ovolval5lem1 - Mathbox for Glauco Siliprandi

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Theorem ovolval5lem1 46643

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 1914	. . 3 ⊢ Ⅎ𝑛𝜑
2		nnex 12168	. . . 4 ⊢ ℕ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
4		volf 25463	. . . . 5 ⊢ vol:dom vol⟶(0[,]+∞)
5	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
6		ioombl 25499	. . . . 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 7039	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ∈ (0[,]+∞))
9	1, 3, 8	sge0xrclmpt 46419	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ∈ ℝ^*)
10		0xr 11197	. . . . 5 ⊢ 0 ∈ ℝ^*
11	10	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
12		pnfxr 11204	. . . . 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 46572	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
17	14, 15, 16	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
18		ovolval5lem1.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
19	18	rpred 12971	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
20	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
21		2nn 12235	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
23		nnnn0 12425	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
24		nnexpcl 14015	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
25	22, 23, 24	syl2anc 584	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
26	25	nnred 12177	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
28	25	nnne0d 12212	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
29	28	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
30	20, 27, 29	redivcld 11986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
31	17, 30	readdcld 11179	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ)
32	31	rexrd 11200	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ^*)
33	15	rexrd 11200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ^*)
34		icombl 25498	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
35	14, 33, 34	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴[,)𝐵) ∈ dom vol)
36		volge0 45952	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ⁺)
39	25	nnrpd 12969	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
40	39	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
41	38, 40	rpdivcld 12988	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ⁺)
42	41	rpge0d 12975	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑊 / (2↑𝑛)))
43	17, 30, 37, 42	addge0d 11730	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
44		rexr 11196	. . . . . 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 13056	. . . . . 6 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) < +∞)
47	44, 45, 46	xrltled 13086	. . . . 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 45518	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ (0[,]+∞))
50	1, 3, 49	sge0xrclmpt 46419	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) ∈ ℝ^*)
51	5, 35	ffvelcdmd 7039	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ (0[,]+∞))
52	1, 3, 51	sge0xrclmpt 46419	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) ∈ ℝ^*)
53	19	rexrd 11200	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
54	52, 53	xaddcld 13237	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊) ∈ ℝ^*)
55	14, 30	resubcld 11582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (𝑊 / (2↑𝑛))) ∈ ℝ)
56		volioore 45981	. . . . . . . 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 584	. . . . . . 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 480	. . . . . 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 4490	. . . . . . 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 481	. . . . . 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 11178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℂ)
62	14	recnd 11178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℂ)
63	30	recnd 11178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℂ)
64	61, 62, 63	subsubd 11537	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
65	64	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
66	58, 60, 65	3eqtrd 2768	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
67	15, 14	resubcld 11582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ∈ ℝ)
68	14, 15	sublevolico 45975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵)))
69	67, 17, 30, 68	leadd1dd 11768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
70	69	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
71	66, 70	eqbrtrd 5124	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
72	57	adantr 480	. . . . . 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 4493	. . . . . . 7 ⊢ (¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵 → if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0) = 0)
74	73	adantl 481	. . . . . 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 2764	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = 0)
76	43	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
77	75, 76	eqbrtrd 5124	. . . 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 812	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
79	1, 3, 8, 49, 78	sge0lempt 46401	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))))
80	17, 30	rexaddd 13170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
81	80	eqcomd 2735	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))
82	81	mpteq2dva 5195	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛)))) = (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛)))))
83	82	fveq2d 6844	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))))
84	30	rexrd 11200	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
85		rexr 11196	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
86	12	a1i 11	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → +∞ ∈ ℝ^*)
87		ltpnf 13056	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) < +∞)
88	85, 86, 87	xrltled 13086	. . . . . . 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 45518	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ (0[,]+∞))
91	1, 3, 51, 90	sge0xadd 46426	. . . 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 12975	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑊)
95	19	ltpnfd 13057	. . . . . . 7 ⊢ (𝜑 → 𝑊 < +∞)
96	92, 93, 53, 94, 95	elicod 13332	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (0[,)+∞))
97	96	sge0ad2en 46422	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑊 / (2↑𝑛)))) = 𝑊)
98	97	oveq2d 7385	. . . 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 2768	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))
100	50, 99	xreqled 45319	. 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 13098	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 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3402 Vcvv 3444 ifcif 4484 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5631 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 + caddc 11047 +∞cpnf 11181 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ℝ⁺crp 12927 +_𝑒 cxad 13046 (,)cioo 13282 [,)cico 13284 [,]cicc 13285 ↑cexp 14002 volcvol 25397 Σ^{^}csumge0 46353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-rest 17361 df-topgen 17382 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22814 df-topon 22831 df-bases 22866 df-cmp 23307 df-ovol 25398 df-vol 25399 df-sumge0 46354

This theorem is referenced by: ovolval5lem2 46644

W3C validator