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Theorem ovolicc1 25040
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 13408 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
41, 2, 3syl2anc 584 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5 ovolcl 25002 . . 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 5149 . . . . . . . . . . 11 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
97, 8sylib 217 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
101, 2opelxpd 5715 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
119, 10elind 4194 . . . . . . . . 9 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
1211adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
13 0le0 12315 . . . . . . . . . 10 0 ≤ 0
14 df-br 5149 . . . . . . . . . 10 (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
1513, 14mpbi 229 . . . . . . . . 9 ⟨0, 0⟩ ∈ ≤
16 0re 11218 . . . . . . . . . 10 0 ∈ ℝ
17 opelxpi 5713 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
1816, 16, 17mp2an 690 . . . . . . . . 9 ⟨0, 0⟩ ∈ (ℝ × ℝ)
1915, 18elini 4193 . . . . . . . 8 ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
20 ifcl 4573 . . . . . . . 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 7115 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24 eqid 2732 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
25 eqid 2732 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
2624, 25ovolsf 24996 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2723, 26syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2827frnd 6725 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
29 icossxr 13411 . . . 4 (0[,)+∞) ⊆ ℝ*
3028, 29sstrdi 3994 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
31 supxrcl 13296 . . 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 11644 . . 3 (𝜑 → (𝐵𝐴) ∈ ℝ)
3433rexrd 11266 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ*)
35 1nn 12225 . . . . . . 7 1 ∈ ℕ
3635a1i 11 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
37 op1stg 7989 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
381, 2, 37syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
3938adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
40 elicc2 13391 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
411, 2, 40syl2anc 584 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
4241biimpa 477 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
4342simp2d 1143 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐴𝑥)
4439, 43eqbrtrd 5170 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
4542simp3d 1144 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
46 op2ndg 7990 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
471, 2, 46syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4847adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4945, 48breqtrrd 5176 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))
50 fveq2 6891 . . . . . . . . . . 11 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
51 iftrue 4534 . . . . . . . . . . . . 13 (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
52 opex 5464 . . . . . . . . . . . . 13 𝐴, 𝐵⟩ ∈ V
5351, 22, 52fvmpt 6998 . . . . . . . . . . . 12 (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
5435, 53ax-mp 5 . . . . . . . . . . 11 (𝐺‘1) = ⟨𝐴, 𝐵
5550, 54eqtrdi 2788 . . . . . . . . . 10 (𝑛 = 1 → (𝐺𝑛) = ⟨𝐴, 𝐵⟩)
5655fveq2d 6895 . . . . . . . . 9 (𝑛 = 1 → (1st ‘(𝐺𝑛)) = (1st ‘⟨𝐴, 𝐵⟩))
5756breq1d 5158 . . . . . . . 8 (𝑛 = 1 → ((1st ‘(𝐺𝑛)) ≤ 𝑥 ↔ (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
5855fveq2d 6895 . . . . . . . . 9 (𝑛 = 1 → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨𝐴, 𝐵⟩))
5958breq2d 5160 . . . . . . . 8 (𝑛 = 1 → (𝑥 ≤ (2nd ‘(𝐺𝑛)) ↔ 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩)))
6057, 59anbi12d 631 . . . . . . 7 (𝑛 = 1 → (((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))) ↔ ((1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))))
6160rspcev 3612 . . . . . 6 ((1 ∈ ℕ ∧ ((1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
6236, 44, 49, 61syl12anc 835 . . . . 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 24992 . . . . 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 256 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺))
6725ovollb2 25013 . . 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 11396 . . . . . . . . 9 (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
7069adantl 482 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
71 nnuz 12867 . . . . . . . . . 10 ℕ = (ℤ‘1)
7235, 71eleqtri 2831 . . . . . . . . 9 1 ∈ (ℤ‘1)
7372a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 1 ∈ (ℤ‘1))
74 simpr 485 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
7574, 71eleqtrdi 2843 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
76 rge0ssre 13435 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ
7727adantr 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
