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Theorem itgneg 23976
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 23940 . . . . . . . 8 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
31, 2syl 17 . . . . . . 7 (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)
4 itgcnval.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝑉)
53, 4mbfmptcl 23809 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
65recld 14318 . . . . 5 ((𝜑𝑥𝐴) → (ℜ‘𝐵) ∈ ℝ)
75iblcn 23971 . . . . . . 7 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)))
81, 7mpbid 224 . . . . . 6 (𝜑 → ((𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1))
98simpld 490 . . . . 5 (𝜑 → (𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1)
106, 9itgcl 23956 . . . 4 (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ)
11 ax-icn 10318 . . . . 5 i ∈ ℂ
125imcld 14319 . . . . . 6 ((𝜑𝑥𝐴) → (ℑ‘𝐵) ∈ ℝ)
138simprd 491 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)
1412, 13itgcl 23956 . . . . 5 (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ)
15 mulcl 10343 . . . . 5 ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
1611, 14, 15sylancr 581 . . . 4 (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
1710, 16negdid 10733 . . 3 (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)))
18 0re 10365 . . . . . . . 8 0 ∈ ℝ
19 ifcl 4352 . . . . . . . 8 (((ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
206, 18, 19sylancl 580 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
216iblre 23966 . . . . . . . . 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 224 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿1))
2322simpld 490 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿1)
2420, 23itgcl 23956 . . . . . 6 (𝜑 → ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
256renegcld 10788 . . . . . . . 8 ((𝜑𝑥𝐴) → -(ℜ‘𝐵) ∈ ℝ)
26 ifcl 4352 . . . . . . . 8 ((-(ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
2725, 18, 26sylancl 580 . . . . . . 7 ((𝜑𝑥𝐴) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
2822simprd 491 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿1)
2927, 28itgcl 23956 . . . . . 6 (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
3024, 29negsubdi2d 10736 . . . . 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 23969 . . . . . 6 (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥))
3231negeqd 10602 . . . . 5 (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = -(∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥))
335negcld 10707 . . . . . . . 8 ((𝜑𝑥𝐴) → -𝐵 ∈ ℂ)
3433recld 14318 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘-𝐵) ∈ ℝ)
354, 1iblneg 23975 . . . . . . . . 9 (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ 𝐿1)
3633iblcn 23971 . . . . . . . . 9 (𝜑 → ((𝑥𝐴 ↦ -𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1)))
3735, 36mpbid 224 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1))
3837simpld 490 . . . . . . 7 (𝜑 → (𝑥𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1)
3934, 38itgreval 23969 . . . . . 6 (𝜑 → ∫𝐴(ℜ‘-𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥))
405renegd 14333 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
4140breq2d 4887 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ (ℜ‘-𝐵) ↔ 0 ≤ -(ℜ‘𝐵)))
4241, 40ifbieq1d 4331 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) = if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0))
4342itgeq2dv 23954 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥)
4440negeqd 10602 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -(ℜ‘-𝐵) = --(ℜ‘𝐵))
456recnd 10392 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (ℜ‘𝐵) ∈ ℂ)
4645negnegd 10711 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → --(ℜ‘𝐵) = (ℜ‘𝐵))
4744, 46eqtrd 2861 . . . . . . . . . 10 ((𝜑𝑥𝐴) → -(ℜ‘-𝐵) = (ℜ‘𝐵))
4847breq2d 4887 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0 ≤ -(ℜ‘-𝐵) ↔ 0 ≤ (ℜ‘𝐵)))
4948, 47ifbieq1d 4331 . . . . . . . 8 ((𝜑𝑥𝐴) → if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) = if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0))
5049itgeq2dv 23954 . . . . . . 7 (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥)
5143, 50oveq12d 6928 . . . . . 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 2861 . . . . 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 2871 . . . 4 (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = ∫𝐴(ℜ‘-𝐵) d𝑥)
54 mulneg2 10798 . . . . . 6 ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
5511, 14, 54sylancr 581 . . . . 5 (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
56 ifcl 4352 . . . . . . . . . . 11 (((ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
