 . . . . . . . . . . . . . . . 16 if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ V
9796, 84fnmpti 6227 . . . . . . . . . . . . . . 15 (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) Fn ℝ
9897a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) Fn ℝ)
9993, 980pledm 23648 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1 → (0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ (ℝ × {0}) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
100 reex 10306 . . . . . . . . . . . . . . 15 ℝ ∈ V
101100a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → ℝ ∈ V)
10295a1i 11 . . . . . . . . . . . . . 14 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → 0 ∈ V)
103 ifcl 4317 . . . . . . . . . . . . . . 15 (((𝑔𝑡) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ)
10488, 86, 103sylancl 576 . . . . . . . . . . . . . 14 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ)
105 fconstmpt 5357 . . . . . . . . . . . . . . 15 (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
106105a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
107 eqidd 2803 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
108101, 102, 104, 106, 107ofrfval2 7139 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1 → ((ℝ × {0}) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
10999, 108bitrd 270 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → (0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
11091, 109mpbird 248 . . . . . . . . . . 11 (𝑔 ∈ dom ∫1 → 0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
111 itg2itg1 23711 . . . . . . . . . . 11 (((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 ∧ 0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
11285, 110, 111syl2anc 575 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
113 itg1cl 23660 . . . . . . . . . . 11 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
11485, 113syl 17 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
115112, 114eqeltrd 2881 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
116115ad2antlr 709 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
117 ltnle 10396 . . . . . . . . . 10 ((((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ ∧ (∫1𝑔) ∈ ℝ) → (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫1𝑔) ↔ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)))
11865, 82, 117syl2an 585 . . . . . . . . 9 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫1𝑔) ↔ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)))
119118biimpar 465 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫1𝑔))
120 max2 12230 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝑔𝑡) ∈ ℝ) → (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
12186, 88, 120sylancr 577 . . . . . . . . . . . . 13 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
122121ralrimiva 3150 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → ∀𝑡 ∈ ℝ (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
12387feqmptd 6464 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1𝑔 = (𝑡 ∈ ℝ ↦ (𝑔𝑡)))
124101, 88, 104, 123, 107ofrfval2 7139 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → (𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ ∀𝑡 ∈ ℝ (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
125122, 124mpbird 248 . . . . . . . . . . 11 (𝑔 ∈ dom ∫1𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
126 itg1le 23688 . . . . . . . . . . 11 ((𝑔 ∈ dom ∫1 ∧ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) → (∫1𝑔) ≤ (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
12785, 125, 126mpd3an23 1580 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (∫1𝑔) ≤ (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
128127, 112breqtrrd 4865 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → (∫1𝑔) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
129128ad2antlr 709 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → (∫1𝑔) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
13081, 83, 116, 119, 129ltletrd 10476 . . . . . . 7 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
131130adantrl 698 . . . . . 6 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ (𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌))) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
132 i1fmbf 23650 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ MblFn)
13385, 132syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ MblFn)
134133adantl 469 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ MblFn)
135 elrege0 12492 . . . . . . . . . . . . . . . . 17 (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
136104, 90, 135sylanbrc 574 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ (0[,)+∞))
137136fmpttd 6601 . . . . . . . . . . . . . . 15 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)):ℝ⟶(0[,)+∞))
138137adantl 469 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)):ℝ⟶(0[,)+∞))
139115adantl 469 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
140104recnd 10347 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ)
141140negcld 10658 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ)
142140, 141ifcld 4318 . . . . . . . . . . . . . . . . . . 19 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℂ)
143 subcl 10559 . . . . . . . . . . . . . . . . . . 19 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ ∧ if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℂ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℂ)
14432, 142, 143syl2an 585 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℂ)
145144anassrs 455 . . . . . . . . . . . . . . . . 17 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℂ)
146145abscld 14392 . . . . . . . . . . . . . . . 16 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) ∈ ℝ)
147145absge0d 14400 . . . . . . . . . . . . . . . 16 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → 0 ≤ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))
148 elrege0 12492 . . . . . . . . . . . . . . . 16 ((abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) ∈ (0[,)+∞) ↔ ((abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) ∈ ℝ ∧ 0 ≤ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))
149146, 147, 148sylanbrc 574 . . . . . . . . . . . . . . 15 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) ∈ (0[,)+∞))
