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Theorem ovolicc1 25551
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 13469 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
41, 2, 3syl2anc 584 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5 ovolcl 25513 . . 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 5144 . . . . . . . . . . 11 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
97, 8sylib 218 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
101, 2opelxpd 5724 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
119, 10elind 4200 . . . . . . . . 9 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
1211adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13 0le0 12367 . . . . . . . . . 10 0 ≤ 0
14 df-br 5144 . . . . . . . . . 10 (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
1513, 14mpbi 230 . . . . . . . . 9 ⟨0, 0⟩ ∈ ≤
16 0re 11263 . . . . . . . . . 10 0 ∈ ℝ
17 opelxpi 5722 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
1816, 16, 17mp2an 692 . . . . . . . . 9 ⟨0, 0⟩ ∈ (ℝ × ℝ)
1915, 18elini 4199 . . . . . . . 8 ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20 ifcl 4571 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
2112, 19, 20sylancl 586 . . . . . . 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 7134 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24 eqid 2737 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
25 eqid 2737 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
2624, 25ovolsf 25507 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2723, 26syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2827frnd 6744 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29 icossxr 13472 . . . 4 (0[,)+∞) ⊆ ℝ*
3028, 29sstrdi 3996 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
31 supxrcl 13357 . . 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 11691 . . 3 (𝜑 → (𝐵𝐴) ∈ ℝ)
3433rexrd 11311 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ*)
35 1nn 12277 . . . . . . 7 1 ∈ ℕ
3635a1i 11 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37 op1stg 8026 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
381, 2, 37syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
3938adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
40 elicc2 13452 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
411, 2, 40syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
4241biimpa 476 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
4342simp2d 1144 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐴𝑥)
4439, 43eqbrtrd 5165 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
4542simp3d 1145 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
46 op2ndg 8027 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
471, 2, 46syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4847adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4945, 48breqtrrd 5171 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))
50 fveq2 6906 . . . . . . . . . . 11 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
51 iftrue 4531 . . . . . . . . . . . . 13 (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52 opex 5469 . . . . . . . . . . . . 13 𝐴, 𝐵⟩ ∈ V
5351, 22, 52fvmpt 7016 . . . . . . . . . . . 12 (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
5435, 53ax-mp 5 . . . . . . . . . . 11 (𝐺‘1) = ⟨𝐴, 𝐵
5550, 54eqtrdi 2793 . . . . . . . . . 10 (𝑛 = 1 → (𝐺𝑛) = ⟨𝐴, 𝐵⟩)
5655fveq2d 6910 . . . . . . . . 9 (𝑛 = 1 → (1st ‘(𝐺𝑛)) = (1st ‘⟨𝐴, 𝐵⟩))
5756breq1d 5153 . . . . . . . 8 (𝑛 = 1 → ((1st ‘(𝐺𝑛)) ≤ 𝑥 ↔ (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
5855fveq2d 6910 . . . . . . . . 9 (𝑛 = 1 → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨𝐴, 𝐵⟩))
5958breq2d 5155 . . . . . . . 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 3622 . . . . . 6 ((1 ∈ ℕ ∧ ((1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
6236, 44, 49, 61syl12anc 837 . . . . 5 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
6362ralrimiva 3146 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
64 ovolficc 25503 . . . . 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 584 . . . 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 25524 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺)) → (vol*‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
6823, 66, 67syl2anc 584 . 2 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
69 addrid 11441 . . . . . . . . 9 (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
7069adantl 481 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71 nnuz 12921 . . . . . . . . . 10 ℕ = (ℤ‘1)
7235, 71eleqtri 2839 . . . . . . . . 9 1 ∈ (ℤ‘1)
7372a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 1 ∈ (ℤ‘1))
74 simpr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
7574, 71eleqtrdi 2851 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
76 rge0ssre 13496 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ
7727adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78 ffvelcdm 7101 . . . . . . . . . . 11 ((seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
