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Theorem ftc1anclem5 32659
 Description: Lemma for ftc1anc 32663, 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 4042 . . . . . . . . 9 (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
21mpteq2ia 4668 . . . . . . . 8 (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
32fveq2i 6106 . . . . . . 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 6267 . . . . . . . . . . . . 13 ((𝜑𝑡𝐷) → (𝐹𝑡) ∈ ℂ)
6 0cnd 9912 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 𝑡𝐷) → 0 ∈ ℂ)
75, 6ifclda 4070 . . . . . . . . . . . 12 (𝜑 → if(𝑡𝐷, (𝐹𝑡), 0) ∈ ℂ)
87recld 13782 . . . . . . . . . . 11 (𝜑 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
98adantr 480 . . . . . . . . . 10 ((𝜑𝑡 ∈ ℝ) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
10 ftc1anc.d . . . . . . . . . . 11 (𝜑𝐷 ⊆ ℝ)
11 rembl 23115 . . . . . . . . . . . 12 ℝ ∈ dom vol
1211a1i 11 . . . . . . . . . . 11 (𝜑 → ℝ ∈ dom vol)
138adantr 480 . . . . . . . . . . 11 ((𝜑𝑡𝐷) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
14 eldifn 3695 . . . . . . . . . . . . 13 (𝑡 ∈ (ℝ ∖ 𝐷) → ¬ 𝑡𝐷)
1514adantl 481 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ (ℝ ∖ 𝐷)) → ¬ 𝑡𝐷)
16 iffalse 4045 . . . . . . . . . . . . . 14 𝑡𝐷 → if(𝑡𝐷, (𝐹𝑡), 0) = 0)
1716fveq2d 6107 . . . . . . . . . . . . 13 𝑡𝐷 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = (ℜ‘0))
18 re0 13740 . . . . . . . . . . . . 13 (ℜ‘0) = 0
1917, 18syl6eq 2660 . . . . . . . . . . . 12 𝑡𝐷 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = 0)
2015, 19syl 17 . . . . . . . . . . 11 ((𝜑𝑡 ∈ (ℝ ∖ 𝐷)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = 0)
21 iftrue 4042 . . . . . . . . . . . . . 14 (𝑡𝐷 → if(𝑡𝐷, (𝐹𝑡), 0) = (𝐹𝑡))
2221fveq2d 6107 . . . . . . . . . . . . 13 (𝑡𝐷 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) = (ℜ‘(𝐹𝑡)))
2322mpteq2ia 4668 . . . . . . . . . . . 12 (𝑡𝐷 ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡)))
244feqmptd 6159 . . . . . . . . . . . . . . 15 (𝜑𝐹 = (𝑡𝐷 ↦ (𝐹𝑡)))
25 ftc1anc.i . . . . . . . . . . . . . . 15 (𝜑𝐹 ∈ 𝐿1)
2624, 25eqeltrrd 2689 . . . . . . . . . . . . . 14 (𝜑 → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ 𝐿1)
275iblcn 23371 . . . . . . . . . . . . . 14 (𝜑 → ((𝑡𝐷 ↦ (𝐹𝑡)) ∈ 𝐿1 ↔ ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1)))
2826, 27mpbid 221 . . . . . . . . . . . . 13 (𝜑 → ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1 ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ 𝐿1))
2928simpld 474 . . . . . . . . . . . 12 (𝜑 → (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ 𝐿1)
3023, 29syl5eqel 2692 . . . . . . . . . . 11 (𝜑 → (𝑡𝐷 ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ 𝐿1)
3110, 12, 13, 20, 30iblss2 23378 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ 𝐿1)
328recnd 9947 . . . . . . . . . . . . 13 (𝜑 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ)
3332adantr 480 . . . . . . . . . . . 12 ((𝜑𝑡 ∈ ℝ) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℂ)
34 eqidd 2611 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
35 absf 13925 . . . . . . . . . . . . . 14 abs:ℂ⟶ℝ
3635a1i 11 . . . . . . . . . . . . 13 (𝜑 → abs:ℂ⟶ℝ)
3736feqmptd 6159 . . . . . . . . . . . 12 (𝜑 → abs = (𝑥 ∈ ℂ ↦ (abs‘𝑥)))
38 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (abs‘𝑥) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3933, 34, 37, 38fmptco 6303 . . . . . . . . . . 11 (𝜑 → (abs ∘ (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) = (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
40 eqid 2610 . . . . . . . . . . . . 13 (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
419, 40fmptd 6292 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))):ℝ⟶ℝ)
42 iblmbf 23340 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ 𝐿1𝐹 ∈ MblFn)
4325, 42syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 ∈ MblFn)
4424, 43eqeltrrd 2689 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑡𝐷 ↦ (𝐹𝑡)) ∈ MblFn)
455ismbfcn2 23212 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑡𝐷 ↦ (𝐹𝑡)) ∈ MblFn ↔ ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn)))
4644, 45mpbid 221 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn ∧ (𝑡𝐷 ↦ (ℑ‘(𝐹𝑡))) ∈ MblFn))
4746simpld 474 . . . . . . . . . . . . . 14 (𝜑 → (𝑡𝐷 ↦ (ℜ‘(𝐹𝑡))) ∈ MblFn)
4823, 47syl5eqel 2692 . . . . . . . . . . . . 13 (𝜑 → (𝑡𝐷 ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ MblFn)
4910, 12, 13, 20, 48mbfss 23219 . . . . . . . . . . . 12 (𝜑 → (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ MblFn)
50 ftc1anclem1 32655 . . . . . . . . . . . 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)
5141, 49, 50syl2anc 691 . . . . . . . . . . 11 (𝜑 → (abs ∘ (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ MblFn)
5239, 51eqeltrrd 2689 . . . . . . . . . 10 (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ MblFn)
539, 31, 52iblabsnc 32644 . . . . . . . . 9 (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∈ 𝐿1)
5432abscld 14023 . . . . . . . . . . 11 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ)
5554adantr 480 . . . . . . . . . 10 ((𝜑𝑡 ∈ ℝ) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ)
5632absge0d 14031 . . . . . . . . . . 11 (𝜑 → 0 ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
5756adantr 480 . . . . . . . . . 10 ((𝜑𝑡 ∈ ℝ) → 0 ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
5855, 57iblpos 23365 . . . . . . . . 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)))
5953, 58mpbid 221 . . . . . . . 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))
6059simprd 478 . . . . . . 7 (𝜑 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))), 0))) ∈ ℝ)
613, 60syl5eqelr 2693 . . . . . 6 (𝜑 → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ)
62 ltsubrp 11742 . . . . . 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))))))
6361, 62sylan 487 . . . . 5 ((𝜑𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) < (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))))
64 rpre 11715 . . . . . . 7 (𝑌 ∈ ℝ+𝑌 ∈ ℝ)
65 resubcl 10224 . . . . . . 7 (((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ ∧ 𝑌 ∈ ℝ) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ)
6661, 64, 65syl2an 493 . . . . . 6 ((𝜑𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ)
6761adantr 480 . . . . . 6 ((𝜑𝑌 ∈ ℝ+) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ∈ ℝ)
6866, 67ltnled 10063 . . . . 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)))
