ovolicc1 - Metamath Proof Explorer

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Theorem ovolicc1 25417

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 13390	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5		ovolcl 25379	. . 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 5108	. . . . . . . . . . 11 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
9	7, 8	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
10	1, 2	opelxpd 5677	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
11	9, 10	elind 4163	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13		0le0 12287	. . . . . . . . . 10 ⊢ 0 ≤ 0
14		df-br 5108	. . . . . . . . . 10 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
15	13, 14	mpbi 230	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ ≤
16		0re 11176	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		opelxpi 5675	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
18	16, 16, 17	mp2an 692	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
19	15, 18	elini 4162	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20		ifcl 4534	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	12, 19, 20	sylancl 586	. . . . . . 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 7086	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		eqid 2729	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
25		eqid 2729	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
26	24, 25	ovolsf 25373	. . . . . 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 6696	. . . 4 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29		icossxr 13393	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ^*
30	28, 29	sstrdi 3959	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ^*)
31		supxrcl 13275	. . 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 11606	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
34	33	rexrd 11224	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
35		1nn 12197	. . . . . . 7 ⊢ 1 ∈ ℕ
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37		op1stg 7980	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
38	1, 2, 37	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
39	38	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
40		elicc2 13372	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
41	1, 2, 40	syl2anc 584	. . . . . . . . 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 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
44	39, 43	eqbrtrd 5129	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
45	42	simp3d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
46		op2ndg 7981	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
47	1, 2, 46	syl2anc 584	. . . . . . . 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 5135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩))
50		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = (𝐺‘1))
51		iftrue 4494	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52		opex 5424	. . . . . . . . . . . . 13 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
53	51, 22, 52	fvmpt 6968	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
54	35, 53	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐺‘1) = ⟨𝐴, 𝐵⟩
55	50, 54	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = ⟨𝐴, 𝐵⟩)
56	55	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑛 = 1 → (1^st ‘(𝐺‘𝑛)) = (1^st ‘⟨𝐴, 𝐵⟩))
57	56	breq1d 5117	. . . . . . . 8 ⊢ (𝑛 = 1 → ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ↔ (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
58	55	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑛 = 1 → (2^nd ‘(𝐺‘𝑛)) = (2^nd ‘⟨𝐴, 𝐵⟩))
59	58	breq2d 5119	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑥 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩)))
60	57, 59	anbi12d 632	. . . . . . 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 3588	. . . . . 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 836	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
63	62	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
64		ovolficc 25369	. . . . 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 584	. . . 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 25390	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ([,] ∘ 𝐺)) → (vol‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ))
68	23, 66, 67	syl2anc 584	. 2 ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ^, < ))
69		addrid 11354	. . . . . . . . 9 ⊢ (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
70	69	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71		nnuz 12836	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
72	35, 71	eleqtri 2826	. . . . . . . . 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 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ_≥‘1))
76		rge0ssre 13417	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
77	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78		ffvelcdm 7053	. . . . . . . . . . 11 ⊢ ((seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
79	77, 35, 78	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
80	76, 79	sselid 3944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
81	80	recnd 11202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
82	23	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
83		elfzuz 13481	. . . . . . . . . . . . 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 12249	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
86	85	fveq2i 6861	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
87	84, 86	eleqtrrdi 2839	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ_≥‘2))
88		eluz2nn 12847	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
89	87, 88	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
90	24	ovolfsval 25371	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑘 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))))
91	82, 89, 90	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))))
92		eqeq1 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
93	92	ifbid 4512	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94		opex 5424	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ V
95	52, 94	ifex 4539	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
96	93, 22, 95	fvmpt 6968	. . . . . . . . . . . . . . 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 12881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ_≥‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
99	98	simprbi 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ≠ 1)
100	87, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ≠ 1)
101	100	neneqd 2930	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ¬ 𝑘 = 1)
102	101	iffalsed 4499	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
103	97, 102	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺‘𝑘) = ⟨0, 0⟩)
104	103	fveq2d 6862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘⟨0, 0⟩))
105		c0ex 11168	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
106	105, 105	op2nd 7977	. . . . . . . . . . . 12 ⊢ (2^nd ‘⟨0, 0⟩) = 0
107	104, 106	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = 0)
108	103	fveq2d 6862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘⟨0, 0⟩))
109	105, 105	op1st 7976	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨0, 0⟩) = 0
110	108, 109	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = 0)
111	107, 110	oveq12d 7405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = (0 − 0))
112		0m0e0 12301	. . . . . . . . . 10 ⊢ (0 − 0) = 0
113	111, 112	eqtrdi 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = 0)
114	91, 113	eqtrd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
115	70, 73, 75, 81, 114	seqid2 14013	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116		1z 12563	. . . . . . . 8 ⊢ 1 ∈ ℤ
117	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
118	24	ovolfsval 25371	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 1 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))))
119	117, 35, 118	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))))
120	54	fveq2i 6861	. . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (2^nd ‘(𝐺‘1)) = 𝐵)
123	54	fveq2i 6861	. . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1^st ‘(𝐺‘1)) = 𝐴)
126	122, 125	oveq12d 7405	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))) = (𝐵 − 𝐴))
127	119, 126	eqtrd 2764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵 − 𝐴))
128	116, 127	seq1i 13980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵 − 𝐴))
129	115, 128	eqtr3d 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵 − 𝐴))
130	33	leidd 11744	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
131	130	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
132	129, 131	eqbrtrd 5129	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
133	132	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
134	27	ffnd 6689	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135		breq1 5110	. . . . . 6 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵 − 𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴)))
136	135	ralrn 7060	. . . . 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 13286	. . . 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 584	. . 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 13122	1 ⊢ (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵 − 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 ⊆ wss 3914 ifcif 4488 ⟨cop 4595 ∪ cuni 4871 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 ran crn 5639 ∘ ccom 5642 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 supcsup 9391 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 ℕcn 12186 2c2 12241 ℤ_≥cuz 12793 [,)cico 13308 [,]cicc 13309 ...cfz 13468 seqcseq 13966 abscabs 15200 vol*covol 25363

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-ovol 25365

This theorem is referenced by: ovolicc 25424

W3C validator