150149fmpttd 6601 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))):ℝ⟶(0[,)+∞))
151 eleq1w 2864 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → (𝑥𝐷𝑡𝐷))
152 fveq2 6402 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → (𝐹𝑥) = (𝐹𝑡))
153151, 152ifbieq1d 4296 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → if(𝑥𝐷, (𝐹𝑥), 0) = if(𝑡𝐷, (𝐹𝑡), 0))
154153fveq2d 6406 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑡 → (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
155 eqid 2802 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) = (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))
156 fvex 6415 . . . . . . . . . . . . . . . . . . . 20 (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ V
157154, 155, 156fvmpt 6497 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
158154breq2d 4849 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
159 fveq2 6402 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
160159breq2d 4849 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → (0 ≤ (𝑔𝑥) ↔ 0 ≤ (𝑔𝑡)))
161160, 159ifbieq1d 4296 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
162161negeqd 10554 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
163158, 161, 162ifbieq12d 4300 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑡 → if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) = if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
164 eqid 2802 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
165 negex 10558 . . . . . . . . . . . . . . . . . . . . 21 -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ V
16696, 165ifex 4321 . . . . . . . . . . . . . . . . . . . 20 if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ V
167163, 164, 166fvmpt 6497 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡) = if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
168157, 167oveq12d 6886 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ ℝ → (((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) − ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡)) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
169168fveq2d 6406 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ ℝ → (abs‘(((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) − ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡))) = (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))
170169mpteq2ia 4927 . . . . . . . . . . . . . . . 16 (𝑡 ∈ ℝ ↦ (abs‘(((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) − ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡)))) = (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))
171170fveq2i 6405 . . . . . . . . . . . . . . 15 (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) − ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡))))) = (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))
172100a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ℝ ∈ V)
173 fvex 6415 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔𝑥) ∈ V
174173, 95ifex 4321 . . . . . . . . . . . . . . . . . . . . 21 if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ V
175174, 95ifex 4321 . . . . . . . . . . . . . . . . . . . 20 if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) ∈ V
176175a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) ∈ V)
177 ovex 6900 . . . . . . . . . . . . . . . . . . . . 21 (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) ∈ V
17895, 177ifex 4321 . . . . . . . . . . . . . . . . . . . 20 if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ V
179178a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ V)
180 ffn 6250 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐷⟶ℂ → 𝐹 Fn 𝐷)
181 frn 6256 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐷⟶ℂ → ran 𝐹 ⊆ ℂ)
182 ref 14069 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ℜ:ℂ⟶ℝ
183 ffn 6250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (ℜ:ℂ⟶ℝ → ℜ Fn ℂ)
184182, 183ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ℜ Fn ℂ
185 fnco 6204 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((ℜ Fn ℂ ∧ 𝐹 Fn 𝐷 ∧ ran 𝐹 ⊆ ℂ) → (ℜ ∘ 𝐹) Fn 𝐷)
186184, 185mp3an1 1565 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹 Fn 𝐷 ∧ ran 𝐹 ⊆ ℂ) → (ℜ ∘ 𝐹) Fn 𝐷)
187180, 181, 186syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:𝐷⟶ℂ → (ℜ ∘ 𝐹) Fn 𝐷)
188 elpreima 6553 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℜ ∘ 𝐹) Fn 𝐷 → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞))))
1894, 187, 1883syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞))))
190 fco 6267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((ℜ:ℂ⟶ℝ ∧ 𝐹:𝐷⟶ℂ) → (ℜ ∘ 𝐹):𝐷⟶ℝ)
191182, 4, 190sylancr 577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (ℜ ∘ 𝐹):𝐷⟶ℝ)
192191ffvelrnda 6575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑥𝐷) → ((ℜ ∘ 𝐹)‘𝑥) ∈ ℝ)
193192biantrurd 524 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑥𝐷) → (0 ≤ ((ℜ ∘ 𝐹)‘𝑥) ↔ (((ℜ ∘ 𝐹)‘𝑥) ∈ ℝ ∧ 0 ≤ ((ℜ ∘ 𝐹)‘𝑥))))
194 elrege0 12492 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞) ↔ (((ℜ ∘ 𝐹)‘𝑥) ∈ ℝ ∧ 0 ≤ ((ℜ ∘ 𝐹)‘𝑥)))
195193, 194syl6bbr 280 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑥𝐷) → (0 ≤ ((ℜ ∘ 𝐹)‘𝑥) ↔ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞)))
196 fvco3 6490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:𝐷⟶ℂ ∧ 𝑥𝐷) → ((ℜ ∘ 𝐹)‘𝑥) = (ℜ‘(𝐹𝑥)))
1974, 196sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑥𝐷) → ((ℜ ∘ 𝐹)‘𝑥) = (ℜ‘(𝐹𝑥)))
198197breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑥𝐷) → (0 ≤ ((ℜ ∘ 𝐹)‘𝑥) ↔ 0 ≤ (ℜ‘(𝐹𝑥))))
199195, 198bitr3d 272 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑥𝐷) → (((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞) ↔ 0 ≤ (ℜ‘(𝐹𝑥))))
200199pm5.32da 570 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝑥𝐷 ∧ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞)) ↔ (𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥)))))
201189, 200bitrd 270 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥)))))
202201adantr 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ ℝ) → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥)))))
203 0le0 11387 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ≤ 0
204203, 18breqtrri 4864 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ≤ (ℜ‘0)