6963, 68mpbid 221 . . . 4 ((𝜑𝑌 ∈ ℝ+) → ¬ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌))
7054rexrd 9968 . . . . . . . . 9 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ*)
71 elxrge0 12152 . . . . . . . . 9 ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ (0[,]+∞) ↔ ((abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℝ* ∧ 0 ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
7270, 56, 71sylanbrc 695 . . . . . . . 8 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ (0[,]+∞))
7372adantr 480 . . . . . . 7 ((𝜑𝑡 ∈ ℝ) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ (0[,]+∞))
74 eqid 2610 . . . . . . 7 (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) = (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
7573, 74fmptd 6292 . . . . . 6 (𝜑 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))):ℝ⟶(0[,]+∞))
7675adantr 480 . . . . 5 ((𝜑𝑌 ∈ ℝ+) → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))):ℝ⟶(0[,]+∞))
7766rexrd 9968 . . . . 5 ((𝜑𝑌 ∈ ℝ+) → ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌) ∈ ℝ*)
78 itg2leub 23307 . . . . 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))))
7976, 77, 78syl2anc 691 . . . 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))))
8069, 79mtbid 313 . . 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)))
81 rexanali 2981 . . 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)))
8280, 81sylibr 223 . 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)))
8366ad2antrr 758 . . . . . . . 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)
84 itg1cl 23258 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → (∫1𝑔) ∈ ℝ)
8584ad2antlr 759 . . . . . . . 8 ((((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) ∧ ¬ (∫1𝑔) ≤ ((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) − 𝑌)) → (∫1𝑔) ∈ ℝ)
86 eqid 2610 . . . . . . . . . . . 12 (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
8786i1fpos 23279 . . . . . . . . . . 11 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1)
88 0re 9919 . . . . . . . . . . . . . 14 0 ∈ ℝ
89 i1ff 23249 . . . . . . . . . . . . . . 15 (𝑔 ∈ dom ∫1𝑔:ℝ⟶ℝ)
9089ffvelrnda 6267 . . . . . . . . . . . . . 14 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ∈ ℝ)
91 max1 11890 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝑔𝑡) ∈ ℝ) → 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
9288, 90, 91sylancr 694 . . . . . . . . . . . . 13 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
9392ralrimiva 2949 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → ∀𝑡 ∈ ℝ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
94 ax-resscn 9872 . . . . . . . . . . . . . . 15 ℝ ⊆ ℂ
9594a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → ℝ ⊆ ℂ)
96 fvex 6113 . . . . . . . . . . . . . . . . 17 (𝑔𝑡) ∈ V
97 c0ex 9913 . . . . . . . . . . . . . . . . 17 0 ∈ V
9896, 97ifex 4106 . . . . . . . . . . . . . . . 16 if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ V
9998, 86fnmpti 5935 . . . . . . . . . . . . . . 15 (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) Fn ℝ
10099a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) Fn ℝ)
10195, 1000pledm 23246 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1 → (0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ (ℝ × {0}) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
102 reex 9906 . . . . . . . . . . . . . . 15 ℝ ∈ V
103102a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → ℝ ∈ V)
10497a1i 11 . . . . . . . . . . . . . 14 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → 0 ∈ V)
105 ifcl 4080 . . . . . . . . . . . . . . 15 (((𝑔𝑡) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ)
10690, 88, 105sylancl 693 . . . . . . . . . . . . . 14 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ)
107 fconstmpt 5085 . . . . . . . . . . . . . . 15 (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0)
108107a1i 11 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → (ℝ × {0}) = (𝑡 ∈ ℝ ↦ 0))
109 eqidd 2611 . . . . . . . . . . . . . 14 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
110103, 104, 106, 108, 109ofrfval2 6813 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1 → ((ℝ × {0}) ∘𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
111101, 110bitrd 267 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → (0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ ∀𝑡 ∈ ℝ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
11293, 111mpbird 246 . . . . . . . . . . 11 (𝑔 ∈ dom ∫1 → 0𝑝𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
113 itg2itg1 23309 . . . . . . . . . . 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))))
11487, 112, 113syl2anc 691 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) = (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
115 itg1cl 23258 . . . . . . . . . . 11 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
11687, 115syl 17 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
117114, 116eqeltrd 2688 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
118117ad2antlr 759 . . . . . . . 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)
119 ltnle 9996 . . . . . . . . . 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)))
12066, 84, 119syl2an 493 . . . . . . . . 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)))
121120biimpar 501 . . . . . . . 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))
122 max2 11892 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝑔𝑡) ∈ ℝ) → (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
12388, 90, 122sylancr 694 . . . . . . . . . . . . 13 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
124123ralrimiva 2949 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → ∀𝑡 ∈ ℝ (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
12589feqmptd 6159 . . . . . . . . . . . . 13 (𝑔 ∈ dom ∫1𝑔 = (𝑡 ∈ ℝ ↦ (𝑔𝑡)))
126103, 90, 106, 125, 109ofrfval2 6813 . . . . . . . . . . . 12 (𝑔 ∈ dom ∫1 → (𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ↔ ∀𝑡 ∈ ℝ (𝑔𝑡) ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
127124, 126mpbird 246 . . . . . . . . . . 11 (𝑔 ∈ dom ∫1𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
128 itg1le 23286 . . . . . . . . . . 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))))
12987, 127, 128mpd3an23 1418 . . . . . . . . . 10 (𝑔 ∈ dom ∫1 → (∫1𝑔) ≤ (∫1‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
130129, 114breqtrrd 4611 . . . . . . . . 9 (𝑔 ∈ dom ∫1 → (∫1𝑔) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))))
