ovolicc1 - Metamath Proof Explorer

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Theorem ovolicc1 25675

Description: The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

**Hypotheses**
Ref	Expression
ovolicc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ovolicc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ovolicc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ovolicc1.4	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))

**Assertion**
Ref	Expression
ovolicc1	⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵 − 𝐴))

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝑛,𝐺 𝜑,𝑛

Proof of Theorem ovolicc1 Dummy variables 𝑘 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolicc.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		ovolicc.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		iccssre 13451	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
4	1, 2, 3	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5		ovolcl 25637	. . 3 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → (vol‘(𝐴[,]𝐵)) ∈ ℝ^)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) ∈ ℝ^)
7		ovolicc.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
8		df-br 5110	. . . . . . . . . . 11 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
9	7, 8	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
10	1, 2	opelxpd 5700	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
11	9, 10	elind 4153	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13		0le0 12337	. . . . . . . . . 10 ⊢ 0 ≤ 0
14		df-br 5110	. . . . . . . . . 10 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
15	13, 14	mpbi 233	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ ≤
16		0re 11205	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		opelxpi 5698	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
18	16, 16, 17	mp2an 704	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
19	15, 18	elini 4152	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20		ifcl 4533	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	12, 19, 20	sylancl 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
22		ovolicc1.4	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
23	21, 22	fmptd 7109	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		eqid 2763	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
25		eqid 2763	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
26	24, 25	ovolsf 25631	. . . . . 6 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
27	23, 26	syl 18	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
28	27	frnd 6714	. . . 4 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29		icossxr 13454	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ^*
30	28, 29	sstrdi 3949	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ^*)
31		supxrcl 13336	. . 3 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ^* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ) ∈ ℝ^)
32	30, 31	syl 18	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ) ∈ ℝ^)
33	2, 1	resubcld 11637	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
34	33	rexrd 11254	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
35		1nn 12239	. . . . . . 7 ⊢ 1 ∈ ℕ
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37		op1stg 7994	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
38	1, 2, 37	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
39	38	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
40		elicc2 13433	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
41	1, 2, 40	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
42	41	biimpa 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
43	42	simp2d 1161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
44	39, 43	eqbrtrd 5133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
45	42	simp3d 1162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
46		op2ndg 7995	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
47	1, 2, 46	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
48	47	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	45, 48	breqtrrd 5139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩))
50		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = (𝐺‘1))
51		iftrue 4493	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52		opex 5445	. . . . . . . . . . . . 13 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
53	51, 22, 52	fvmpt 6989	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
54	35, 53	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐺‘1) = ⟨𝐴, 𝐵⟩
55	50, 54	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = ⟨𝐴, 𝐵⟩)
56	55	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑛 = 1 → (1^st ‘(𝐺‘𝑛)) = (1^st ‘⟨𝐴, 𝐵⟩))
57	56	breq1d 5119	. . . . . . . 8 ⊢ (𝑛 = 1 → ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ↔ (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
58	55	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑛 = 1 → (2^nd ‘(𝐺‘𝑛)) = (2^nd ‘⟨𝐴, 𝐵⟩))
59	58	breq2d 5121	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑥 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩)))
60	57, 59	anbi12d 643	. . . . . . 7 ⊢ (𝑛 = 1 → (((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))) ↔ ((1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩))))
61	60	rspcev 3581	. . . . . 6 ⊢ ((1 ∈ ℕ ∧ ((1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩))) → ∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
62	36, 44, 49, 61	syl12anc 849	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
63	62	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
64		ovolficc 25627	. . . . 5 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐴[,]𝐵) ⊆ ∪ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛)))))
65	4, 23, 64	syl2anc 595	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ⊆ ∪ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛)))))
66	63, 65	mpbird 260	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ ran ([,] ∘ 𝐺))
67	25	ovollb2 25648	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ([,] ∘ 𝐺)) → (vol‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ))
68	23, 66, 67	syl2anc 595	. 2 ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ))
69		addrid 11385	. . . . . . . . 9 ⊢ (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
70	69	adantl 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71		nnuz 12896	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
72	35, 71	eleqtri 2861	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘1)
73	72	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 1 ∈ (ℤ_≥‘1))
