ovolval5lem1 - Mathbox for Glauco Siliprandi

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Theorem ovolval5lem1 46178

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 1909	. . 3 ⊢ Ⅎ𝑛𝜑
2		nnex 12251	. . . 4 ⊢ ℕ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
4		volf 25502	. . . . 5 ⊢ vol:dom vol⟶(0[,]+∞)
5	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
6		ioombl 25538	. . . . 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 7094	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ∈ (0[,]+∞))
9	1, 3, 8	sge0xrclmpt 45954	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ∈ ℝ^*)
10		0xr 11293	. . . . 5 ⊢ 0 ∈ ℝ^*
11	10	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ^*)
12		pnfxr 11300	. . . . 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 46107	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
17	14, 15, 16	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
18		ovolval5lem1.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℝ⁺)
19	18	rpred 13051	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
20	19	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
21		2nn 12318	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
23		nnnn0 12512	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
24		nnexpcl 14075	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
25	22, 23, 24	syl2anc 582	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
26	25	nnred 12260	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
27	26	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
28	25	nnne0d 12295	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
29	28	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
30	20, 27, 29	redivcld 12075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
31	17, 30	readdcld 11275	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ)
32	31	rexrd 11296	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ^*)
33	15	rexrd 11296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ^*)
34		icombl 25537	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
35	14, 33, 34	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴[,)𝐵) ∈ dom vol)
36		volge0 45487	. . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑊 ∈ ℝ⁺)
39	25	nnrpd 13049	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
40	39	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
41	38, 40	rpdivcld 13068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ⁺)
42	41	rpge0d 13055	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑊 / (2↑𝑛)))
43	17, 30, 37, 42	addge0d 11822	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
44		rexr 11292	. . . . . 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 13135	. . . . . 6 ⊢ (((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ ℝ → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) < +∞)
47	44, 45, 46	xrltled 13164	. . . . 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 45050	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) ∈ (0[,]+∞))
50	1, 3, 49	sge0xrclmpt 45954	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) ∈ ℝ^*)
51	5, 35	ffvelcdmd 7094	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴[,)𝐵)) ∈ (0[,]+∞))
52	1, 3, 51	sge0xrclmpt 45954	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) ∈ ℝ^*)
53	19	rexrd 11296	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
54	52, 53	xaddcld 13315	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊) ∈ ℝ^*)
55	14, 30	resubcld 11674	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 − (𝑊 / (2↑𝑛))) ∈ ℝ)
56		volioore 45516	. . . . . . . 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 582	. . . . . . 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 479	. . . . . 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 4536	. . . . . . 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 480	. . . . . 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 11274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℂ)
62	14	recnd 11274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℂ)
63	30	recnd 11274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℂ)
64	61, 62, 63	subsubd 11631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
65	64	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
66	58, 60, 65	3eqtrd 2769	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))))
67	15, 14	resubcld 11674	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ∈ ℝ)
68	14, 15	sublevolico 45510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵)))
69	67, 17, 30, 68	leadd1dd 11860	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
70	69	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → ((𝐵 − 𝐴) + (𝑊 / (2↑𝑛))) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
71	66, 70	eqbrtrd 5171	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
72	57	adantr 479	. . . . . 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 4539	. . . . . . 7 ⊢ (¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵 → if((𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵, (𝐵 − (𝐴 − (𝑊 / (2↑𝑛)))), 0) = 0)
74	73	adantl 480	. . . . . 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 2765	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) = 0)
76	43	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ (𝐴 − (𝑊 / (2↑𝑛))) ≤ 𝐵) → 0 ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
77	75, 76	eqbrtrd 5171	. . . 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 811	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)) ≤ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
79	1, 3, 8, 49, 78	sge0lempt 45936	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))))
80	17, 30	rexaddd 13248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))
81	80	eqcomd 2731	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))) = ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))
82	81	mpteq2dva 5249	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛)))) = (𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛)))))
83	82	fveq2d 6900	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) +_𝑒 (𝑊 / (2↑𝑛))))))
84	30	rexrd 11296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
85		rexr 11292	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) ∈ ℝ^*)
86	12	a1i 11	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → +∞ ∈ ℝ^*)
87		ltpnf 13135	. . . . . . . 8 ⊢ ((𝑊 / (2↑𝑛)) ∈ ℝ → (𝑊 / (2↑𝑛)) < +∞)
88	85, 86, 87	xrltled 13164	. . . . . . 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 45050	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ (0[,]+∞))
91	1, 3, 51, 90	sge0xadd 45961	. . . 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 13055	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑊)
95	19	ltpnfd 13136	. . . . . . 7 ⊢ (𝜑 → 𝑊 < +∞)
96	92, 93, 53, 94, 95	elicod 13409	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (0[,)+∞))
97	96	sge0ad2en 45957	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑊 / (2↑𝑛)))) = 𝑊)
98	97	oveq2d 7435	. . . 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 2769	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ((vol‘(𝐴[,)𝐵)) + (𝑊 / (2↑𝑛))))) = ((Σ^{^}‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +_𝑒 𝑊))
100	50, 99	xreqled 44850	. 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 13176	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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 {crab 3418 Vcvv 3461 ifcif 4530 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 0cc0 11140 + caddc 11143 +∞cpnf 11277 ℝ^*cxr 11279 < clt 11280 ≤ cle 11281 − cmin 11476 / cdiv 11903 ℕcn 12245 2c2 12300 ℕ₀cn0 12505 ℝ⁺crp 13009 +_𝑒 cxad 13125 (,)cioo 13359 [,)cico 13361 [,]cicc 13362 ↑cexp 14062 volcvol 25436 Σ^{^}csumge0 45888

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 df-rest 17407 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22840 df-topon 22857 df-bases 22893 df-cmp 23335 df-ovol 25437 df-vol 25438 df-sumge0 45889

This theorem is referenced by: ovolval5lem2 46179

W3C validator