ovolval5lem1 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ovolval5lem1		Structured version Visualization version GIF version

Theorem ovolval5lem1 47258

Description: (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛) ))(,)𝐵)))) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵) ))) +_𝑒 𝑊)). (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval5lem1.a	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
ovolval5lem1.b	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)
ovolval5lem1.w	⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
ovolval5lem1.c	⊢ 𝐶 = {𝑛 ∈ ℕ ∣ 𝐴 < 𝐵}

**Assertion**
Ref	Expression
ovolval5lem1	⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))

Distinct variable groups: 𝐶,𝑛 𝑛,𝑊 𝜑,𝑛 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑛)

**Proof of Theorem ovolval5lem1**
Step	Hyp	Ref	Expression
1		nfv 1941	. . 3 ⊢ Ⅎ𝑛𝜑
2		nnex 12239	. . . 4 ⊢ ℕ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
4		volf 25657	. . . . 5 ⊢ vol:dom vol⟶(0[,]+∞)
5	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
6		ioombl 25693	. . . . 5 ⊢ ((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵) ∈ dom vol
7	6	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵) ∈ dom vol)
8	5, 7	ffvelcdmd 7081	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ∈ (0[,]+∞))
9	1, 3, 8	sge0xrclmpt 47034	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ∈ ℝ^*)
10		0xr 11256	. . . . 5 ⊢ 0 ∈ ℝ^*
11	10	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
12		pnfxr 11263	. . . . 5 ⊢ +∞ ∈ ℝ^*
13	12	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ^*)
14		ovolval5lem1.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)
15		ovolval5lem1.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)
16		volicore 47187	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
17	14, 15, 16	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
18		ovolval5lem1.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
19	18	rpred 13060	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
20	19	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
21		2nn 12314	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
23		nnnn0 12511	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
24		nnexpcl 14110	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
25	22, 23, 24	syl2anc 595	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
26	25	nnred 12248	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
27	26	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
28	25	nnne0d 12286	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
29	28	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
30	20, 27, 29	redivcld 12043	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
31	17, 30	readdcld 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ)
32	31	rexrd 11259	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ^*)
33	15	rexrd 11259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ^*)
34		icombl 25692	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
35	14, 33, 34	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴[,)𝐵) ∈ dom vol)
36		volge0 46567	. . . . . 6 ⊢ ((𝐴[,)𝐵) ∈ dom vol → 0 ≤ (vol‘(𝐴[,)𝐵)))
37	35, 36	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐴[,)𝐵)))
38	18	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ⁺)
39	25	nnrpd 13058	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
40	39	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
41	38, 40	rpdivcld 13077	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ⁺)
42	41	rpge0d 13064	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑊 / (2↑𝑛)))
43	17, 30, 37, 42	addge0d 11790	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
44		rexr 11255	. . . . . 6 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ^*)
45	12	a1i 11	. . . . . 6 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → +∞ ∈ ℝ^*)
46		ltpnf 13145	. . . . . 6 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) < +∞)
47	44, 45, 46	xrltled 13175	. . . . 5 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ≤ +∞)
48	31, 47	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ≤ +∞)
49	11, 13, 32, 43, 48	eliccxrd 46135	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ (0[,]+∞))
50	1, 3, 49	sge0xrclmpt 47034	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) ∈ ℝ^*)
51	5, 35	ffvelcdmd 7081	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ (0[,]+∞))
52	1, 3, 51	sge0xrclmpt 47034	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) ∈ ℝ^*)
53	19	rexrd 11259	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
54	52, 53	xaddcld 13327	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊) ∈ ℝ^*)
55	14, 30	resubcld 11642	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (𝑊 / (2↑𝑛))) ∈ ℝ)
56		volioore 46596	. . . . . . . 8 ⊢ (((𝐴 − (𝑊 / (2↑𝑛))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0))
57	55, 15, 56	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0))
58	57	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0))
59		iftrue 4498	. . . . . . 7 ⊢ ((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵 → if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0) = (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))))
60	59	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0) = (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))))
61	15	recnd 11237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℂ)
62	14	recnd 11237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℂ)
63	30	recnd 11237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℂ)
64	61, 62, 63	subsubd 11597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
65	64	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
66	58, 60, 65	3eqtrd 2808	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
67	15, 14	resubcld 11642	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ∈ ℝ)
68	14, 15	sublevolico 46590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵)))
69	67, 17, 30, 68	leadd1dd 11828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
70	69	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
71	66, 70	eqbrtrd 5137	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
72	57	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0))
73		iffalse 4501	. . . . . . 7 ⊢ (¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵 → if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0) = 0)
74	73	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0) = 0)
75	72, 74	eqtrd 2804	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = 0)
76	43	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
77	75, 76	eqbrtrd 5137	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
78	71, 77	pm2.61dan 824	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
79	1, 3, 8, 49, 78	sge0lempt 47016	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))))
80	17, 30	rexaddd 13260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
81	80	eqcomd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))
82	81	mpteq2dva 5208	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛)))) = (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛)))))
83	82	fveq2d 6886	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))))
84	30	rexrd 11259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
85		rexr 11255	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
86	12	a1i 11	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → +∞ ∈ ℝ^*)
87		ltpnf 13145	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) < +∞)
88	85, 86, 87	xrltled 13175	. . . . . . 7 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) ≤ +∞)
89	30, 88	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ≤ +∞)
90	11, 13, 84, 42, 89	eliccxrd 46135	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ (0[,]+∞))
91	1, 3, 51, 90	sge0xadd 47041	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑊 / (2↑𝑛))))))
92	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ^*)
93	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
94	18	rpge0d 13064	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑊)
95	19	ltpnfd 13146	. . . . . . 7 ⊢ (𝜑 → 𝑊 < +∞)
96	92, 93, 53, 94, 95	elicod 13422	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (0[,)+∞))
97	96	sge0ad2en 47037	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑊 / (2↑𝑛)))) = 𝑊)
98	97	oveq2d 7427	. . . 4 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑊 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))
99	83, 91, 98	3eqtrd 2808	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))
100	50, 99	xreqled 45938	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))
101	9, 50, 54, 79, 100	xrletrd 13187	1 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 {crab 3423 Vcvv 3463 ifcif 4492 class class class wbr 5113 ↦ cmpt 5196 dom cdm 5662 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ℝcr 11099 0cc0 11100 + caddc 11103 +∞cpnf 11240 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 − cmin 11441 / cdiv 11871 ℕcn 12233 2c2 12295 ℕ₀cn0 12504 ℝ⁺crp 13016 +_𝑒 cxad 13135 (,)cioo 13372 [,)cico 13374 [,]cicc 13375 ↑cexp 14097 volcvol 25591 Σ^{^}csumge0 46968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-rest 17475 df-topgen 17496 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-top 23020 df-topon 23037 df-bases 23072 df-cmp 23513 df-ovol 25592 df-vol 25593 df-sumge0 46969

This theorem is referenced by: ovolval5lem2 47259

W3C validator