131130ad2antlr 759 . . . . . . . 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))))
13283, 85, 118, 121, 131ltletrd 10076 . . . . . . 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))))
133132adantrl 748 . . . . . 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))))
134 i1fmbf 23248 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ MblFn)
13587, 134syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ MblFn)
136135adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ MblFn)
137 elrege0 12149 . . . . . . . . . . . . . . . . 17 (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ (0[,)+∞) ↔ (if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
138106, 92, 137sylanbrc 695 . . . . . . . . . . . . . . . 16 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ (0[,)+∞))
139138, 86fmptd 6292 . . . . . . . . . . . . . . 15 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)):ℝ⟶(0[,)+∞))
140139adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)):ℝ⟶(0[,)+∞))
141117adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
142106recnd 9947 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ)
143142negcld 10258 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℂ)
144142, 143ifcld 4081 . . . . . . . . . . . . . . . . . . 19 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℂ)
145 subcl 10159 . . . . . . . . . . . . . . . . . . 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)
14632, 144, 145syl2an 493 . . . . . . . . . . . . . . . . . 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)
147146anassrs 678 . . . . . . . . . . . . . . . . 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)
148147abscld 14023 . . . . . . . . . . . . . . . 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)
149147absge0d 14031 . . . . . . . . . . . . . . . 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)))))
150 elrege0 12149 . . . . . . . . . . . . . . . 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))))))
151148, 149, 150sylanbrc 695 . . . . . . . . . . . . . . 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[,)+∞))
152 eqid 2610 . . . . . . . . . . . . . . 15 (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)))))
153151, 152fmptd 6292 . . . . . . . . . . . . . 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[,)+∞))
154 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → (𝑥𝐷𝑡𝐷))
155 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → (𝐹𝑥) = (𝐹𝑡))
156154, 155ifbieq1d 4059 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → if(𝑥𝐷, (𝐹𝑥), 0) = if(𝑡𝐷, (𝐹𝑡), 0))
157156fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑡 → (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
158 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) = (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))
159 fvex 6113 . . . . . . . . . . . . . . . . . . . 20 (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ V
160157, 158, 159fvmpt 6191 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ ℝ → ((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)))‘𝑡) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
161157breq2d 4595 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
162 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
163162breq2d 4595 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → (0 ≤ (𝑔𝑥) ↔ 0 ≤ (𝑔𝑡)))
164163, 162ifbieq1d 4059 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
165164negeqd 10154 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
166161, 164, 165ifbieq12d 4063 . . . . . . . . . . . . . . . . . . . 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)))
167 eqid 2610 . . . . . . . . . . . . . . . . . . . 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)))
168 negex 10158 . . . . . . . . . . . . . . . . . . . . 21 -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ V
16998, 168ifex 4106 . . . . . . . . . . . . . . . . . . . 20 if(0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)), if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0), -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ V
170166, 167, 169fvmpt 6191 . . . . . . . . . . . . . . . . . . 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)))
171160, 170oveq12d 6567 . . . . . . . . . . . . . . . . . 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))))
172171fveq2d 6107 . . . . . . . . . . . . . . . . 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)))))
173172mpteq2ia 4668 . . . . . . . . . . . . . . . 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)))))
174173fveq2i 6106 . . . . . . . . . . . . . . 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))))))
175102a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ℝ ∈ V)
176 fvex 6113 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔𝑥) ∈ V
177176, 97ifex 4106 . . . . . . . . . . . . . . . . . . . . 21 if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ V
178177, 97ifex 4106 . . . . . . . . . . . . . . . . . . . 20 if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) ∈ V
179178a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0), 0) ∈ V)
180 ovex 6577 . . . . . . . . . . . . . . . . . . . . 21 (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) ∈ V
18197, 180ifex 4106 . . . . . . . . . . . . . . . . . . . 20 if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ V
182181a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ℝ) → if(0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)), 0, (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) ∈ V)
183 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐷⟶ℂ → 𝐹 Fn 𝐷)
184 frn 5966 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐹:𝐷⟶ℂ → ran 𝐹 ⊆ ℂ)
185 ref 13700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ℜ:ℂ⟶ℝ
186 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (ℜ:ℂ⟶ℝ → ℜ Fn ℂ)
187185, 186ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ℜ Fn ℂ
188 fnco 5913 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((ℜ Fn ℂ ∧ 𝐹 Fn 𝐷 ∧ ran 𝐹 ⊆ ℂ) → (ℜ ∘ 𝐹) Fn 𝐷)
189187, 188mp3an1 1403 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹 Fn 𝐷 ∧ ran 𝐹 ⊆ ℂ) → (ℜ ∘ 𝐹) Fn 𝐷)
190183, 184, 189syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:𝐷⟶ℂ → (ℜ ∘ 𝐹) Fn 𝐷)
191 elpreima 6245 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℜ ∘ 𝐹) Fn 𝐷 → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞))))
1924, 190, 1913syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞))))
