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Theorem itgneg 23789
Description: Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)
Hypotheses
Ref Expression
itgcnval.1 ((𝜑𝑥𝐴) → 𝐵𝑉)
itgcnval.2 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
Assertion
Ref Expression
itgneg (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑉
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem itgneg
StepHypRef Expression
1 itgcnval.2 . . . . . . . 8 (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
2 iblmbf 23753 . . . . . . . 8 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
31, 2syl 17 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
4 itgcnval.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝑉)
53, 4mbfmptcl 23623 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
65recld 14141 . . . . 5 ((𝜑𝑥𝐴) → (ℜ‘𝐵) ∈ ℝ)
75iblcn 23784 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)))
81, 7mpbid 222 . . . . . 6 (𝜑 → ((𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1))
98simpld 476 . . . . 5 (𝜑 → (𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1)
106, 9itgcl 23769 . . . 4 (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ)
11 ax-icn 10196 . . . . 5 i ∈ ℂ
125imcld 14142 . . . . . 6 ((𝜑𝑥𝐴) → (ℑ‘𝐵) ∈ ℝ)
138simprd 477 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)
1412, 13itgcl 23769 . . . . 5 (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ)
15 mulcl 10221 . . . . 5 ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
1611, 14, 15sylancr 567 . . . 4 (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
1710, 16negdid 10606 . . 3 (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)))
18 0re 10241 . . . . . . . 8 0 ∈ ℝ
19 ifcl 4267 . . . . . . . 8 (((ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
206, 18, 19sylancl 566 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
216iblre 23779 . . . . . . . . 9 (𝜑 → ((𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿1)))
229, 21mpbid 222 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿1))
2322simpld 476 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿1)
2420, 23itgcl 23769 . . . . . 6 (𝜑 → ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
256renegcld 10658 . . . . . . . 8 ((𝜑𝑥𝐴) → -(ℜ‘𝐵) ∈ ℝ)
26 ifcl 4267 . . . . . . . 8 ((-(ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
2725, 18, 26sylancl 566 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
2822simprd 477 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿1)
2927, 28itgcl 23769 . . . . . 6 (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
3024, 29negsubdi2d 10609 . . . . 5 (𝜑 → -(∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥))
316, 9itgreval 23782 . . . . . 6 (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥))
3231negeqd 10476 . . . . 5 (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = -(∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥))
335negcld 10580 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝐵 ∈ ℂ)
3433recld 14141 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘-𝐵) ∈ ℝ)
354, 1iblneg 23788 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ 𝐿1)
3633iblcn 23784 . . . . . . . . 9 (𝜑 → ((𝑥𝐴 ↦ -𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1)))
3735, 36mpbid 222 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1))
3837simpld 476 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1)
3934, 38itgreval 23782 . . . . . 6 (𝜑 → ∫𝐴(ℜ‘-𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥))
405renegd 14156 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
4140breq2d 4796 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ (ℜ‘-𝐵) ↔ 0 ≤ -(ℜ‘𝐵)))
4241, 40ifbieq1d 4246 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) = if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0))
4342itgeq2dv 23767 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥)
4440negeqd 10476 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -(ℜ‘-𝐵) = --(ℜ‘𝐵))
456recnd 10269 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (ℜ‘𝐵) ∈ ℂ)
4645negnegd 10584 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → --(ℜ‘𝐵) = (ℜ‘𝐵))
4744, 46eqtrd 2804 . . . . . . . . . 10 ((𝜑𝑥𝐴) → -(ℜ‘-𝐵) = (ℜ‘𝐵))
4847breq2d 4796 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ -(ℜ‘-𝐵) ↔ 0 ≤ (ℜ‘𝐵)))
4948, 47ifbieq1d 4246 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) = if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0))
5049itgeq2dv 23767 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥)
5143, 50oveq12d 6810 . . . . . 6 (𝜑 → (∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥))
5239, 51eqtrd 2804 . . . . 5 (𝜑 → ∫𝐴(ℜ‘-𝐵) d𝑥 = (∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥))