205204biantru 521 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑥𝐷 ↔ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0)))
206 eldif 3773 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (ℝ ∖ 𝐷) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥𝐷))
207206baibr 528 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℝ → (¬ 𝑥𝐷𝑥 ∈ (ℝ ∖ 𝐷)))
208205, 207syl5rbbr 277 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℝ → (𝑥 ∈ (ℝ ∖ 𝐷) ↔ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0))))
209208adantl 469 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ 𝐷) ↔ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0))))
210202, 209orbi12d 933 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥 ∈ ℝ) → ((𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ∨ 𝑥 ∈ (ℝ ∖ 𝐷)) ↔ ((𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥))) ∨ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0)))))
211 elun 3946 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ↔ (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ∨ 𝑥 ∈ (ℝ ∖ 𝐷)))
212 fveq2 6402 . . . . . . . . . . . . . . . . . . . . . . . 24 (if(𝑥𝐷, (𝐹𝑥), 0) = (𝐹𝑥) → (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) = (ℜ‘(𝐹𝑥)))
213212breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . 23 (if(𝑥𝐷, (𝐹𝑥), 0) = (𝐹𝑥) → (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ 0 ≤ (ℜ‘(𝐹𝑥))))
214 fveq2 6402 . . . . . . . . . . . . . . . . . . . . . . . 24 (if(𝑥𝐷, (𝐹𝑥), 0) = 0 → (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) = (ℜ‘0))
215214breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . 23 (if(𝑥𝐷, (𝐹𝑥), 0) = 0 → (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ 0 ≤ (ℜ‘0)))
216213, 215elimif 4309 . . . . . . . . . . . . . . . . . . . . . 22 (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ ((𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥))) ∨ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0))))
217210, 211, 2163bitr4g 305 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ ℝ) → (𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ↔ 0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))))
218217ifbid 4295 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ℝ) → if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) = if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0))
219218mpteq2dva 4931 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)) = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)))
220217ifbid 4295 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ℝ) → if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) = if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))
221220mpteq2dva 4931 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))) = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))))
222172, 176, 179, 219, 221offval2 7138 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)) ∘𝑓 + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))) = (𝑥 ∈ ℝ ↦ (if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) + if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))))
223 ovif12 6963 . . . . . . . . . . . . . . . . . . . 20 (if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) + if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))) = if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), (if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) + 0), (0 + (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))
22487ffvelrnda 6575 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℝ)
225224recnd 10347 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℂ)
226 0cn 10311 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
227 ifcl 4317 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ ℂ)
228225, 226, 227sylancl 576 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ ℂ)
229228addid1d 10515 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) + 0) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
230228mulm1d 10761 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) = -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
231230oveq2d 6884 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (0 + (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) = (0 + -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
232228negcld 10658 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ ℂ)
233232addid2d 10516 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (0 + -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) = -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
234231, 233eqtrd 2836 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (0 + (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) = -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
235229, 234ifeq12d 4293 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), (if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) + 0), (0 + (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))) = if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
236223, 235syl5eq 2848 . . . . . . . . . . . . . . . . . . 19 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) + if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))) = if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
237236mpteq2dva 4931 . . . . . . . . . . . . . . . . . 18 (𝑔 ∈ dom ∫1 → (𝑥 ∈ ℝ ↦ (if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) + if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))) = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))
238222, 237sylan9eq 2856 . . . . . . . . . . . . . . . . 17 ((𝜑𝑔 ∈ dom ∫1) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)) ∘𝑓 + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))) = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))
239 0xr 10365 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ*
240 pnfxr 10371 . . . . . . . . . . . . . . . . . . . . . . 23 +∞ ∈ ℝ*
241 0ltpnf 12166 . . . . . . . . . . . . . . . . . . . . . . 23 0 < +∞
242 snunioo 12515 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 < +∞) → ({0} ∪ (0(,)+∞)) = (0[,)+∞))
243239, 240, 241, 242mp3an 1578 . . . . . . . . . . . . . . . . . . . . . 22 ({0} ∪ (0(,)+∞)) = (0[,)+∞)
244243imaeq2i 5668 . . . . . . . . . . . . . . . . . . . . 21 ((ℜ ∘ 𝐹) “ ({0} ∪ (0(,)+∞))) = ((ℜ ∘ 𝐹) “ (0[,)+∞))
