Theorem ioombl1lem3 25084

Description: Lemma for ioombl1 25086. (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 24990	. . . . . . 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 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
5	4	simp2d 1143	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
6	1, 5	eqeltrid 2837	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ)
7	6	recnd 11244	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℂ)
8		ioombl1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
10		ioombl1.p	. . . . . . 7 ⊢ 𝑃 = (1st ‘(𝐹‘𝑛))
11	4	simp1d 1142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
12	10, 11	eqeltrid 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ)
13	9, 12	ifcld 4574	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ)
14	13, 6	ifcld 4574	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ)
15	14	recnd 11244	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℂ)
16	12	recnd 11244	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℂ)
17	7, 15, 16	npncand 11597	. 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 25082	. . . . . 6 ⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
30	29	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31		eqid 2732	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
32	31	ovolfsval 24994	. . . . 5 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
33	30, 32	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
34		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
35		opex 5464	. . . . . . . 8 ⊢ ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩ ∈ V
36	27	fvmpt2 7009	. . . . . . . 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 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩)
38	37	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
39		op2ndg 7990	. . . . . . 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 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩) = 𝑄)
41	38, 40	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = 𝑄)
42	37	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄⟩))
43		op1stg 7989	. . . . . . 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 584	. . . . . 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 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
46	41, 45	oveq12d 7429	. . . 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 2772	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (𝑄 − if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)))
48	29	simprd 496	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
49		eqid 2732	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
50	49	ovolfsval 24994	. . . . 5 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻‘𝑛)) − (1st ‘(𝐻‘𝑛))))
51	48, 50	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻‘𝑛)) − (1st ‘(𝐻‘𝑛))))
52		opex 5464	. . . . . . . 8 ⊢ ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩ ∈ V
53	28	fvmpt2 7009	. . . . . . . 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 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) = ⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩)
55	54	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐻‘𝑛)) = (2nd ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩))
56		op2ndg 7990	. . . . . . 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 584	. . . . . 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 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))
59	54	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐻‘𝑛)) = (1st ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩))
60		op1stg 7989	. . . . . . 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 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘⟨𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)⟩) = 𝑃)
62	59, 61	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐻‘𝑛)) = 𝑃)
63	58, 62	oveq12d 7429	. . . 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 2772	. . 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 7429	. 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 2732	. . . . 5 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
67	66	ovolfsval 24994	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
68	2, 67	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
69	1, 10	oveq12i 7423	. . 3 ⊢ (𝑄 − 𝑃) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))
70	68, 69	eqtr4di 2790	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = (𝑄 − 𝑃))
71	17, 65, 70	3eqtr4d 2782	1 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947   ⊆ wss 3948  ifcif 4528  ⟨cop 4634  ∪ cuni 4908   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674  ran crn 5677   ∘ ccom 5680  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411  1^st c1st 7975  2^nd c2nd 7976  supcsup 9437  ℝcr 11111  1c1 11113   + caddc 11115  +∞cpnf 11247  ℝ^*cxr 11249   < clt 11250   ≤ cle 11251   − cmin 11446  ℕcn 12214  ℝ⁺crp 12976  (,)cioo 13326  seqcseq 13968  abscabs 15183  vol*covol 24986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-seq 13969  df-exp 14030  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185

This theorem is referenced by:  ioombl1lem4  25085