ovolicc1 - Metamath Proof Explorer

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Theorem ovolicc1 25467

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 13444	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5		ovolcl 25429	. . 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 5120	. . . . . . . . . . 11 ⊢ (𝐴 ≤ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
9	7, 8	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
10	1, 2	opelxpd 5693	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
11	9, 10	elind 4175	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13		0le0 12339	. . . . . . . . . 10 ⊢ 0 ≤ 0
14		df-br 5120	. . . . . . . . . 10 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
15	13, 14	mpbi 230	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ ≤
16		0re 11235	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
17		opelxpi 5691	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
18	16, 16, 17	mp2an 692	. . . . . . . . 9 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
19	15, 18	elini 4174	. . . . . . . 8 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20		ifcl 4546	. . . . . . . 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 7103	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24		eqid 2735	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
25		eqid 2735	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
26	24, 25	ovolsf 25423	. . . . . 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 6713	. . . 4 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29		icossxr 13447	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ^*
30	28, 29	sstrdi 3971	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ^*)
31		supxrcl 13329	. . 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 11663	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
34	33	rexrd 11283	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
35		1nn 12249	. . . . . . 7 ⊢ 1 ∈ ℕ
36	35	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37		op1stg 7998	. . . . . . . . 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 13426	. . . . . . . . . 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 5141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
45	42	simp3d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
46		op2ndg 7999	. . . . . . . . 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 5147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2^nd ‘⟨𝐴, 𝐵⟩))
50		fveq2 6875	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = (𝐺‘1))
51		iftrue 4506	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52		opex 5439	. . . . . . . . . . . . 13 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
53	51, 22, 52	fvmpt 6985	. . . . . . . . . . . 12 ⊢ (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
54	35, 53	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐺‘1) = ⟨𝐴, 𝐵⟩
55	50, 54	eqtrdi 2786	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = ⟨𝐴, 𝐵⟩)
56	55	fveq2d 6879	. . . . . . . . 9 ⊢ (𝑛 = 1 → (1^st ‘(𝐺‘𝑛)) = (1^st ‘⟨𝐴, 𝐵⟩))
57	56	breq1d 5129	. . . . . . . 8 ⊢ (𝑛 = 1 → ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ↔ (1^st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
58	55	fveq2d 6879	. . . . . . . . 9 ⊢ (𝑛 = 1 → (2^nd ‘(𝐺‘𝑛)) = (2^nd ‘⟨𝐴, 𝐵⟩))
59	58	breq2d 5131	. . . . . . . 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 3601	. . . . . 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 3132	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1^st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2^nd ‘(𝐺‘𝑛))))
64		ovolficc 25419	. . . . 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 25440	. . 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 11413	. . . . . . . . 9 ⊢ (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
70	69	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71		nnuz 12893	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
72	35, 71	eleqtri 2832	. . . . . . . . 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 2844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ_≥‘1))
76		rge0ssre 13471	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
77	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78		ffvelcdm 7070	. . . . . . . . . . 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 3956	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
81	80	recnd 11261	. . . . . . . 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 13535	. . . . . . . . . . . . 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 12301	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
86	85	fveq2i 6878	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
87	84, 86	eleqtrrdi 2845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ_≥‘2))
88		eluz2nn 12896	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
89	87, 88	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
90	24	ovolfsval 25421	. . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
93	92	ifbid 4524	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94		opex 5439	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ V
95	52, 94	ifex 4551	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
96	93, 22, 95	fvmpt 6985	. . . . . . . . . . . . . . 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 12936	. . . . . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ¬ 𝑘 = 1)
102	101	iffalsed 4511	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
103	97, 102	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺‘𝑘) = ⟨0, 0⟩)
104	103	fveq2d 6879	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = (2^nd ‘⟨0, 0⟩))
105		c0ex 11227	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
106	105, 105	op2nd 7995	. . . . . . . . . . . 12 ⊢ (2^nd ‘⟨0, 0⟩) = 0
107	104, 106	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2^nd ‘(𝐺‘𝑘)) = 0)
108	103	fveq2d 6879	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = (1^st ‘⟨0, 0⟩))
109	105, 105	op1st 7994	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨0, 0⟩) = 0
110	108, 109	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1^st ‘(𝐺‘𝑘)) = 0)
111	107, 110	oveq12d 7421	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = (0 − 0))
112		0m0e0 12358	. . . . . . . . . 10 ⊢ (0 − 0) = 0
113	111, 112	eqtrdi 2786	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2^nd ‘(𝐺‘𝑘)) − (1^st ‘(𝐺‘𝑘))) = 0)
114	91, 113	eqtrd 2770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
115	70, 73, 75, 81, 114	seqid2 14064	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116		1z 12620	. . . . . . . 8 ⊢ 1 ∈ ℤ
117	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
118	24	ovolfsval 25421	. . . . . . . . . 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 6878	. . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (2^nd ‘(𝐺‘1)) = 𝐵)
123	54	fveq2i 6878	. . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (1^st ‘(𝐺‘1)) = 𝐴)
126	122, 125	oveq12d 7421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2^nd ‘(𝐺‘1)) − (1^st ‘(𝐺‘1))) = (𝐵 − 𝐴))
127	119, 126	eqtrd 2770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵 − 𝐴))
128	116, 127	seq1i 14031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵 − 𝐴))
129	115, 128	eqtr3d 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵 − 𝐴))
130	33	leidd 11801	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
131	130	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐵 − 𝐴) ≤ (𝐵 − 𝐴))
132	129, 131	eqbrtrd 5141	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
133	132	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴))
134	27	ffnd 6706	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135		breq1 5122	. . . . . 6 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵 − 𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵 − 𝐴)))
136	135	ralrn 7077	. . . . 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 13340	. . . 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 13176	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 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ∩ cin 3925 ⊆ wss 3926 ifcif 4500 ⟨cop 4607 ∪ cuni 4883 class class class wbr 5119 ↦ cmpt 5201 × cxp 5652 ran crn 5655 ∘ ccom 5658 Fn wfn 6525 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 1^st c1st 7984 2^nd c2nd 7985 supcsup 9450 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11464 ℕcn 12238 2c2 12293 ℤ_≥cuz 12850 [,)cico 13362 [,]cicc 13363 ...cfz 13522 seqcseq 14017 abscabs 15251 vol*covol 25413

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8717 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-n0 12500 df-z 12587 df-uz 12851 df-q 12963 df-rp 13007 df-ioo 13364 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-clim 15502 df-sum 15701 df-ovol 25415

This theorem is referenced by: ovolicc 25474

W3C validator