245 imaundi 5750 . . . . . . . . . . . . . . . . . . . . 21 ((ℜ ∘ 𝐹) “ ({0} ∪ (0(,)+∞))) = (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞)))
246244, 245eqtr3i 2826 . . . . . . . . . . . . . . . . . . . 20 ((ℜ ∘ 𝐹) “ (0[,)+∞)) = (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞)))
247 ismbfcn 23604 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝐷⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
2484, 247syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
24942, 248mpbid 223 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))
250249simpld 484 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (ℜ ∘ 𝐹) ∈ MblFn)
251 mbfimasn 23607 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ ∧ 0 ∈ ℝ) → ((ℜ ∘ 𝐹) “ {0}) ∈ dom vol)
25286, 251mp3an3 1567 . . . . . . . . . . . . . . . . . . . . . 22 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ) → ((ℜ ∘ 𝐹) “ {0}) ∈ dom vol)
253 mbfima 23605 . . . . . . . . . . . . . . . . . . . . . 22 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ) → ((ℜ ∘ 𝐹) “ (0(,)+∞)) ∈ dom vol)
254 unmbl 23512 . . . . . . . . . . . . . . . . . . . . . 22 ((((ℜ ∘ 𝐹) “ {0}) ∈ dom vol ∧ ((ℜ ∘ 𝐹) “ (0(,)+∞)) ∈ dom vol) → (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞))) ∈ dom vol)
255252, 253, 254syl2anc 575 . . . . . . . . . . . . . . . . . . . . 21 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ) → (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞))) ∈ dom vol)
256250, 191, 255syl2anc 575 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞))) ∈ dom vol)
257246, 256syl5eqel 2885 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((ℜ ∘ 𝐹) “ (0[,)+∞)) ∈ dom vol)
2584fdmd 6259 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → dom 𝐹 = 𝐷)
259 mbfdm 23601 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
26042, 259syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → dom 𝐹 ∈ dom vol)
261258, 260eqeltrrd 2882 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐷 ∈ dom vol)
262 difmbl 23518 . . . . . . . . . . . . . . . . . . . 20 ((ℝ ∈ dom vol ∧ 𝐷 ∈ dom vol) → (ℝ ∖ 𝐷) ∈ dom vol)
26311, 261, 262sylancr 577 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ∖ 𝐷) ∈ dom vol)
264 unmbl 23512 . . . . . . . . . . . . . . . . . . 19 ((((ℜ ∘ 𝐹) “ (0[,)+∞)) ∈ dom vol ∧ (ℝ ∖ 𝐷) ∈ dom vol) → (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol)
265257, 263, 264syl2anc 575 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol)
266 fveq2 6402 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑥 → (𝑔𝑡) = (𝑔𝑥))
267266breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑥 → (0 ≤ (𝑔𝑡) ↔ 0 ≤ (𝑔𝑥)))
268267, 266ifbieq1d 4296 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑥 → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
269268, 84, 174fvmpt 6497 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ℝ → ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
270269eqcomd 2808 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) = ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥))
271270ifeq1d 4291 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℝ → if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) = if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥), 0))
272271mpteq2ia 4927 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥), 0))
273272i1fres 23680 . . . . . . . . . . . . . . . . . . 19 (((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 ∧ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)) ∈ dom ∫1)
274 id 22 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1)
275 neg1rr 11401 . . . . . . . . . . . . . . . . . . . . . 22 -1 ∈ ℝ
276275a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → -1 ∈ ℝ)
277274, 276i1fmulc 23678 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → ((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ dom ∫1)
278 cmmbl 23509 . . . . . . . . . . . . . . . . . . . 20 ((((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol → (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))) ∈ dom vol)
279 ifnot 4323 . . . . . . . . . . . . . . . . . . . . . . 23 if(¬ 𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)), 0) = if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
280 eldif 3773 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))))
281280baibr 528 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℝ → (¬ 𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ↔ 𝑥 ∈ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)))))
282 tru 1642 . . . . . . . . . . . . . . . . . . . . . . . . . 26
283 negex 10558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 -1 ∈ V
284283fconst 6300 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ℝ × {-1}):ℝ⟶{-1}
285 ffn 6250 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((ℝ × {-1}):ℝ⟶{-1} → (ℝ × {-1}) Fn ℝ)
286284, 285mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⊤ → (ℝ × {-1}) Fn ℝ)
28797a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⊤ → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) Fn ℝ)
288100a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⊤ → ℝ ∈ V)
289 inidm 4013 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (ℝ ∩ ℝ) = ℝ
290283fvconst2 6688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ ℝ → ((ℝ × {-1})‘𝑥) = -1)
291290adantl 469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⊤ ∧ 𝑥 ∈ ℝ) → ((ℝ × {-1})‘𝑥) = -1)
292269adantl 469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⊤ ∧ 𝑥 ∈ ℝ) → ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
293286, 287, 288, 288, 289, 291, 292ofval 7130 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((⊤ ∧ 𝑥 ∈ ℝ) → (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥) = (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
294282, 293mpan 673 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℝ → (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥) = (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
295294eqcomd 2808 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℝ → (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) = (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥))
296281, 295ifbieq1d 4296 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ℝ → if(¬ 𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)), 0) = if(𝑥 ∈ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))), (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥), 0))