78 ffvelcdm 7083 . . . . . . . . . . 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 3980 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
8180recnd 11244 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
8223ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
83 elfzuz 13499 . . . . . . . . . . . . 13 (𝑘 ∈ ((1 + 1)...𝑥) → 𝑘 ∈ (ℤ‘(1 + 1)))
8483adantl 482 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘(1 + 1)))
85 df-2 12277 . . . . . . . . . . . . 13 2 = (1 + 1)
8685fveq2i 6894 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
8784, 86eleqtrrdi 2844 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘2))
88 eluz2nn 12870 . . . . . . . . . . 11 (𝑘 ∈ (ℤ‘2) → 𝑘 ∈ ℕ)
8987, 88syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
9024ovolfsval 24994 . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
9392ifbid 4551 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
94 opex 5464 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ V
9552, 94ifex 4578 . . . . . . . . . . . . . . . 16 if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
9693, 22, 95fvmpt 6998 . . . . . . . . . . . . . . 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 12908 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ℤ‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
9998simprbi 497 . . . . . . . . . . . . . . . . 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 4539 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
10397, 102eqtrd 2772 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺𝑘) = ⟨0, 0⟩)
104103fveq2d 6895 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = (2nd ‘⟨0, 0⟩))
105 c0ex 11210 . . . . . . . . . . . . 13 0 ∈ V
106105, 105op2nd 7986 . . . . . . . . . . . 12 (2nd ‘⟨0, 0⟩) = 0
107104, 106eqtrdi 2788 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = 0)
108103fveq2d 6895 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = (1st ‘⟨0, 0⟩))
109105, 105op1st 7985 . . . . . . . . . . . 12 (1st ‘⟨0, 0⟩) = 0
110108, 109eqtrdi 2788 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = 0)
111107, 110oveq12d 7429 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = (0 − 0))
112 0m0e0 12334 . . . . . . . . . 10 (0 − 0) = 0
113111, 112eqtrdi 2788 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = 0)
11491, 113eqtrd 2772 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
11570, 73, 75, 81, 114seqid2 14016 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
116 1z 12594 . . . . . . . 8 1 ∈ ℤ
11723adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
11824ovolfsval 24994 . . . . . . . . . 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 6894 . . . . . . . . . . 11 (2nd ‘(𝐺‘1)) = (2nd ‘⟨𝐴, 𝐵⟩)
12147adantr 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
122120, 121eqtrid 2784 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (2nd ‘(𝐺‘1)) = 𝐵)
12354fveq2i 6894 . . . . . . . . . . 11 (1st ‘(𝐺‘1)) = (1st ‘⟨𝐴, 𝐵⟩)
12438adantr 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
125123, 124eqtrid 2784 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (1st ‘(𝐺‘1)) = 𝐴)
126122, 125oveq12d 7429 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))) = (𝐵𝐴))
127119, 126eqtrd 2772 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵𝐴))
128116, 127seq1i 13982 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵𝐴))
129115, 128eqtr3d 2774 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵𝐴))
13033leidd 11782 . . . . . . 7 (𝜑 → (𝐵𝐴) ≤ (𝐵𝐴))
131130adantr 481 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (𝐵𝐴) ≤ (𝐵𝐴))
132129, 131eqbrtrd 5170 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
133132ralrimiva 3146 . . . 4 (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
13427ffnd 6718 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
135 breq1 5151 . . . . . 6 (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
136135ralrn 7089 . . . . 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 256 . . 3 (𝜑 → ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴))
139 supxrleub 13307 . . . 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 256 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (𝐵𝐴))
1426, 32, 34, 68, 141xrletrd 13143 1 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  cin 3947  wss 3948  ifcif 4528  cop 4634   cuni 4908   class class class wbr 5148  cmpt 5231   × cxp 5674  ran crn 5677  ccom 5680   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  supcsup 9437  cc 11110  cr 11111  0cc0 11112  1c1 11113   + caddc 11115  +∞cpnf 11247  *cxr 11249   < clt 11250  cle 11251  cmin 11446  cn 12214  2c2 12269  cuz 12824  [,)cico 13328  [,]cicc 13329  ...cfz 13486  seqcseq 13968  abscabs 15183  vol*covol 24986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-ioo 13330  df-ico 13332  df-icc 13333  df-fz 13487  df-fzo 13630  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-sum 15635  df-ovol 24988
This theorem is referenced by:  ovolicc  25047
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