193 fco 5971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((ℜ:ℂ⟶ℝ ∧ 𝐹:𝐷⟶ℂ) → (ℜ ∘ 𝐹):𝐷⟶ℝ)
194185, 4, 193sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (ℜ ∘ 𝐹):𝐷⟶ℝ)
195194ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑥𝐷) → ((ℜ ∘ 𝐹)‘𝑥) ∈ ℝ)
196195biantrurd 528 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑥𝐷) → (0 ≤ ((ℜ ∘ 𝐹)‘𝑥) ↔ (((ℜ ∘ 𝐹)‘𝑥) ∈ ℝ ∧ 0 ≤ ((ℜ ∘ 𝐹)‘𝑥))))
197 elrege0 12149 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞) ↔ (((ℜ ∘ 𝐹)‘𝑥) ∈ ℝ ∧ 0 ≤ ((ℜ ∘ 𝐹)‘𝑥)))
198196, 197syl6bbr 277 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑥𝐷) → (0 ≤ ((ℜ ∘ 𝐹)‘𝑥) ↔ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞)))
199 fvco3 6185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:𝐷⟶ℂ ∧ 𝑥𝐷) → ((ℜ ∘ 𝐹)‘𝑥) = (ℜ‘(𝐹𝑥)))
2004, 199sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑥𝐷) → ((ℜ ∘ 𝐹)‘𝑥) = (ℜ‘(𝐹𝑥)))
201200breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑥𝐷) → (0 ≤ ((ℜ ∘ 𝐹)‘𝑥) ↔ 0 ≤ (ℜ‘(𝐹𝑥))))
202198, 201bitr3d 269 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑥𝐷) → (((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞) ↔ 0 ≤ (ℜ‘(𝐹𝑥))))
203202pm5.32da 671 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝑥𝐷 ∧ ((ℜ ∘ 𝐹)‘𝑥) ∈ (0[,)+∞)) ↔ (𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥)))))
204192, 203bitrd 267 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥)))))
205204adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ ℝ) → (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ↔ (𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥)))))
206 0le0 10987 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ≤ 0
207206, 18breqtrri 4610 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ≤ (ℜ‘0)
208207biantru 525 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑥𝐷 ↔ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0)))
209 eldif 3550 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (ℝ ∖ 𝐷) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥𝐷))
210209baibr 943 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℝ → (¬ 𝑥𝐷𝑥 ∈ (ℝ ∖ 𝐷)))
211208, 210syl5rbbr 274 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℝ → (𝑥 ∈ (ℝ ∖ 𝐷) ↔ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0))))
212211adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ ℝ) → (𝑥 ∈ (ℝ ∖ 𝐷) ↔ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0))))
213205, 212orbi12d 742 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥 ∈ ℝ) → ((𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ∨ 𝑥 ∈ (ℝ ∖ 𝐷)) ↔ ((𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥))) ∨ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0)))))
214 elun 3715 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ↔ (𝑥 ∈ ((ℜ ∘ 𝐹) “ (0[,)+∞)) ∨ 𝑥 ∈ (ℝ ∖ 𝐷)))
215 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (if(𝑥𝐷, (𝐹𝑥), 0) = (𝐹𝑥) → (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) = (ℜ‘(𝐹𝑥)))
216215breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . 23 (if(𝑥𝐷, (𝐹𝑥), 0) = (𝐹𝑥) → (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ 0 ≤ (ℜ‘(𝐹𝑥))))
217 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (if(𝑥𝐷, (𝐹𝑥), 0) = 0 → (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) = (ℜ‘0))
218217breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . 23 (if(𝑥𝐷, (𝐹𝑥), 0) = 0 → (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ 0 ≤ (ℜ‘0)))
219216, 218elimif 4072 . . . . . . . . . . . . . . . . . . . . . 22 (0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0)) ↔ ((𝑥𝐷 ∧ 0 ≤ (ℜ‘(𝐹𝑥))) ∨ (¬ 𝑥𝐷 ∧ 0 ≤ (ℜ‘0))))
220213, 214, 2193bitr4g 302 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ ℝ) → (𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ↔ 0 ≤ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))))
221220ifbid 4058 . . . . . . . . . . . . . . . . . . . 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))
222221mpteq2dva 4672 . . . . . . . . . . . . . . . . . . 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)))
223220ifbid 4058 . . . . . . . . . . . . . . . . . . . 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))))
224223mpteq2dva 4672 . . . . . . . . . . . . . . . . . . 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)))))
225175, 179, 182, 222, 224offval2 6812 . . . . . . . . . . . . . . . . . 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))))))
226 ovif12 6637 . . . . . . . . . . . . . . . . . . . 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))))
22789ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℝ)
228227recnd 9947 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (𝑔𝑥) ∈ ℂ)
229 0cn 9911 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
230 ifcl 4080 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ ℂ)
231228, 229, 230sylancl 693 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ ℂ)
232231addid1d 10115 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) + 0) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
233231mulm1d 10361 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) = -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
234233oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (0 + (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) = (0 + -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
235231negcld 10258 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) ∈ ℂ)
236235addid2d 10116 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (0 + -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) = -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
237234, 236eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔 ∈ dom ∫1𝑥 ∈ ℝ) → (0 + (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))) = -if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
238232, 237ifeq12d 4056 . . . . . . . . . . . . . . . . . . . 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)))
239226, 238syl5eq 2656 . . . . . . . . . . . . . . . . . . 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)))
240239mpteq2dva 4672 . . . . . . . . . . . . . . . . . 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))))
241225, 240sylan9eq 2664 . . . . . . . . . . . . . . . . 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))))
242 0xr 9965 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ*
243 pnfxr 9971 . . . . . . . . . . . . . . . . . . . . . . 23 +∞ ∈ ℝ*
244 0ltpnf 11832 . . . . . . . . . . . . . . . . . . . . . . 23 0 < +∞