297279, 296syl5eqr 2850 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) = if(𝑥 ∈ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))), (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥), 0))
298297mpteq2ia 4927 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))), (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥), 0))
299298i1fres 23680 . . . . . . . . . . . . . . . . . . . 20 ((((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ dom ∫1 ∧ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))) ∈ dom ∫1)
300277, 278, 299syl2an 585 . . . . . . . . . . . . . . . . . . 19 (((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 ∧ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))) ∈ dom ∫1)
301273, 300i1fadd 23670 . . . . . . . . . . . . . . . . . 18 (((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 ∧ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)) ∘𝑓 + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))) ∈ dom ∫1)
30285, 265, 301syl2anr 586 . . . . . . . . . . . . . . . . 17 ((𝜑𝑔 ∈ dom ∫1) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0)) ∘𝑓 + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))))) ∈ dom ∫1)
303238, 302eqeltrrd 2882 . . . . . . . . . . . . . . . 16 ((𝜑𝑔 ∈ dom ∫1) → (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ dom ∫1)
304154cbvmptv 4937 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) = (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
305304, 31syl5eqel 2885 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1)
3069, 304fmptd 6600 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ)
307305, 306jca 503 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1 ∧ (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ))
308307adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑔 ∈ dom ∫1) → ((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1 ∧ (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ))
309 ftc1anclem4 33794 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ dom ∫1 ∧ (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1 ∧ (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) − ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡))))) ∈ ℝ)
3103093expb 1142 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ dom ∫1 ∧ ((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1 ∧ (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ)) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) − ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡))))) ∈ ℝ)
311303, 308, 310syl2anc 575 . . . . . . . . . . . . . . 15 ((𝜑𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) − ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡))))) ∈ ℝ)
312171, 311syl5eqelr 2886 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) ∈ ℝ)
313134, 138, 139, 150, 312itg2addnc 33770 . . . . . . . . . . . . 13 ((𝜑𝑔 ∈ dom ∫1) → (∫2‘((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∘𝑓 + (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) = ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))))
314100a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑔 ∈ dom ∫1) → ℝ ∈ V)
31596a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ V)
316 eqidd 2803 . . . . . . . . . . . . . . 15 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
317 eqidd 2803 . . . . . . . . . . . . . . 15 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))) = (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))
318314, 315, 146, 316, 317offval2 7138 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∘𝑓 + (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) = (𝑡 ∈ ℝ ↦ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))))
319318fveq2d 6406 . . . . . . . . . . . . 13 ((𝜑𝑔 ∈ dom ∫1) → (∫2‘((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∘𝑓 + (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) = (∫2‘(𝑡 ∈ ℝ ↦ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))))
320313, 319eqtr3d 2838 . . . . . . . . . . . 12 ((𝜑𝑔 ∈ dom ∫1) → ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) = (∫2‘(𝑡 ∈ ℝ ↦ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))))
321320adantr 468 . . . . . . . . . . 11 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) = (∫2‘(𝑡 ∈ ℝ ↦ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))))
322 nfv 2005 . . . . . . . . . . . . . 14 𝑡(𝜑𝑔 ∈ dom ∫1)
323 nfcv 2944 . . . . . . . . . . . . . . 15 𝑡𝑔
324 nfcv 2944 . . . . . . . . . . . . . . 15 𝑡𝑟
325 nfmpt1 4934 . . . . . . . . . . . . . . 15 𝑡(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
326323, 324, 325nfbr 4884 . . . . . . . . . . . . . 14 𝑡 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
327322, 326nfan 1990 . . . . . . . . . . . . 13 𝑡((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
328 anass 456 . . . . . . . . . . . . . . 15 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) ↔ (𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)))
32987ffnd 6251 . . . . . . . . . . . . . . . . . . . 20 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
330 fvex 6415 . . . . . . . . . . . . . . . . . . . . . 22 (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ V
331 eqid 2802 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) = (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
332330, 331fnmpti 6227 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) Fn ℝ
333332a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) Fn ℝ)
334 eqidd 2803 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) = (𝑔𝑡))
335331fvmpt2 6506 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ V) → ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))‘𝑡) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
336330, 335mpan2 674 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ ℝ → ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))‘𝑡) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
337336adantl 469 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))‘𝑡) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
