ovolicc1 - Metamath Proof Explorer

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Theorem ovolicc1 25485

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 13357	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
4	1, 2, 3	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5		ovolcl 25447	. . 3 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → (vol‘(𝐴[,]𝐵)) ∈ ℝ^)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) ∈ ℝ^)
7		ovolicc.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
8		df-br 5101	. . . . . . . . . . 11 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
9	7, 8	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
10	1, 2	opelxpd 5671	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
11	9, 10	elind 4154	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13		0le0 12258	. . . . . . . . . 10 ⊢ 0 ≤ 0
14		df-br 5101	. . . . . . . . . 10 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
15	13, 14	mpbi 230	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ ≤
16		0re 11146	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		opelxpi 5669	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
18	16, 16, 17	mp2an 693	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
19	15, 18	elini 4153	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20		ifcl 4527	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	12, 19, 20	sylancl 587	. . . . . . 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 7068	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		eqid 2737	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
25		eqid 2737	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
26	24, 25	ovolsf 25441	. . . . . 6 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
27	23, 26	syl 17	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
28	27	frnd 6678	. . . 4 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29		icossxr 13360	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ^*
30	28, 29	sstrdi 3948	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ^*)
31		supxrcl 13242	. . 3 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ^* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ) ∈ ℝ^)
32	30, 31	syl 17	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ) ∈ ℝ^)
33	2, 1	resubcld 11577	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
34	33	rexrd 11194	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
35		1nn 12168	. . . . . . 7 ⊢ 1 ∈ ℕ
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37		op1stg 7955	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
38	1, 2, 37	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
39	38	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
40		elicc2 13339	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
41	1, 2, 40	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
42	41	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
43	42	simp2d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
44	39, 43	eqbrtrd 5122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
45	42	simp3d 1145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
46		op2ndg 7956	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
47	1, 2, 46	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
48	47	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	45, 48	breqtrrd 5128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩))
50		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = (𝐺‘1))
51		iftrue 4487	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52		opex 5419	. . . . . . . . . . . . 13 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
53	51, 22, 52	fvmpt 6949	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
54	35, 53	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐺‘1) = ⟨𝐴, 𝐵⟩
55	50, 54	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = ⟨𝐴, 𝐵⟩)
56	55	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑛 = 1 → (1^st ‘(𝐺‘𝑛)) = (1^st ‘⟨𝐴, 𝐵⟩))
57	56	breq1d 5110	. . . . . . . 8 ⊢ (𝑛 = 1 → ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ↔ (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
58	55	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑛 = 1 → (2^nd ‘(𝐺‘𝑛)) = (2^nd ‘⟨𝐴, 𝐵⟩))
59	58	breq2d 5112	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑥 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩)))
60	57, 59	anbi12d 633	. . . . . . 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 3578	. . . . . 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 837	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
63	62	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
64		ovolficc 25437	. . . . 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 585	. . . 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 257	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ ran ([,] ∘ 𝐺))
67	25	ovollb2 25458	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ([,] ∘ 𝐺)) → (vol‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ))
68	23, 66, 67	syl2anc 585	. 2 ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ))
69		addrid 11325	. . . . . . . . 9 ⊢ (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
70	69	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71		nnuz 12802	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
72	35, 71	eleqtri 2835	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘1)
73	72	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 1 ∈ (ℤ_≥‘1))
74		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
75	74, 71	eleqtrdi 2847	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ_≥‘1))
76		rge0ssre 13384	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
77	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78		ffvelcdm 7035	. . . . . . . . . . 11 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