245 snunioo 12169 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 < +∞) → ({0} ∪ (0(,)+∞)) = (0[,)+∞))
246242, 243, 244, 245mp3an 1416 . . . . . . . . . . . . . . . . . . . . . 22 ({0} ∪ (0(,)+∞)) = (0[,)+∞)
247246imaeq2i 5383 . . . . . . . . . . . . . . . . . . . . 21 ((ℜ ∘ 𝐹) “ ({0} ∪ (0(,)+∞))) = ((ℜ ∘ 𝐹) “ (0[,)+∞))
248 imaundi 5464 . . . . . . . . . . . . . . . . . . . . 21 ((ℜ ∘ 𝐹) “ ({0} ∪ (0(,)+∞))) = (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞)))
249247, 248eqtr3i 2634 . . . . . . . . . . . . . . . . . . . 20 ((ℜ ∘ 𝐹) “ (0[,)+∞)) = (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞)))
250 ismbfcn 23204 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝐷⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
2514, 250syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
25243, 251mpbid 221 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))
253252simpld 474 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (ℜ ∘ 𝐹) ∈ MblFn)
254 mbfimasn 23207 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ ∧ 0 ∈ ℝ) → ((ℜ ∘ 𝐹) “ {0}) ∈ dom vol)
25588, 254mp3an3 1405 . . . . . . . . . . . . . . . . . . . . . 22 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ) → ((ℜ ∘ 𝐹) “ {0}) ∈ dom vol)
256 mbfima 23205 . . . . . . . . . . . . . . . . . . . . . 22 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ) → ((ℜ ∘ 𝐹) “ (0(,)+∞)) ∈ dom vol)
257 unmbl 23112 . . . . . . . . . . . . . . . . . . . . . 22 ((((ℜ ∘ 𝐹) “ {0}) ∈ dom vol ∧ ((ℜ ∘ 𝐹) “ (0(,)+∞)) ∈ dom vol) → (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞))) ∈ dom vol)
258255, 256, 257syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℜ ∘ 𝐹):𝐷⟶ℝ) → (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞))) ∈ dom vol)
259253, 194, 258syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((ℜ ∘ 𝐹) “ {0}) ∪ ((ℜ ∘ 𝐹) “ (0(,)+∞))) ∈ dom vol)
260249, 259syl5eqel 2692 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((ℜ ∘ 𝐹) “ (0[,)+∞)) ∈ dom vol)
261 fdm 5964 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:𝐷⟶ℂ → dom 𝐹 = 𝐷)
2624, 261syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → dom 𝐹 = 𝐷)
263 mbfdm 23201 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
26443, 263syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → dom 𝐹 ∈ dom vol)
265262, 264eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐷 ∈ dom vol)
266 difmbl 23118 . . . . . . . . . . . . . . . . . . . 20 ((ℝ ∈ dom vol ∧ 𝐷 ∈ dom vol) → (ℝ ∖ 𝐷) ∈ dom vol)
26711, 265, 266sylancr 694 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ∖ 𝐷) ∈ dom vol)
268 unmbl 23112 . . . . . . . . . . . . . . . . . . 19 ((((ℜ ∘ 𝐹) “ (0[,)+∞)) ∈ dom vol ∧ (ℝ ∖ 𝐷) ∈ dom vol) → (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol)
269260, 267, 268syl2anc 691 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol)
270 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑥 → (𝑔𝑡) = (𝑔𝑥))
271270breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑥 → (0 ≤ (𝑔𝑡) ↔ 0 ≤ (𝑔𝑥)))
272271, 270ifbieq1d 4059 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑥 → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
273272, 86, 177fvmpt 6191 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ℝ → ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
274273eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0) = ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥))
275274ifeq1d 4054 . . . . . . . . . . . . . . . . . . . . 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))
276275mpteq2ia 4668 . . . . . . . . . . . . . . . . . . . 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))
277276i1fres 23278 . . . . . . . . . . . . . . . . . . 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)
278 id 22 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1)
279 neg1rr 11002 . . . . . . . . . . . . . . . . . . . . . 22 -1 ∈ ℝ
280279a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → -1 ∈ ℝ)
281278, 280i1fmulc 23276 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ dom ∫1 → ((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ dom ∫1)
282 cmmbl 23109 . . . . . . . . . . . . . . . . . . . 20 ((((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ∈ dom vol → (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))) ∈ dom vol)
283 ifnot 4083 . . . . . . . . . . . . . . . . . . . . . . 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)))
284 eldif 3550 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷))))
285284baibr 943 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℝ → (¬ 𝑥 ∈ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)) ↔ 𝑥 ∈ (ℝ ∖ (((ℜ ∘ 𝐹) “ (0[,)+∞)) ∪ (ℝ ∖ 𝐷)))))
286 tru 1479 . . . . . . . . . . . . . . . . . . . . . . . . . 26
287 negex 10158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 -1 ∈ V
288287fconst 6004 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ℝ × {-1}):ℝ⟶{-1}
289 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((ℝ × {-1}):ℝ⟶{-1} → (ℝ × {-1}) Fn ℝ)
290288, 289mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⊤ → (ℝ × {-1}) Fn ℝ)
29199a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⊤ → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) Fn ℝ)
292102a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⊤ → ℝ ∈ V)
293 inidm 3784 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (ℝ ∩ ℝ) = ℝ
294287fvconst2 6374 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ ℝ → ((ℝ × {-1})‘𝑥) = -1)
295294adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⊤ ∧ 𝑥 ∈ ℝ) → ((ℝ × {-1})‘𝑥) = -1)
296273adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⊤ ∧ 𝑥 ∈ ℝ) → ((𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))‘𝑥) = if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0))
297290, 291, 292, 292, 293, 295, 296ofval 6804 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((⊤ ∧ 𝑥 ∈ ℝ) → (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥) = (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
298286, 297mpan 702 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℝ → (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥) = (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)))
299298eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℝ → (-1 · if(0 ≤ (𝑔𝑥), (𝑔𝑥), 0)) = (((ℝ × {-1}) ∘𝑓 · (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))‘𝑥))
300285, 299ifbieq1d 4059 . . . . . . . . . . . . . . . . . . . . . . 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))