74		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
75	74, 71	eleqtrdi 2873	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ_≥‘1))
76		rge0ssre 13478	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
77	27	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78		ffvelcdm 7076	. . . . . . . . . . 11 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
79	77, 35, 78	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
80	76, 79	sselid 3935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
81	80	recnd 11232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
82	23	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
83		elfzuz 13543	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((1 + 1)...𝑥) → 𝑘 ∈ (ℤ_≥‘(1 + 1)))
84	83	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ_≥‘(1 + 1)))
85		df-2 12298	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
86	85	fveq2i 6884	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
87	84, 86	eleqtrrdi 2874	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ_≥‘2))
88		eluz2nn 12907	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
89	87, 88	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
90	24	ovolfsval 25629	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑘 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))))
91	82, 89, 90	syl2anc 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))))
92		eqeq1 2767	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
93	92	ifbid 4511	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94		opex 5445	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ V
95	52, 94	ifex 4538	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
96	93, 22, 95	fvmpt 6989	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
97	89, 96	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺‘𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
98		eluz2b3 12941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ_≥‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
99	98	simprbi 502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ≠ 1)
100	87, 99	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ≠ 1)
101	100	neneqd 2963	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ¬ 𝑘 = 1)
102	101	iffalsed 4498	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
103	97, 102	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺‘𝑘) = ⟨0, 0⟩)
104	103	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘⟨0, 0⟩))
105		c0ex 11195	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
106	105, 105	op2nd 7991	. . . . . . . . . . . 12 ⊢ (2^nd ‘⟨0, 0⟩) = 0
107	104, 106	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = 0)
108	103	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘⟨0, 0⟩))
109	105, 105	op1st 7990	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨0, 0⟩) = 0
110	108, 109	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = 0)
111	107, 110	oveq12d 7428	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = (0 − 0))
112		0m0e0 12354	. . . . . . . . . 10 ⊢ (0 − 0) = 0
113	111, 112	eqtrdi 2814	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = 0)
114	91, 113	eqtrd 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
115	70, 73, 75, 81, 114	seqid2 14080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116		1z 12619	. . . . . . . 8 ⊢ 1 ∈ ℤ
117	23	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
118	24	ovolfsval 25629	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 1 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))))
119	117, 35, 118	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))))
120	54	fveq2i 6884	. . . . . . . . . . 11 ⊢ (2^nd ‘(𝐺‘1)) = (2^nd ‘⟨𝐴, 𝐵⟩)
121	47	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
122	120, 121	eqtrid 2810	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (2^nd ‘(𝐺‘1)) = 𝐵)
123	54	fveq2i 6884	. . . . . . . . . . 11 ⊢ (1^st ‘(𝐺‘1)) = (1^st ‘⟨𝐴, 𝐵⟩)
124	38	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
125	123, 124	eqtrid 2810	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1^st ‘(𝐺‘1)) = 𝐴)
126	122, 125	oveq12d 7428	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))) = (𝐵 − 𝐴))
127	119, 126	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵 − 𝐴))
128	116, 127	seq1i 14047	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵 − 𝐴))
129	115, 128	eqtr3d 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵 − 𝐴))
130	33	leidd 11775	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
131	130	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
132	129, 131	eqbrtrd 5133	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
133	132	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
134	27	ffnd 6706	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135		breq1 5112	. . . . . 6 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵 − 𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴)))
136	135	ralrn 7083	. . . . 5 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴) ↔ ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴)))
137	134, 136	syl 18	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴) ↔ ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴)))
138	133, 137	mpbird 260	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴))
139		supxrleub 13347	. . . 4 ⊢ ((ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ^* ∧ (𝐵 − 𝐴) ∈ ℝ^) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ) ≤ (𝐵 − 𝐴) ↔ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴)))
140	30, 34, 139	syl2anc 595	. . 3 ⊢ (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^*, < ) ≤ (𝐵 − 𝐴) ↔ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴)))
141	138, 140	mpbird 260	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^*, < ) ≤ (𝐵 − 𝐴))
142	6, 32, 34, 68, 141	xrletrd 13182	1 ⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵 − 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ∩ cin 3904 ⊆ wss 3905 ifcif 4487 ⟨cop 4595 ∪ cuni 4872 class class class wbr 5109 ↦ cmpt 5192 × cxp 5659 ran crn 5662 ∘ ccom 5665 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 2^nd c2nd 7981 supcsup 9396 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 +∞cpnf 11235 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 − cmin 11436 ℕcn 12228 2c2 12290 ℤ_≥cuz 12857 [,)cico 13369 [,]cicc 13370 ...cfz 13530 seqcseq 14033 abscabs 15281 vol*covol 25621

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-q 12968 df-rp 13012 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-ovol 25623

This theorem is referenced by: ovolicc 25682

W3C validator