7977, 35, 78sylancl 586 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
8076, 79sselid 3981 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
8180recnd 11289 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
8223ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
83 elfzuz 13560 . . . . . . . . . . . . 13 (𝑘 ∈ ((1 + 1)...𝑥) → 𝑘 ∈ (ℤ‘(1 + 1)))
8483adantl 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘(1 + 1)))
85 df-2 12329 . . . . . . . . . . . . 13 2 = (1 + 1)
8685fveq2i 6909 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
8784, 86eleqtrrdi 2852 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘2))
88 eluz2nn 12924 . . . . . . . . . . 11 (𝑘 ∈ (ℤ‘2) → 𝑘 ∈ ℕ)
8987, 88syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
9024ovolfsval 25505 . . . . . . . . . 10 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑘 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))))
9182, 89, 90syl2anc 584 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))))
92 eqeq1 2741 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
9392ifbid 4549 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94 opex 5469 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ V
9552, 94ifex 4576 . . . . . . . . . . . . . . . 16 if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
9693, 22, 95fvmpt 7016 . . . . . . . . . . . . . . 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 12964 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ℤ‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
9998simprbi 496 . . . . . . . . . . . . . . . . 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 4536 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
10397, 102eqtrd 2777 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺𝑘) = ⟨0, 0⟩)
104103fveq2d 6910 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = (2nd ‘⟨0, 0⟩))
105 c0ex 11255 . . . . . . . . . . . . 13 0 ∈ V
106105, 105op2nd 8023 . . . . . . . . . . . 12 (2nd ‘⟨0, 0⟩) = 0
107104, 106eqtrdi 2793 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = 0)
108103fveq2d 6910 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = (1st ‘⟨0, 0⟩))
109105, 105op1st 8022 . . . . . . . . . . . 12 (1st ‘⟨0, 0⟩) = 0
110108, 109eqtrdi 2793 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = 0)
111107, 110oveq12d 7449 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = (0 − 0))
112 0m0e0 12386 . . . . . . . . . 10 (0 − 0) = 0
113111, 112eqtrdi 2793 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = 0)
11491, 113eqtrd 2777 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
11570, 73, 75, 81, 114seqid2 14089 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116 1z 12647 . . . . . . . 8 1 ∈ ℤ
11723adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
11824ovolfsval 25505 . . . . . . . . . 10 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 1 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))))
119117, 35, 118sylancl 586 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))))
12054fveq2i 6909 . . . . . . . . . . 11 (2nd ‘(𝐺‘1)) = (2nd ‘⟨𝐴, 𝐵⟩)
12147adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
122120, 121eqtrid 2789 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (2nd ‘(𝐺‘1)) = 𝐵)
12354fveq2i 6909 . . . . . . . . . . 11 (1st ‘(𝐺‘1)) = (1st ‘⟨𝐴, 𝐵⟩)
12438adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
125123, 124eqtrid 2789 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (1st ‘(𝐺‘1)) = 𝐴)
126122, 125oveq12d 7449 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))) = (𝐵𝐴))
127119, 126eqtrd 2777 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵𝐴))
128116, 127seq1i 14056 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵𝐴))
129115, 128eqtr3d 2779 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵𝐴))
13033leidd 11829 . . . . . . 7 (𝜑 → (𝐵𝐴) ≤ (𝐵𝐴))
131130adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (𝐵𝐴) ≤ (𝐵𝐴))
132129, 131eqbrtrd 5165 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
133132ralrimiva 3146 . . . 4 (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
13427ffnd 6737 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135 breq1 5146 . . . . . 6 (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
136135ralrn 7108 . . . . 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 13368 . . . 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 584 . . 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 13204 1 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  ifcif 4525  cop 4632   cuni 4907   class class class wbr 5143  cmpt 5225   × cxp 5683  ran crn 5686  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cmin 11492  cn 12266  2c2 12321  cuz 12878  [,)cico 13389  [,]cicc 13390  ...cfz 13547  seqcseq 14042  abscabs 15273  vol*covol 25497
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-ovol 25499
This theorem is referenced by:  ovolicc  25558
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