301283, 300syl5eqr 2658 . . . . . . . . . . . . . . . . . . . . . 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))
302301mpteq2ia 4668 . . . . . . . . . . . . . . . . . . . . 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))
303302i1fres 23278 . . . . . . . . . . . . . . . . . . . 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)
304281, 282, 303syl2an 493 . . . . . . . . . . . . . . . . . . 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)
305277, 304i1fadd 23268 . . . . . . . . . . . . . . . . . 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)
30687, 269, 305syl2anr 494 . . . . . . . . . . . . . . . . 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)
307241, 306eqeltrrd 2689 . . . . . . . . . . . . . . . 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)
308157cbvmptv 4678 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) = (𝑡 ∈ ℝ ↦ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
309308, 31syl5eqel 2692 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1)
3109, 308fmptd 6292 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ)
311309, 310jca 553 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1 ∧ (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ))
312311adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑔 ∈ dom ∫1) → ((𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))) ∈ 𝐿1 ∧ (𝑥 ∈ ℝ ↦ (ℜ‘if(𝑥𝐷, (𝐹𝑥), 0))):ℝ⟶ℝ))
313 ftc1anclem4 32658 . . . . . . . . . . . . . . . . 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)
3143133expb 1258 . . . . . . . . . . . . . . . 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)
315307, 312, 314syl2anc 691 . . . . . . . . . . . . . . 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)
316174, 315syl5eqelr 2693 . . . . . . . . . . . . . 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)
317136, 140, 141, 153, 316itg2addnc 32634 . . . . . . . . . . . . 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))))))))
318102a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑔 ∈ dom ∫1) → ℝ ∈ V)
31998a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ V)
320 eqidd 2611 . . . . . . . . . . . . . . 15 ((𝜑𝑔 ∈ dom ∫1) → (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) = (𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
321 eqidd 2611 . . . . . . . . . . . . . . 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))))))
322318, 319, 148, 320, 321offval2 6812 . . . . . . . . . . . . . 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)))))))
323322fveq2d 6107 . . . . . . . . . . . . 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))))))))
324317, 323eqtr3d 2646 . . . . . . . . . . . 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))))))))
325324adantr 480 . . . . . . . . . . 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))))))))
326 nfv 1830 . . . . . . . . . . . . . 14 𝑡(𝜑𝑔 ∈ dom ∫1)
327 nfcv 2751 . . . . . . . . . . . . . . 15 𝑡𝑔
328 nfcv 2751 . . . . . . . . . . . . . . 15 𝑡𝑟
329 nfmpt1 4675 . . . . . . . . . . . . . . 15 𝑡(𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
330327, 328, 329nfbr 4629 . . . . . . . . . . . . . 14 𝑡 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
331326, 330nfan 1816 . . . . . . . . . . . . 13 𝑡((𝜑𝑔 ∈ dom ∫1) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))))
332 anass 679 . . . . . . . . . . . . . . 15 (((𝜑𝑔 ∈ dom ∫1) ∧ 𝑡 ∈ ℝ) ↔ (𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)))
333 ffn 5958 . . . . . . . . . . . . . . . . . . . . 21 (𝑔:ℝ⟶ℝ → 𝑔 Fn ℝ)
33489, 333syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑔 ∈ dom ∫1𝑔 Fn ℝ)
335 fvex 6113 . . . . . . . . . . . . . . . . . . . . . 22 (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ V
336335, 74fnmpti 5935 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) Fn ℝ
337336a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑔 ∈ dom ∫1 → (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) Fn ℝ)
338 eqidd 2611 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → (𝑔𝑡) = (𝑔𝑡))
33974fvmpt2 6200 . . . . . . . . . . . . . . . . . . . . . 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))))
340335, 339mpan2 703 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 ∈ ℝ → ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))‘𝑡) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
341340adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → ((𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))‘𝑡) = (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
342334, 337, 103, 103, 293, 338, 341ofrval 6805 . . . . . . . . . . . . . . . . . . 19 ((𝑔 ∈ dom ∫1𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 𝑡 ∈ ℝ) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3433423com23 1263 . . . . . . . . . . . . . . . . . 18 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3443433expa 1257 . . . . . . . . . . . . . . . . 17 (((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
345344adantll 746 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ 𝑔𝑟 ≤ (𝑡 ∈ ℝ ↦ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))) → (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
346 resubcl 10224 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
3478, 106, 346syl2an 493 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
348347ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 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)
349 absid 13884 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
3508, 349sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
351350breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → ((𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ↔ (𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
352351biimpa 500 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
353352an32s 842 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
354353adantllr 751 . . . . . . . . . . . . . . . . . . . . . . 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)))
355 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔𝑡) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → ((𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ↔ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
356 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → (0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ↔ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