79	77, 35, 78	sylancl 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
80	76, 79	sselid 3933	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
81	80	recnd 11172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
82	23	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
83		elfzuz 13448	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((1 + 1)...𝑥) → 𝑘 ∈ (ℤ_≥‘(1 + 1)))
84	83	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ_≥‘(1 + 1)))
85		df-2 12220	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
86	85	fveq2i 6845	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
87	84, 86	eleqtrrdi 2848	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ_≥‘2))
88		eluz2nn 12813	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
89	87, 88	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
90	24	ovolfsval 25439	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑘 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))))
91	82, 89, 90	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))))
92		eqeq1 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
93	92	ifbid 4505	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94		opex 5419	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ V
95	52, 94	ifex 4532	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
96	93, 22, 95	fvmpt 6949	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
97	89, 96	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺‘𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
98		eluz2b3 12847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ_≥‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
99	98	simprbi 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ≠ 1)
100	87, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ≠ 1)
101	100	neneqd 2938	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ¬ 𝑘 = 1)
102	101	iffalsed 4492	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
103	97, 102	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺‘𝑘) = ⟨0, 0⟩)
104	103	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘⟨0, 0⟩))
105		c0ex 11138	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
106	105, 105	op2nd 7952	. . . . . . . . . . . 12 ⊢ (2^nd ‘⟨0, 0⟩) = 0
107	104, 106	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = 0)
108	103	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘⟨0, 0⟩))
109	105, 105	op1st 7951	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨0, 0⟩) = 0
110	108, 109	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = 0)
111	107, 110	oveq12d 7386	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = (0 − 0))
112		0m0e0 12272	. . . . . . . . . 10 ⊢ (0 − 0) = 0
113	111, 112	eqtrdi 2788	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = 0)
114	91, 113	eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
115	70, 73, 75, 81, 114	seqid2 13983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116		1z 12533	. . . . . . . 8 ⊢ 1 ∈ ℤ
117	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
118	24	ovolfsval 25439	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 1 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))))
119	117, 35, 118	sylancl 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))))
120	54	fveq2i 6845	. . . . . . . . . . 11 ⊢ (2^nd ‘(𝐺‘1)) = (2^nd ‘⟨𝐴, 𝐵⟩)
121	47	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
122	120, 121	eqtrid 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (2^nd ‘(𝐺‘1)) = 𝐵)
123	54	fveq2i 6845	. . . . . . . . . . 11 ⊢ (1^st ‘(𝐺‘1)) = (1^st ‘⟨𝐴, 𝐵⟩)
124	38	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
125	123, 124	eqtrid 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1^st ‘(𝐺‘1)) = 𝐴)
126	122, 125	oveq12d 7386	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))) = (𝐵 − 𝐴))
127	119, 126	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵 − 𝐴))
128	116, 127	seq1i 13950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵 − 𝐴))
129	115, 128	eqtr3d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵 − 𝐴))
130	33	leidd 11715	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
131	130	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
132	129, 131	eqbrtrd 5122	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
133	132	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
134	27	ffnd 6671	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135		breq1 5103	. . . . . 6 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵 − 𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴)))
136	135	ralrn 7042	. . . . 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 17	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴) ↔ ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴)))
138	133, 137	mpbird 257	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴))
139		supxrleub 13253	. . . 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 585	. . 3 ⊢ (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^*, < ) ≤ (𝐵 − 𝐴) ↔ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵 − 𝐴)))
141	138, 140	mpbird 257	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^*, < ) ≤ (𝐵 − 𝐴))
142	6, 32, 34, 68, 141	xrletrd 13088	1 ⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵 − 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∩ cin 3902 ⊆ wss 3903 ifcif 4481 ⟨cop 4588 ∪ cuni 4865 class class class wbr 5100 ↦ cmpt 5181 × cxp 5630 ran crn 5633 ∘ ccom 5636 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 supcsup 9355 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 +∞cpnf 11175 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 − cmin 11376 ℕcn 12157 2c2 12212 ℤ_≥cuz 12763 [,)cico 13275 [,]cicc 13276 ...cfz 13435 seqcseq 13936 abscabs 15169 vol*covol 25431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-ovol 25433

This theorem is referenced by: ovolicc 25492

W3C validator