338329, 333, 101, 101, 289, 334, 337ofrval 7131 . . . . . . . . . . . . . . . . . . 19 ((𝑔 ∈ dom ∫1𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3393383com23 1149 . . . . . . . . . . . . . . . . . 18 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3403393expa 1140 . . . . . . . . . . . . . . . . 17 (((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
341340adantll 696 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
342 resubcl 10624 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
3438, 104, 342syl2an 585 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
344343ad2antrr 708 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
345 absid 14253 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
3468, 345sylan 571 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
347346breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ↔ (𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
348347biimpa 464 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
349348an32s 634 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
350349adantllr 701 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
351 breq1 4840 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔𝑡) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → ((𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ↔ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
352 breq1 4840 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → (0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ↔ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
353351, 352ifboth 4311 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
354350, 353sylancom 578 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
355 subge0 10820 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ) → (0 ≤ ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3568, 104, 355syl2an 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (0 ≤ ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
357356ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (0 ≤ ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
358354, 357mpbird 248 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → 0 ≤ ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
359344, 358absidd 14378 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
360 iftrue 4279 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
361360oveq2d 6884 . . . . . . . . . . . . . . . . . . . . . 22 (0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
362361fveq2d 6406 . . . . . . . . . . . . . . . . . . . . 21 (0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) = (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
363362adantl 469 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) = (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
3648ad2antrr 708 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
365345oveq1d 6883 . . . . . . . . . . . . . . . . . . . . 21 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
366364, 365sylan 571 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
367359, 363, 3663eqtr4d 2846 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) = ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
368104renegcld 10736 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ)
369 resubcl 10624 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
3708, 368, 369syl2an 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
371370ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
37288ad3antlr 713 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (𝑔𝑡) ∈ ℝ)
3738ad3antrrr 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
3748adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
375 ltnle 10396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 ↔ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
37686, 375mpan2 674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 ↔ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
377 ltle 10405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
37886, 377mpan2 674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
379376, 378sylbird 251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ → (¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
380379imp 395 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0)
381 absnid 14255 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
382380, 381syldan 581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
383382breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ↔ (𝑔𝑡) ≤ -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
384383biimpa 464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (𝑔𝑡) ≤ -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
385384an32s 634 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (𝑔𝑡) ≤ -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
386374, 385sylanl1 662 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (𝑔𝑡) ≤ -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
387372, 373, 386lenegcon2d 10889 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -(𝑔𝑡))
388 simpll 774 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → 𝜑)
38986a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → 0 ∈ ℝ)
3908, 389ltnled 10463 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 ↔ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3918, 86, 377sylancl 576 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
392390, 391sylbird 251 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
393392imp 395 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0)
394388, 393sylan 571 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0)
395 negeq 10552 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔𝑡) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → -(𝑔𝑡) = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
396395breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔𝑡) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -(𝑔𝑡) ↔ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
397 neg0 10606 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -0 = 0
398 negeq 10552 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → -0 = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
399397, 398syl5eqr 2850 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → 0 = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
400399breq2d 4849 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0 ↔ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