357355, 356ifboth 4074 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑔𝑡) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∧ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
358354, 357sylancom 698 . . . . . . . . . . . . . . . . . . . . . 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)))
359 subge0 10420 . . . . . . . . . . . . . . . . . . . . . . . 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))))
3608, 106, 359syl2an 493 . . . . . . . . . . . . . . . . . . . . . . 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))))
361360ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 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))))
362358, 361mpbird 246 . . . . . . . . . . . . . . . . . . . . 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)))
363348, 362absidd 14009 . . . . . . . . . . . . . . . . . . . 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)))
364 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . . 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))
365364oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 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)))
366365fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 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))))
367366adantl 481 . . . . . . . . . . . . . . . . . . . 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))))
3688ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
369349oveq1d 6564 . . . . . . . . . . . . . . . . . . . . 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)))
370368, 369sylan 487 . . . . . . . . . . . . . . . . . . . 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)))
371363, 367, 3703eqtr4d 2654 . . . . . . . . . . . . . . . . . . 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)))
372106renegcld 10336 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔 ∈ dom ∫1𝑡 ∈ ℝ) → -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ)
373 resubcl 10224 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
3748, 372, 373syl2an 493 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) − -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)) ∈ ℝ)
375374ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 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)
37690ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (𝑔𝑡) ∈ ℝ)
3778ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 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)
3788adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ)
379 ltnle 9996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 ↔ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
38088, 379mpan2 703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 ↔ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
381 ltle 10005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
38288, 381mpan2 703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
383380, 382sylbird 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ → (¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
384383imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0)
385 absnid 13886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
386384, 385syldan 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ∈ ℝ ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
387386breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
388387biimpa 500 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
389388an32s 842 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
390378, 389sylanl1 680 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
391376, 377, 390lenegcon2d 10489 . . . . . . . . . . . . . . . . . . . . . . . 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))
392 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑔 ∈ dom ∫1𝑡 ∈ ℝ)) ∧ (𝑔𝑡) ≤ (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))) → 𝜑)
39388a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → 0 ∈ ℝ)
3948, 393ltnled 10063 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 ↔ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))))
3958, 88, 381sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) < 0 → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
396394, 395sylbird 249 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0))
397396imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0)
398392, 397sylan 487 . . . . . . . . . . . . . . . . . . . . . . . 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)
399 negeq 10152 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑔𝑡) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → -(𝑔𝑡) = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
400399breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑔𝑡) = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -(𝑔𝑡) ↔ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
401 neg0 10206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -0 = 0
402 negeq 10152 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → -0 = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
403401, 402syl5eqr 2658 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → 0 = -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
404403breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 = if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0) → ((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0 ↔ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0)))
405400, 404ifboth 4074 . . . . . . . . . . . . . . . . . . . . . . . 24 (((ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -(𝑔𝑡) ∧ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0) → (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ -if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))
406391, 398, 405syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 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))
407 suble0 10421 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
4088, 372, 407syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . 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)))
409408ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . 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)))
410406, 409mpbird 246 . . . . . . . . . . . . . . . . . . . . . 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)
411375, 410absnidd 14000 . . . . . . . . . . . . . . . . . . . . 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)))
412 subneg 10209 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
