ovolval5lem1 - Mathbox for Glauco Siliprandi

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Theorem ovolval5lem1 41612

Description: |- ( ph -> ( sum^ (𝑛 ∈ ℕ ↦ (vol ( ( A - ( W / ( 2 ^ n ) ) ) (,) B ) ) ) ) <_ ( ( sum^ (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 2010	. . 3 ⊢ Ⅎ𝑛𝜑
2		nnex 11319	. . . 4 ⊢ ℕ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
4		volf 23637	. . . . 5 ⊢ vol:dom vol⟶(0[,]+∞)
5	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
6		ioombl 23673	. . . . 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	ffvelrnd 6586	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ∈ (0[,]+∞))
9	1, 3, 8	sge0xrclmpt 41388	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ∈ ℝ^*)
10		0xr 10375	. . . . 5 ⊢ 0 ∈ ℝ^*
11	10	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
12		pnfxr 10382	. . . . 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 41541	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
17	14, 15, 16	syl2anc 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
18		ovolval5lem1.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
19	18	rpred 12117	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
20	19	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
21		2nn 11386	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
23		nnnn0 11588	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
24		nnexpcl 13127	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
25	22, 23, 24	syl2anc 580	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
26	25	nnred 11329	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
27	26	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
28	25	nnne0d 11363	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
29	28	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
30	20, 27, 29	redivcld 11145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
31	17, 30	readdcld 10358	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ)
32	31	rexrd 10378	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ^*)
33	15	rexrd 10378	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ^*)
34		icombl 23672	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
35	14, 33, 34	syl2anc 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴[,)𝐵) ∈ dom vol)
36		volge0 40920	. . . . . 6 ⊢ ((𝐴[,)𝐵) ∈ dom vol → 0 ≤ (vol‘(𝐴[,)𝐵)))
37	35, 36	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐴[,)𝐵)))
38	18	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ⁺)
39	25	nnrpd 12115	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
40	39	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
41	38, 40	rpdivcld 12134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ⁺)
42	41	rpge0d 12121	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑊 / (2↑𝑛)))
43	17, 30, 37, 42	addge0d 10895	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
44		rexr 10374	. . . . . 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 12201	. . . . . 6 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) < +∞)
47	44, 45, 46	xrltled 12230	. . . . 5 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ≤ +∞)
48	31, 47	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ≤ +∞)
49	11, 13, 32, 43, 48	eliccxrd 40498	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ (0[,]+∞))
50	1, 3, 49	sge0xrclmpt 41388	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) ∈ ℝ^*)
51	5, 35	ffvelrnd 6586	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ (0[,]+∞))
52	1, 3, 51	sge0xrclmpt 41388	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) ∈ ℝ^*)
53	19	rexrd 10378	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
54	52, 53	xaddcld 12380	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊) ∈ ℝ^*)
55	14, 30	resubcld 10750	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (𝑊 / (2↑𝑛))) ∈ ℝ)
56		volioore 40950	. . . . . . . 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 580	. . . . . . 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 473	. . . . . 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 4283	. . . . . . 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 474	. . . . . 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 10357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℂ)
62	14	recnd 10357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℂ)
63	30	recnd 10357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℂ)
64	61, 62, 63	subsubd 10712	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
65	64	adantr 473	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
66	58, 60, 65	3eqtrd 2837	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
67	15, 14	resubcld 10750	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ∈ ℝ)
68	14, 15	sublevolico 40944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵)))
69	67, 17, 30, 68	leadd1dd 10933	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
70	69	adantr 473	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
71	66, 70	eqbrtrd 4865	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
72	57	adantr 473	. . . . . 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 4286	. . . . . . 7 ⊢ (¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵 → if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0) = 0)
74	73	adantl 474	. . . . . 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 2833	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = 0)
76	43	adantr 473	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
77	75, 76	eqbrtrd 4865	. . . 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 848	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
79	1, 3, 8, 49, 78	sge0lempt 41370	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))))
80	17, 30	rexaddd 12314	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
81	80	eqcomd 2805	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))
82	81	mpteq2dva 4937	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛)))) = (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛)))))
83	82	fveq2d 6415	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))))
84	30	rexrd 10378	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
85		rexr 10374	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
86	12	a1i 11	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → +∞ ∈ ℝ^*)
87		ltpnf 12201	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) < +∞)
88	85, 86, 87	xrltled 12230	. . . . . . 7 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) ≤ +∞)
89	30, 88	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ≤ +∞)
90	11, 13, 84, 42, 89	eliccxrd 40498	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ (0[,]+∞))
91	1, 3, 51, 90	sge0xadd 41395	. . . 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 12121	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑊)
95	19	ltpnfd 12202	. . . . . . 7 ⊢ (𝜑 → 𝑊 < +∞)
96	92, 93, 53, 94, 95	elicod 12473	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (0[,)+∞))
97	96	sge0ad2en 41391	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑊 / (2↑𝑛)))) = 𝑊)
98	97	oveq2d 6894	. . . 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 2837	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))
100	50, 99	xreqled 40290	. 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 12242	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 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 {crab 3093 Vcvv 3385 ifcif 4277 class class class wbr 4843 ↦ cmpt 4922 dom cdm 5312 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 0cc0 10224 + caddc 10227 +∞cpnf 10360 ℝ^*cxr 10362 < clt 10363 ≤ cle 10364 − cmin 10556 / cdiv 10976 ℕcn 11312 2c2 11368 ℕ₀cn0 11580 ℝ⁺crp 12074 +_𝑒 cxad 12191 (,)cioo 12424 [,)cico 12426 [,]cicc 12427 ↑cexp 13114 volcvol 23571 Σ^{^}csumge0 41322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-rlim 14561 df-sum 14758 df-rest 16398 df-topgen 16419 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-top 21027 df-topon 21044 df-bases 21079 df-cmp 21519 df-ovol 23572 df-vol 23573 df-sumge0 41323

This theorem is referenced by: ovolval5lem2 41613

W3C validator