401396, 400ifboth 4311 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -(𝑔𝑡) ∧ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
402387, 394, 401syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
403 suble0 10821 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ) → (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ≤ 0 ↔ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
4048, 368, 403syl2an 585 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ≤ 0 ↔ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
405404ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ≤ 0 ↔ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
406402, 405mpbird 248 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ≤ 0)
407371, 406absnidd 14369 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = -((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
408 subneg 10609 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ ∧ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) + if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
409408negeqd 10554 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ ∧ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ) → -((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = -((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) + if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
410 negdi2 10618 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ ∧ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ) → -((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) + if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (-(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
411409, 410eqtrd 2836 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ ∧ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ) → -((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (-(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
41232, 140, 411syl2an 585 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → -((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (-(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
413412ad2antrr 708 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → -((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (-(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
414407, 413eqtrd 2836 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = (-(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
415 iffalse 4282 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
416415oveq2d 6884 . . . . . . . . . . . . . . . . . . . . . 22 (¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
417416fveq2d 6406 . . . . . . . . . . . . . . . . . . . . 21 (¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) = (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
418417adantl 469 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) = (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
4198, 381sylan 571 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
420393, 419syldan 581 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
421420oveq1d 6883 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (-(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
422388, 421sylan 571 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (-(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
423414, 418, 4223eqtr4d 2846 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) = ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
424367, 423pm2.61dan 838 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))) = ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
425424oveq2d 6884 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))) = (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
42653recnd 10347 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℂ)
427 pncan3 10568 . . . . . . . . . . . . . . . . . . 19 ((if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ ∧ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℂ) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
428140, 426, 427syl2anr 586 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
429428adantr 468 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
430425, 429eqtrd 2836 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
431341, 430syldan 581 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
432328, 431sylanb 572 . . . . . . . . . . . . . 14 ((((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
433432an32s 634 . . . . . . . . . . . . 13 ((((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∧ 𝑡 ∈ ℝ) → (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
434327, 433mpteq2da 4930 . . . . . . . . . . . 12 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (𝑡 ∈ ℝ ↦ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) = (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
435434fveq2d 6406 . . . . . . . . . . 11 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (∫2‘(𝑡 ∈ ℝ ↦ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) + (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) = (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))))
436321, 435eqtrd 2836 . . . . . . . . . 10 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) = (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))))
437436breq1d 4847 . . . . . . . . 9 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌) ↔ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
438437adantllr 701 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌) ↔ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
439312adantlr 697 . . . . . . . . . 10 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) ∈ ℝ)
44063ad2antlr 709 . . . . . . . . . 10 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → 𝑌 ∈ ℝ)
441115adantl 469 . . . . . . . . . 10 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
442439, 440, 441ltadd2d 10472 . . . . . . . . 9 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 ↔ ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
443442adantr 468 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 ↔ ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
