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Theorem ovolicc1 25033
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 13406 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
41, 2, 3syl2anc 585 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5 ovolcl 24995 . . 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 5150 . . . . . . . . . . 11 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
97, 8sylib 217 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
101, 2opelxpd 5716 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
119, 10elind 4195 . . . . . . . . 9 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
1211adantr 482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13 0le0 12313 . . . . . . . . . 10 0 ≤ 0
14 df-br 5150 . . . . . . . . . 10 (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
1513, 14mpbi 229 . . . . . . . . 9 ⟨0, 0⟩ ∈ ≤
16 0re 11216 . . . . . . . . . 10 0 ∈ ℝ
17 opelxpi 5714 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
1816, 16, 17mp2an 691 . . . . . . . . 9 ⟨0, 0⟩ ∈ (ℝ × ℝ)
1915, 18elini 4194 . . . . . . . 8 ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20 ifcl 4574 . . . . . . . 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 7114 . . . . . 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 24989 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2723, 26syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2827frnd 6726 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29 icossxr 13409 . . . 4 (0[,)+∞) ⊆ ℝ*
3028, 29sstrdi 3995 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
31 supxrcl 13294 . . 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 11642 . . 3 (𝜑 → (𝐵𝐴) ∈ ℝ)
3433rexrd 11264 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ*)
35 1nn 12223 . . . . . . 7 1 ∈ ℕ
3635a1i 11 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37 op1stg 7987 . . . . . . . . 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 13389 . . . . . . . . . 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 5171 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
4542simp3d 1145 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
46 op2ndg 7988 . . . . . . . . 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 5177 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))
50 fveq2 6892 . . . . . . . . . . 11 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
51 iftrue 4535 . . . . . . . . . . . . 13 (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52 opex 5465 . . . . . . . . . . . . 13 𝐴, 𝐵⟩ ∈ V
5351, 22, 52fvmpt 6999 . . . . . . . . . . . 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 6896 . . . . . . . . 9 (𝑛 = 1 → (1st ‘(𝐺𝑛)) = (1st ‘⟨𝐴, 𝐵⟩))
5756breq1d 5159 . . . . . . . 8 (𝑛 = 1 → ((1st ‘(𝐺𝑛)) ≤ 𝑥 ↔ (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
5855fveq2d 6896 . . . . . . . . 9 (𝑛 = 1 → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨𝐴, 𝐵⟩))
5958breq2d 5161 . . . . . . . 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 3613 . . . . . 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 3147 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
64 ovolficc 24985 . . . . 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 25006 . . 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 11394 . . . . . . . . 9 (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
7069adantl 483 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71 nnuz 12865 . . . . . . . . . 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 13433 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ
7727adantr 482 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78 ffvelcdm 7084 . . . . . . . . . . 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 3981 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
8180recnd 11242 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
8223ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
83 elfzuz 13497 . . . . . . . . . . . . 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 12275 . . . . . . . . . . . . 13 2 = (1 + 1)
8685fveq2i 6895 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
8784, 86eleqtrrdi 2845 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘2))
88 eluz2nn 12868 . . . . . . . . . . 11 (𝑘 ∈ (ℤ‘2) → 𝑘 ∈ ℕ)
8987, 88syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
9024ovolfsval 24987 . . . . . . . . . 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 4552 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94 opex 5465 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ V
9552, 94ifex 4579 . . . . . . . . . . . . . . . 16 if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
9693, 22, 95fvmpt 6999 . . . . . . . . . . . . . . 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 12906 . . . . . . . . . . . . . . . . . 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 2946 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ¬ 𝑘 = 1)
102101iffalsed 4540 . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = (2nd ‘⟨0, 0⟩))
105 c0ex 11208 . . . . . . . . . . . . 13 0 ∈ V
106105, 105op2nd 7984 . . . . . . . . . . . 12 (2nd ‘⟨0, 0⟩) = 0
107104, 106eqtrdi 2789 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = 0)
108103fveq2d 6896 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = (1st ‘⟨0, 0⟩))
109105, 105op1st 7983 . . . . . . . . . . . 12 (1st ‘⟨0, 0⟩) = 0
110108, 109eqtrdi 2789 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = 0)
111107, 110oveq12d 7427 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = (0 − 0))
112 0m0e0 12332 . . . . . . . . . 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 14014 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116 1z 12592 . . . . . . . 8 1 ∈ ℤ
11723adantr 482 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
11824ovolfsval 24987 . . . . . . . . . 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 6895 . . . . . . . . . . 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 6895 . . . . . . . . . . 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 7427 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))) = (𝐵𝐴))
127119, 126eqtrd 2773 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵𝐴))
128116, 127seq1i 13980 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵𝐴))
129115, 128eqtr3d 2775 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵𝐴))
13033leidd 11780 . . . . . . 7 (𝜑 → (𝐵𝐴) ≤ (𝐵𝐴))
131130adantr 482 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (𝐵𝐴) ≤ (𝐵𝐴))
132129, 131eqbrtrd 5171 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
133132ralrimiva 3147 . . . 4 (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
13427ffnd 6719 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135 breq1 5152 . . . . . 6 (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
136135ralrn 7090 . . . . 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 13305 . . . 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 13141 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 2941  wral 3062  wrex 3071  cin 3948  wss 3949  ifcif 4529  cop 4635   cuni 4909   class class class wbr 5149  cmpt 5232   × cxp 5675  ran crn 5678  ccom 5681   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  supcsup 9435  cc 11108  cr 11109  0cc0 11110  1c1 11111   + caddc 11113  +∞cpnf 11245  *cxr 11247   < clt 11248  cle 11249  cmin 11444  cn 12212  2c2 12267  cuz 12822  [,)cico 13326  [,]cicc 13327  ...cfz 13484  seqcseq 13966  abscabs 15181  vol*covol 24979
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-ovol 24981
This theorem is referenced by:  ovolicc  25040
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