413412negeqd 10154 . . . . . . . . . . . . . . . . . . . . . . . 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)))
414 negdi2 10218 . . . . . . . . . . . . . . . . . . . . . . . 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)))
415413, 414eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . 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)))
41632, 142, 415syl2an 493 . . . . . . . . . . . . . . . . . . . . . 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)))
417416ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 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)))
418411, 417eqtrd 2644 . . . . . . . . . . . . . . . . . . . 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)))
419 iffalse 4045 . . . . . . . . . . . . . . . . . . . . . . 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))
420419oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 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)))
421420fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 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))))
422421adantl 481 . . . . . . . . . . . . . . . . . . . 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))))
4238, 385sylan 487 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)) ≤ 0) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
424397, 423syldan 486 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ¬ 0 ≤ (ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) = -(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0)))
425424oveq1d 6564 . . . . . . . . . . . . . . . . . . . . 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)))
426392, 425sylan 487 . . . . . . . . . . . . . . . . . . . 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)))
427418, 422, 4263eqtr4d 2654 . . . . . . . . . . . . . . . . . . 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)))
428371, 427pm2.61dan 828 . . . . . . . . . . . . . . . . . 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)))
429428oveq2d 6565 . . . . . . . . . . . . . . . . 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))))
43054recnd 9947 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (abs‘(ℜ‘if(𝑡𝐷, (𝐹𝑡), 0))) ∈ ℂ)
431 pncan3 10168 . . . . . . . . . . . . . . . . . . 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))))
432142, 430, 431syl2anr 494 . . . . . . . . . . . . . . . . . 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))))
433432adantr 480 . . . . . . . . . . . . . . . . 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))))
434429, 433eqtrd 2644 . . . . . . . . . . . . . . . 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))))
435345, 434syldan 486 . . . . . . . . . . . . . . 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))))
436332, 435sylanb 488 . . . . . . . . . . . . . 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))))
437436an32s 842 . . . . . . . . . . . . 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))))
438331, 437mpteq2da 4671 . . . . . . . . . . . 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)))))
439438fveq2d 6107 . . . . . . . . . . 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))))))
440325, 439eqtrd 2644 . . . . . . . . . 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))))))
441440breq1d 4593 . . . . . . . . 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)))
442441adantllr 751 . . . . . . . 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)))
443316adantlr 747 . . . . . . . . . 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)
44464ad2antlr 759 . . . . . . . . . 10 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → 𝑌 ∈ ℝ)
445117adantl 481 . . . . . . . . . 10 (((𝜑𝑌 ∈ ℝ+) ∧ 𝑔 ∈ dom ∫1) → (∫2‘(𝑡 ∈ ℝ ↦ if(0 ≤ (𝑔𝑡), (𝑔𝑡), 0))) ∈ ℝ)
446443, 444, 445ltadd2d 10072 . . . . . . . . 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)))
447446adantr 480 . . . . . . . 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)))
448 ltsubadd 10377 . . . . . . . . . . 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)))
44961, 64, 117, 448syl3an 1360 . . . . . . . . . 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)))
4504493expa 1257 . . . . . . . . 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)))
451450adantr 480 . . . . . . . 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)))
452442, 447, 4513bitr4d 299 . . . . . . 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)))))
453452adantrr 749 . . . . . 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)))))
454133, 453mpbird 246 . . . . 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)
455454ex 449 . . . 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))
456455reximdva 3000 . . 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))
457 fveq1 6102 . . . . . . . . . . . . . 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))
458457, 170sylan9eq 2664 . . . . . . . . . . . . 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)))
459458oveq2d 6565 . . . . . . . . . . . 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))))
460459fveq2d 6107 . . . . . . . . . . 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)))))
461460mpteq2dva 4672 . . . . . . . . . 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))))))
462461fveq2d 6107 . . . . . . . . 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)))))))
463462breq1d 4593 . . . . . . . 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))
464463rspcev 3282 . . . . . . 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)
465464ex 449 . . . . . 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))
466307, 465syl 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))
467466rexlimdva 3013 . . . 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))
468467adantr 480 . . 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))
469456, 468syld 46 . 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))
47082, 469mpd 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ∘𝑟 cofr 6794  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146  ℝ+crp 11708  (,)cioo 12046  [,)cico 12048  [,]cicc 12049  ℜcre 13685  ℑcim 13686  abscabs 13822  volcvol 23039  MblFncmbf 23189  ∫1citg1 23190  ∫2citg2 23191  𝐿1cibl 23192  ∫citg 23193  0𝑝c0p 23242 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197  df-0p 23243 This theorem is referenced by:  ftc1anclem6  32660
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