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Theorem ovolicc1 24903
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 (𝜑𝐴 ∈ ℝ)
ovolicc.2 (𝜑𝐵 ∈ ℝ)
ovolicc.3 (𝜑𝐴𝐵)
ovolicc1.4 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
Assertion
Ref Expression
ovolicc1 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵𝐴))
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.
StepHypRef Expression
1 ovolicc.1 . . . 4 (𝜑𝐴 ∈ ℝ)
2 ovolicc.2 . . . 4 (𝜑𝐵 ∈ ℝ)
3 iccssre 13355 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
41, 2, 3syl2anc 585 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5 ovolcl 24865 . . 3 ((𝐴[,]𝐵) ⊆ ℝ → (vol*‘(𝐴[,]𝐵)) ∈ ℝ*)
64, 5syl 17 . 2 (𝜑 → (vol*‘(𝐴[,]𝐵)) ∈ ℝ*)
7 ovolicc.3 . . . . . . . . . . 11 (𝜑𝐴𝐵)
8 df-br 5110 . . . . . . . . . . 11 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
97, 8sylib 217 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
101, 2opelxpd 5675 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
119, 10elind 4158 . . . . . . . . 9 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
1211adantr 482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13 0le0 12262 . . . . . . . . . 10 0 ≤ 0
14 df-br 5110 . . . . . . . . . 10 (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
1513, 14mpbi 229 . . . . . . . . 9 ⟨0, 0⟩ ∈ ≤
16 0re 11165 . . . . . . . . . 10 0 ∈ ℝ
17 opelxpi 5674 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
1816, 16, 17mp2an 691 . . . . . . . . 9 ⟨0, 0⟩ ∈ (ℝ × ℝ)
1915, 18elini 4157 . . . . . . . 8 ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20 ifcl 4535 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
2112, 19, 20sylancl 587 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
22 ovolicc1.4 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
2321, 22fmptd 7066 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24 eqid 2733 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
25 eqid 2733 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
2624, 25ovolsf 24859 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2723, 26syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2827frnd 6680 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29 icossxr 13358 . . . 4 (0[,)+∞) ⊆ ℝ*
3028, 29sstrdi 3960 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
31 supxrcl 13243 . . 3 (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
3230, 31syl 17 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
332, 1resubcld 11591 . . 3 (𝜑 → (𝐵𝐴) ∈ ℝ)
3433rexrd 11213 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ*)
35 1nn 12172 . . . . . . 7 1 ∈ ℕ
3635a1i 11 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37 op1stg 7937 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
381, 2, 37syl2anc 585 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
3938adantr 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
40 elicc2 13338 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
411, 2, 40syl2anc 585 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
4241biimpa 478 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
4342simp2d 1144 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐴𝑥)
4439, 43eqbrtrd 5131 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
4542simp3d 1145 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
46 op2ndg 7938 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
471, 2, 46syl2anc 585 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4847adantr 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4945, 48breqtrrd 5137 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))
50 fveq2 6846 . . . . . . . . . . 11 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
51 iftrue 4496 . . . . . . . . . . . . 13 (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52 opex 5425 . . . . . . . . . . . . 13 𝐴, 𝐵⟩ ∈ V
5351, 22, 52fvmpt 6952 . . . . . . . . . . . 12 (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
5435, 53ax-mp 5 . . . . . . . . . . 11 (𝐺‘1) = ⟨𝐴, 𝐵
5550, 54eqtrdi 2789 . . . . . . . . . 10 (𝑛 = 1 → (𝐺𝑛) = ⟨𝐴, 𝐵⟩)
5655fveq2d 6850 . . . . . . . . 9 (𝑛 = 1 → (1st ‘(𝐺𝑛)) = (1st ‘⟨𝐴, 𝐵⟩))
5756breq1d 5119 . . . . . . . 8 (𝑛 = 1 → ((1st ‘(𝐺𝑛)) ≤ 𝑥 ↔ (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
5855fveq2d 6850 . . . . . . . . 9 (𝑛 = 1 → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨𝐴, 𝐵⟩))
5958breq2d 5121 . . . . . . . 8 (𝑛 = 1 → (𝑥 ≤ (2nd ‘(𝐺𝑛)) ↔ 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩)))
6057, 59anbi12d 632 . . . . . . 7 (𝑛 = 1 → (((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))) ↔ ((1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))))
6160rspcev 3583 . . . . . 6 ((1 ∈ ℕ ∧ ((1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
6236, 44, 49, 61syl12anc 836 . . . . 5 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
6362ralrimiva 3140 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
64 ovolficc 24855 . . . . 5 (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
654, 23, 64syl2anc 585 . . . 4 (𝜑 → ((𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
6663, 65mpbird 257 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺))
6725ovollb2 24876 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺)) → (vol*‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
6823, 66, 67syl2anc 585 . 2 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
69 addrid 11343 . . . . . . . . 9 (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
7069adantl 483 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71 nnuz 12814 . . . . . . . . . 10 ℕ = (ℤ‘1)
7235, 71eleqtri 2832 . . . . . . . . 9 1 ∈ (ℤ‘1)
7372a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 1 ∈ (ℤ‘1))
74 simpr 486 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
7574, 71eleqtrdi 2844 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
76 rge0ssre 13382 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ
7727adantr 482 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78 ffvelcdm 7036 . . . . . . . . . . 11 ((seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
7977, 35, 78sylancl 587 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
8076, 79sselid 3946 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
