Theorem ioombl1lem3 25936

Description: Lemma for ioombl1 25938. (Contributed by Mario Carneiro, 18-Aug-2014.)

Hypotheses

Ref	Expression
ioombl1.b	⊢ 𝐵 = (𝐴(,)+∞)
ioombl1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ioombl1.e	⊢ (𝜑 → 𝐸 ⊆ ℝ)
ioombl1.v	⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
ioombl1.c	⊢ (𝜑 → 𝐶 ∈ ℝ+)
ioombl1.s	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ioombl1.t	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ioombl1.u	⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ioombl1.f1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ioombl1.f2	⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹))
ioombl1.f3	⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
ioombl1.p	⊢ 𝑃 = (1st ‘(𝐹‘𝑛))
ioombl1.q	⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))
ioombl1.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
ioombl1.h	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)

Assertion

Ref	Expression
ioombl1lem3	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))

Distinct variable groups:   𝐵,𝑛   𝐶,𝑛   𝑛,𝐸   𝑛,𝐹   𝑛,𝐺   𝑛,𝐻   𝜑,𝑛   𝑆,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝑃(𝑛)   𝑄(𝑛)   𝑇(𝑛)   𝑈(𝑛)

Proof of Theorem ioombl1lem3

Step	Hyp	Ref	Expression
1		ioombl1.q	. . . . 5 ⊢ 𝑄 = (2nd ‘(𝐹‘𝑛))
2		ioombl1.f1	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 25842	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
4	2, 3	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
5	4	simp2d 1161	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
6	1, 5	eqeltrid 2868	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
7	6	recnd 11376	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℂ)
8		ioombl1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
10		ioombl1.p	. . . . . . 7 ⊢ 𝑃 = (1st ‘(𝐹‘𝑛))
11	4	simp1d 1160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
12	10, 11	eqeltrid 2868	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
13	9, 12	ifcld 4532	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ)
14	13, 6	ifcld 4532	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
15	14	recnd 11376	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℂ)
16	12	recnd 11376	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℂ)
17	7, 15, 16	npncand 11732	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑄 − if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) + (if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) − 𝑃)) = (𝑄 − 𝑃))
18		ioombl1.b	. . . . . . 7 ⊢ 𝐵 = (𝐴(,)+∞)
19		ioombl1.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
20		ioombl1.v	. . . . . . 7 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
21		ioombl1.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
22		ioombl1.s	. . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
23		ioombl1.t	. . . . . . 7 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
24		ioombl1.u	. . . . . . 7 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
25		ioombl1.f2	. . . . . . 7 ⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐹))
26		ioombl1.f3	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
27		ioombl1.g	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
28		ioombl1.h	. . . . . . 7 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)
29	18, 8, 19, 20, 21, 22, 23, 24, 2, 25, 26, 10, 1, 27, 28	ioombl1lem1 25934	. . . . . 6 ⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
30	29	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31		eqid 2763	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
32	31	ovolfsval 25846	. . . . 5 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
33	30, 32	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
34		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
35		opex 5439	. . . . . . . 8 ⊢ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V
36	27	fvmpt2 7036	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V) → (𝐺‘𝑛) = ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
37	34, 35, 36	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
38	37	fveq2d 6920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
39		op2ndg 8048	. . . . . . 7 ⊢ ((if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (2nd ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
40	14, 6, 39	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
41	38, 40	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = 𝑄)
42	37	fveq2d 6920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
43		op1stg 8047	. . . . . . 7 ⊢ ((if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (1st ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
44	14, 6, 43	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
45	42, 44	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
46	41, 45	oveq12d 7469	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (𝑄 − if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)))
47	33, 46	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (𝑄 − if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)))
48	29	simprd 501	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49		eqid 2763	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
50	49	ovolfsval 25846	. . . . 5 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻‘𝑛)) − (1st ‘(𝐻‘𝑛))))
51	48, 50	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻‘𝑛)) − (1st ‘(𝐻‘𝑛))))
52		opex 5439	. . . . . . . 8 ⊢ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V
53	28	fvmpt2 7036	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V) → (𝐻‘𝑛) = ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)
54	34, 52, 53	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) = ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)
55	54	fveq2d 6920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐻‘𝑛)) = (2nd ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩))
56		op2ndg 8048	. . . . . . 7 ⊢ ((𝑃 ∈ ℝ ∧ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (2nd ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
57	12, 14, 56	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
58	55, 57	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
59	54	fveq2d 6920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐻‘𝑛)) = (1st ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩))
60		op1stg 8047	. . . . . . 7 ⊢ ((𝑃 ∈ ℝ ∧ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (1st ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
61	12, 14, 60	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
62	59, 61	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐻‘𝑛)) = 𝑃)
63	58, 62	oveq12d 7469	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐻‘𝑛)) − (1st ‘(𝐻‘𝑛))) = (if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) − 𝑃))
64	51, 63	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = (if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) − 𝑃))
65	47, 64	oveq12d 7469	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = ((𝑄 − if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) + (if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) − 𝑃)))
66		eqid 2763	. . . . 5 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
67	66	ovolfsval 25846	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
68	2, 67	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
69	1, 10	oveq12i 7463	. . 3 ⊢ (𝑄 − 𝑃) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))
70	68, 69	eqtr4di 2816	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = (𝑄 − 𝑃))
71	17, 65, 70	3eqtr4d 2808	1 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ifcif 4485  ⟨cop 4840  ∪ cuni 4872   class class class wbr 5108   ↦ cmpt 5192   × cxp 5656  ran crn 5659   ∘ ccom 5662  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  supcsup 9469  ℝcr 11238  1c1 11240   + caddc 11242  +∞cpnf 11379  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383   − cmin 11580  ℕcn 12375  ℝ⁺crp 13160  (,)cioo 13516  seqcseq 14184  abscabs 15444  vol*covol 25838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-seq 14185  df-exp 14245  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446

This theorem is used by:  ioombl1lem4  25937