5330, 32, 523eqtr4d 2814 . . . 4 (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = ∫𝐴(ℜ‘-𝐵) d𝑥)
54 mulneg2 10668 . . . . . 6 ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
5511, 14, 54sylancr 567 . . . . 5 (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
56 ifcl 4267 . . . . . . . . . . 11 (((ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
5712, 18, 56sylancl 566 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
5812iblre 23779 . . . . . . . . . . . 12 (𝜑 → ((𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿1)))
5913, 58mpbid 222 . . . . . . . . . . 11 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿1))
6059simpld 476 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿1)
6157, 60itgcl 23769 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
6212renegcld 10658 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -(ℑ‘𝐵) ∈ ℝ)
63 ifcl 4267 . . . . . . . . . . 11 ((-(ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
6462, 18, 63sylancl 566 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
6559simprd 477 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿1)
6664, 65itgcl 23769 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
6761, 66negsubdi2d 10609 . . . . . . . 8 (𝜑 → -(∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥))
685imnegd 14157 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
6968breq2d 4796 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (0 ≤ (ℑ‘-𝐵) ↔ 0 ≤ -(ℑ‘𝐵)))
7069, 68ifbieq1d 4246 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) = if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0))
7170itgeq2dv 23767 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥)
7268negeqd 10476 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → -(ℑ‘-𝐵) = --(ℑ‘𝐵))
7312recnd 10269 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (ℑ‘𝐵) ∈ ℂ)
7473negnegd 10584 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → --(ℑ‘𝐵) = (ℑ‘𝐵))
7572, 74eqtrd 2804 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -(ℑ‘-𝐵) = (ℑ‘𝐵))
7675breq2d 4796 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (0 ≤ -(ℑ‘-𝐵) ↔ 0 ≤ (ℑ‘𝐵)))
7776, 75ifbieq1d 4246 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) = if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0))
7877itgeq2dv 23767 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥)
7971, 78oveq12d 6810 . . . . . . . 8 (𝜑 → (∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥))
8067, 79eqtr4d 2807 . . . . . . 7 (𝜑 → -(∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥))
8112, 13itgreval 23782 . . . . . . . 8 (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥))
8281negeqd 10476 . . . . . . 7 (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = -(∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥))
8333imcld 14142 . . . . . . . 8 ((𝜑𝑥𝐴) → (ℑ‘-𝐵) ∈ ℝ)
8437simprd 477 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1)
8583, 84itgreval 23782 . . . . . . 7 (𝜑 → ∫𝐴(ℑ‘-𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥))
8680, 82, 853eqtr4d 2814 . . . . . 6 (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = ∫𝐴(ℑ‘-𝐵) d𝑥)
8786oveq2d 6808 . . . . 5 (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
8855, 87eqtr3d 2806 . . . 4 (𝜑 → -(i · ∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
8953, 88oveq12d 6810 . . 3 (𝜑 → (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
9017, 89eqtrd 2804 . 2 (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
914, 1itgcnval 23785 . . 3 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
9291negeqd 10476 . 2 (𝜑 → -∫𝐴𝐵 d𝑥 = -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
9333, 35itgcnval 23785 . 2 (𝜑 → ∫𝐴-𝐵 d𝑥 = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
9490, 92, 933eqtr4d 2814 1 (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  ifcif 4223   class class class wbr 4784  cmpt 4861  cfv 6031  (class class class)co 6792  cc 10135  cr 10136  0cc0 10137  ici 10139   + caddc 10140   · cmul 10142  cle 10276  cmin 10467  -cneg 10468  cre 14044  cim 14045  MblFncmbf 23601  𝐿1cibl 23604  citg 23605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215  ax-addf 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-disj 4753  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-ofr 7044  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-rp 12035  df-xadd 12151  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-fl 12800  df-mod 12876  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624  df-xmet 19953  df-met 19954  df-ovol 23451  df-vol 23452  df-mbf 23606  df-itg1 23607  df-itg2 23608  df-ibl 23609  df-itg 23610  df-0p 23656
This theorem is referenced by:  itgsub  23811  itgsubnc  33797  itgmulc2nc  33803  sqwvfourb  40957
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