8180recnd 11191 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
8223ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
83 elfzuz 13446 . . . . . . . . . . . . 13 (𝑘 ∈ ((1 + 1)...𝑥) → 𝑘 ∈ (ℤ‘(1 + 1)))
8483adantl 483 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘(1 + 1)))
85 df-2 12224 . . . . . . . . . . . . 13 2 = (1 + 1)
8685fveq2i 6849 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
8784, 86eleqtrrdi 2845 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘2))
88 eluz2nn 12817 . . . . . . . . . . 11 (𝑘 ∈ (ℤ‘2) → 𝑘 ∈ ℕ)
8987, 88syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
9024ovolfsval 24857 . . . . . . . . . 10 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑘 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))))
9182, 89, 90syl2anc 585 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))))
92 eqeq1 2737 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
9392ifbid 4513 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94 opex 5425 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ V
9552, 94ifex 4540 . . . . . . . . . . . . . . . 16 if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
9693, 22, 95fvmpt 6952 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝐺𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
9789, 96syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
98 eluz2b3 12855 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ℤ‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
9998simprbi 498 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ‘2) → 𝑘 ≠ 1)
10087, 99syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ≠ 1)
101100neneqd 2945 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ¬ 𝑘 = 1)
102101iffalsed 4501 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
10397, 102eqtrd 2773 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺𝑘) = ⟨0, 0⟩)
104103fveq2d 6850 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = (2nd ‘⟨0, 0⟩))
105 c0ex 11157 . . . . . . . . . . . . 13 0 ∈ V
106105, 105op2nd 7934 . . . . . . . . . . . 12 (2nd ‘⟨0, 0⟩) = 0
107104, 106eqtrdi 2789 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = 0)
108103fveq2d 6850 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = (1st ‘⟨0, 0⟩))
109105, 105op1st 7933 . . . . . . . . . . . 12 (1st ‘⟨0, 0⟩) = 0
110108, 109eqtrdi 2789 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = 0)
111107, 110oveq12d 7379 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = (0 − 0))
112 0m0e0 12281 . . . . . . . . . 10 (0 − 0) = 0
113111, 112eqtrdi 2789 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = 0)
11491, 113eqtrd 2773 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
11570, 73, 75, 81, 114seqid2 13963 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116 1z 12541 . . . . . . . 8 1 ∈ ℤ
11723adantr 482 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
11824ovolfsval 24857 . . . . . . . . . 10 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 1 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))))
119117, 35, 118sylancl 587 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))))
12054fveq2i 6849 . . . . . . . . . . 11 (2nd ‘(𝐺‘1)) = (2nd ‘⟨𝐴, 𝐵⟩)
12147adantr 482 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
122120, 121eqtrid 2785 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (2nd ‘(𝐺‘1)) = 𝐵)
12354fveq2i 6849 . . . . . . . . . . 11 (1st ‘(𝐺‘1)) = (1st ‘⟨𝐴, 𝐵⟩)
12438adantr 482 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
125123, 124eqtrid 2785 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (1st ‘(𝐺‘1)) = 𝐴)
126122, 125oveq12d 7379 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))) = (𝐵𝐴))
127119, 126eqtrd 2773 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵𝐴))
128116, 127seq1i 13929 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵𝐴))
129115, 128eqtr3d 2775 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵𝐴))
13033leidd 11729 . . . . . . 7 (𝜑 → (𝐵𝐴) ≤ (𝐵𝐴))
131130adantr 482 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (𝐵𝐴) ≤ (𝐵𝐴))
132129, 131eqbrtrd 5131 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
133132ralrimiva 3140 . . . 4 (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
13427ffnd 6673 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135 breq1 5112 . . . . . 6 (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
136135ralrn 7042 . . . . 5 (seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴) ↔ ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
137134, 136syl 17 . . . 4 (𝜑 → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴) ↔ ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
138133, 137mpbird 257 . . 3 (𝜑 → ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴))
139 supxrleub 13254 . . . 4 ((ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* ∧ (𝐵𝐴) ∈ ℝ*) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (𝐵𝐴) ↔ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴)))
14030, 34, 139syl2anc 585 . . 3 (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (𝐵𝐴) ↔ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴)))
141138, 140mpbird 257 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (𝐵𝐴))
1426, 32, 34, 68, 141xrletrd 13090 1 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  cin 3913  wss 3914  ifcif 4490  cop 4596   cuni 4869   class class class wbr 5109  cmpt 5192   × cxp 5635  ran crn 5638  ccom 5641   Fn wfn 6495  wf 6496  cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  supcsup 9384  cc 11057  cr 11058  0cc0 11059  1c1 11060   + caddc 11062  +∞cpnf 11194  *cxr 11196   < clt 11197  cle 11198  cmin 11393  cn 12161  2c2 12216  cuz 12771  [,)cico 13275  [,]cicc 13276  ...cfz 13433  seqcseq 13915  abscabs 15128  vol*covol 24849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  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 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13434  df-fzo 13577  df-seq 13916  df-exp 13977  df-hash 14240  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-clim 15379  df-sum 15580  df-ovol 24851
This theorem is referenced by:  ovolicc  24910
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