444 ltsubadd 10777 . . . . . . . . . . 11 (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ ∧ 𝑌 ∈ ℝ ∧ (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ) → (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ↔ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
44560, 63, 115, 444syl3an 1192 . . . . . . . . . 10 ((𝜑𝑌 ∈ ℝ+𝑔 ∈ dom ∫1) → (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ↔ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
4464453expa 1140 . . . . . . . . 9 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ↔ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
447446adantr 468 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ↔ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) < ((∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) + 𝑌)))
448438, 443, 4473bitr4d 302 . . . . . . 7 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 ↔ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))
449448adantrr 699 . . . . . 6 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ (𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌))) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 ↔ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))
450131, 449mpbird 248 . . . . 5 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ (𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌))) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌)
451450ex 399 . . . 4 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → ((𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌))
452451reximdva 3200 . . 3 ((𝜑𝑌 ∈ ℝ+) → (∃𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → ∃𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌))
453 fveq1 6401 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) → (𝑓𝑡) = ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))‘𝑡))
454453, 167sylan9eq 2856 . . . . . . . . . . . . 13 ((𝑓 = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∧ 𝑡 ∈ ℝ) → (𝑓𝑡) = if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
455454oveq2d 6884 . . . . . . . . . . . 12 ((𝑓 = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∧ 𝑡 ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡)) = ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
456455fveq2d 6406 . . . . . . . . . . 11 ((𝑓 = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∧ 𝑡 ∈ ℝ) → (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))) = (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))
457456mpteq2dva 4931 . . . . . . . . . 10 (𝑓 = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) → (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡)))) = (𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))))
458457fveq2d 6406 . . . . . . . . 9 (𝑓 = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) = (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))))
459458breq1d 4847 . . . . . . . 8 (𝑓 = (𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌 ↔ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌))
460459rspcev 3498 . . . . . . 7 (((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ dom ∫1 ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌) → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌)
461460ex 399 . . . . . 6 ((𝑥 ∈ ℝ ↦ if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ dom ∫1 → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌))
462303, 461syl 17 . . . . 5 ((𝜑𝑔 ∈ dom ∫1) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌))
463462rexlimdva 3215 . . . 4 (𝜑 → (∃𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌))
464463adantr 468 . . 3 ((𝜑𝑌 ∈ ℝ+) → (∃𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))))) < 𝑌 → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌))
465452, 464syld 47 . 2 ((𝜑𝑌 ∈ ℝ+) → (∃𝑔 ∈ dom ∫1(𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌))
46680, 465mpd 15 1 ((𝜑𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − (𝑓𝑡))))) < 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wtru 1638  wcel 2155  wral 3092  wrex 3093  Vcvv 3387  cdif 3760  cun 3761  wss 3763  ifcif 4273  {csn 4364   class class class wbr 4837  cmpt 4916   × cxp 5303  ccnv 5304  dom cdm 5305  ran crn 5306  cima 5308  ccom 5309   Fn wfn 6090  wf 6091  cfv 6095  (class class class)co 6868  𝑓 cof 7119  𝑟 cofr 7120  cc 10213  cr 10214  0cc0 10215  1c1 10216   + caddc 10218   · cmul 10220  +∞cpnf 10350  *cxr 10352   < clt 10353  cle 10354  cmin 10545  -cneg 10546  +crp 12040  (,)cioo 12387  [,)cico 12389  [,]cicc 12390  cre 14054  cim 14055  abscabs 14191  volcvol 23438  MblFncmbf 23589  1citg1 23590  2citg2 23591  𝐿1cibl 23592  citg 23593  0𝑝c0p 23644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-inf2 8779  ax-cnex 10271  ax-resscn 10272  ax-1cn 10273  ax-icn 10274  ax-addcl 10275  ax-addrcl 10276  ax-mulcl 10277  ax-mulrcl 10278  ax-mulcom 10279  ax-addass 10280  ax-mulass 10281  ax-distr 10282  ax-i2m1 10283  ax-1ne0 10284  ax-1rid 10285  ax-rnegex 10286  ax-rrecex 10287  ax-cnre 10288  ax-pre-lttri 10289  ax-pre-lttrn 10290  ax-pre-ltadd 10291  ax-pre-mulgt0 10292  ax-pre-sup 10293  ax-addf 10294
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-disj 4806  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-isom 6104  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-of 7121  df-ofr 7122  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-2o 7791  df-oadd 7794  df-er 7973  df-map 8088  df-pm 8089  df-en 8187  df-dom 8188  df-sdom 8189  df-fin 8190  df-fi 8550  df-sup 8581  df-inf 8582  df-oi 8648  df-card 9042  df-cda 9269  df-pnf 10355  df-mnf 10356  df-xr 10357  df-ltxr 10358  df-le 10359  df-sub 10547  df-neg 10548  df-div 10964  df-nn 11300  df-2 11358  df-3 11359  df-4 11360  df-n0 11554  df-z 11638  df-uz 11899  df-q 12002  df-rp 12041  df-xneg 12156  df-xadd 12157  df-xmul 12158  df-ioo 12391  df-ico 12393  df-icc 12394  df-fz 12544  df-fzo 12684  df-fl 12811  df-mod 12887  df-seq 13019  df-exp 13078  df-hash 13332  df-cj 14056  df-re 14057  df-im 14058  df-sqrt 14192  df-abs 14193  df-clim 14436  df-sum 14634  df-rest 16282  df-topgen 16303  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-top 20906  df-topon 20923  df-bases 20958  df-cmp 21398  df-ovol 23439  df-vol 23440  df-mbf 23594  df-itg1 23595  df-itg2 23596  df-ibl 23597  df-0p 23645
This theorem is referenced by:  ftc1anclem6  33796
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