5712, 18, 56sylancl 580 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
5812iblre 23966 . . . . . . . . . . . 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 224 . . . . . . . . . . 11 (𝜑 → ((𝑥𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿1))
6059simpld 490 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿1)
6157, 60itgcl 23956 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
6212renegcld 10788 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → -(ℑ‘𝐵) ∈ ℝ)
63 ifcl 4352 . . . . . . . . . . 11 ((-(ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
6462, 18, 63sylancl 580 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
6559simprd 491 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿1)
6664, 65itgcl 23956 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
6761, 66negsubdi2d 10736 . . . . . . . 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 14334 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
6968breq2d 4887 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (0 ≤ (ℑ‘-𝐵) ↔ 0 ≤ -(ℑ‘𝐵)))
7069, 68ifbieq1d 4331 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) = if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0))
7170itgeq2dv 23954 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥)
7268negeqd 10602 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → -(ℑ‘-𝐵) = --(ℑ‘𝐵))
7312recnd 10392 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (ℑ‘𝐵) ∈ ℂ)
7473negnegd 10711 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → --(ℑ‘𝐵) = (ℑ‘𝐵))
7572, 74eqtrd 2861 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → -(ℑ‘-𝐵) = (ℑ‘𝐵))
7675breq2d 4887 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (0 ≤ -(ℑ‘-𝐵) ↔ 0 ≤ (ℑ‘𝐵)))
7776, 75ifbieq1d 4331 . . . . . . . . . 10 ((𝜑𝑥𝐴) → if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) = if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0))
7877itgeq2dv 23954 . . . . . . . . 9 (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥)
7971, 78oveq12d 6928 . . . . . . . 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 2864 . . . . . . 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 23969 . . . . . . . 8 (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥))
8281negeqd 10602 . . . . . . 7 (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = -(∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥))
8333imcld 14319 . . . . . . . 8 ((𝜑𝑥𝐴) → (ℑ‘-𝐵) ∈ ℝ)
8437simprd 491 . . . . . . . 8 (𝜑 → (𝑥𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1)
8583, 84itgreval 23969 . . . . . . 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 2871 . . . . . 6 (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = ∫𝐴(ℑ‘-𝐵) d𝑥)
8786oveq2d 6926 . . . . 5 (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
8855, 87eqtr3d 2863 . . . 4 (𝜑 → -(i · ∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
8953, 88oveq12d 6928 . . 3 (𝜑 → (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
9017, 89eqtrd 2861 . 2 (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
914, 1itgcnval 23972 . . 3 (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
9291negeqd 10602 . 2 (𝜑 → -∫𝐴𝐵 d𝑥 = -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
9333, 35itgcnval 23972 . 2 (𝜑 → ∫𝐴-𝐵 d𝑥 = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
9490, 92, 933eqtr4d 2871 1 (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  ifcif 4308   class class class wbr 4875  cmpt 4954  cfv 6127  (class class class)co 6910  cc 10257  cr 10258  0cc0 10259  ici 10261   + caddc 10262   · cmul 10264  cle 10399  cmin 10592  -cneg 10593  cre 14221  cim 14222  MblFncmbf 23787  𝐿1cibl 23790  citg 23791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337  ax-addf 10338
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-disj 4844  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-of 7162  df-ofr 7163  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-sup 8623  df-inf 8624  df-oi 8691  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-n0 11626  df-z 11712  df-uz 11976  df-q 12079  df-rp 12120  df-xadd 12240  df-ioo 12474  df-ico 12476  df-icc 12477  df-fz 12627  df-fzo 12768  df-fl 12895  df-mod 12971  df-seq 13103  df-exp 13162  df-hash 13418  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-clim 14603  df-sum 14801  df-xmet 20106  df-met 20107  df-ovol 23637  df-vol 23638  df-mbf 23792  df-itg1 23793  df-itg2 23794  df-ibl 23795  df-itg 23796  df-0p 23843
This theorem is referenced by:  itgsub  23998  itgsubnc  34010  itgmulc2nc  34016  sqwvfourb  41234
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