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Theorem ovolicc1 23330
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 12293 . . . 4 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
41, 2, 3syl2anc 694 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
5 ovolcl 23292 . . 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 4686 . . . . . . . . . . 11 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≤ )
97, 8sylib 208 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ≤ )
10 opelxpi 5182 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
111, 2, 10syl2anc 694 . . . . . . . . . 10 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (ℝ × ℝ))
129, 11elind 3831 . . . . . . . . 9 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
1312adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
14 0le0 11148 . . . . . . . . . 10 0 ≤ 0
15 df-br 4686 . . . . . . . . . 10 (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
1614, 15mpbi 220 . . . . . . . . 9 ⟨0, 0⟩ ∈ ≤
17 0re 10078 . . . . . . . . . 10 0 ∈ ℝ
18 opelxpi 5182 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
1917, 17, 18mp2an 708 . . . . . . . . 9 ⟨0, 0⟩ ∈ (ℝ × ℝ)
20 elin 3829 . . . . . . . . 9 (⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨0, 0⟩ ∈ ≤ ∧ ⟨0, 0⟩ ∈ (ℝ × ℝ)))
2116, 19, 20mpbir2an 975 . . . . . . . 8 ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
22 ifcl 4163 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
2313, 21, 22sylancl 695 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
24 ovolicc1.4 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
2523, 24fmptd 6425 . . . . . 6 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
26 eqid 2651 . . . . . . 7 ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
27 eqid 2651 . . . . . . 7 seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
2826, 27ovolsf 23287 . . . . . 6 (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
2925, 28syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
30 frn 6091 . . . . 5 (seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
3129, 30syl 17 . . . 4 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
32 icossxr 12296 . . . 4 (0[,)+∞) ⊆ ℝ*
3331, 32syl6ss 3648 . . 3 (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
34 supxrcl 12183 . . 3 (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
3533, 34syl 17 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
362, 1resubcld 10496 . . 3 (𝜑 → (𝐵𝐴) ∈ ℝ)
3736rexrd 10127 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ*)
38 1nn 11069 . . . . . . 7 1 ∈ ℕ
3938a1i 11 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 1 ∈ ℕ)
40 op1stg 7222 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
411, 2, 40syl2anc 694 . . . . . . . 8 (𝜑 → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
4241adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
43 elicc2 12276 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
441, 2, 43syl2anc 694 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
4544biimpa 500 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
4645simp2d 1094 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐴𝑥)
4742, 46eqbrtrd 4707 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥)
4845simp3d 1095 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
49 op2ndg 7223 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
501, 2, 49syl2anc 694 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
5150adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
5248, 51breqtrrd 4713 . . . . . 6 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))
53 fveq2 6229 . . . . . . . . . . 11 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
54 iftrue 4125 . . . . . . . . . . . . 13 (𝑛 = 1 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨𝐴, 𝐵⟩)
55 opex 4962 . . . . . . . . . . . . 13 𝐴, 𝐵⟩ ∈ V
5654, 24, 55fvmpt 6321 . . . . . . . . . . . 12 (1 ∈ ℕ → (𝐺‘1) = ⟨𝐴, 𝐵⟩)
5738, 56ax-mp 5 . . . . . . . . . . 11 (𝐺‘1) = ⟨𝐴, 𝐵
5853, 57syl6eq 2701 . . . . . . . . . 10 (𝑛 = 1 → (𝐺𝑛) = ⟨𝐴, 𝐵⟩)
5958fveq2d 6233 . . . . . . . . 9 (𝑛 = 1 → (1st ‘(𝐺𝑛)) = (1st ‘⟨𝐴, 𝐵⟩))
6059breq1d 4695 . . . . . . . 8 (𝑛 = 1 → ((1st ‘(𝐺𝑛)) ≤ 𝑥 ↔ (1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥))
6158fveq2d 6233 . . . . . . . . 9 (𝑛 = 1 → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨𝐴, 𝐵⟩))
6261breq2d 4697 . . . . . . . 8 (𝑛 = 1 → (𝑥 ≤ (2nd ‘(𝐺𝑛)) ↔ 𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩)))
6360, 62anbi12d 747 . . . . . . 7 (𝑛 = 1 → (((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))) ↔ ((1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))))
6463rspcev 3340 . . . . . 6 ((1 ∈ ℕ ∧ ((1st ‘⟨𝐴, 𝐵⟩) ≤ 𝑥𝑥 ≤ (2nd ‘⟨𝐴, 𝐵⟩))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
6539, 47, 52, 64syl12anc 1364 . . . . 5 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
6665ralrimiva 2995 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛))))
67 ovolficc 23283 . . . . 5 (((𝐴[,]𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → ((𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
684, 25, 67syl2anc 694 . . . 4 (𝜑 → ((𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∃𝑛 ∈ ℕ ((1st ‘(𝐺𝑛)) ≤ 𝑥𝑥 ≤ (2nd ‘(𝐺𝑛)))))
6966, 68mpbird 247 . . 3 (𝜑 → (𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺))
7027ovollb2 23303 . . 3 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴[,]𝐵) ⊆ ran ([,] ∘ 𝐺)) → (vol*‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
7125, 69, 70syl2anc 694 . 2 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
72 addid1 10254 . . . . . . . . 9 (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘)
7372adantl 481 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ℂ) → (𝑘 + 0) = 𝑘)
74 nnuz 11761 . . . . . . . . . 10 ℕ = (ℤ‘1)
7538, 74eleqtri 2728 . . . . . . . . 9 1 ∈ (ℤ‘1)
7675a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 1 ∈ (ℤ‘1))
77 simpr 476 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
7877, 74syl6eleq 2740 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
79 rge0ssre 12318 . . . . . . . . . 10 (0[,)+∞) ⊆ ℝ
8029adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
81 ffvelrn 6397 . . . . . . . . . . 11 ((seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) ∧ 1 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
8280, 38, 81sylancl 695 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ (0[,)+∞))
8379, 82sseldi 3634 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℝ)
8483recnd 10106 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) ∈ ℂ)
8525ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
86 elfzuz 12376 . . . . . . . . . . . . 13 (𝑘 ∈ ((1 + 1)...𝑥) → 𝑘 ∈ (ℤ‘(1 + 1)))
8786adantl 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘(1 + 1)))
88 df-2 11117 . . . . . . . . . . . . 13 2 = (1 + 1)
8988fveq2i 6232 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
9087, 89syl6eleqr 2741 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ (ℤ‘2))
91 eluz2nn 11764 . . . . . . . . . . 11 (𝑘 ∈ (ℤ‘2) → 𝑘 ∈ ℕ)
9290, 91syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ∈ ℕ)
9326ovolfsval 23285 . . . . . . . . . 10 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑘 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))))
9485, 92, 93syl2anc 694 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))))
95 eqeq1 2655 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑛 = 1 ↔ 𝑘 = 1))
9695ifbid 4141 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → if(𝑛 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
97 opex 4962 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ V
9855, 97ifex 4189 . . . . . . . . . . . . . . . 16 if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) ∈ V
9996, 24, 98fvmpt 6321 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝐺𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
10092, 99syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺𝑘) = if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩))
101 eluz2b3 11800 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ℤ‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
102101simprbi 479 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ‘2) → 𝑘 ≠ 1)
10390, 102syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → 𝑘 ≠ 1)
104103neneqd 2828 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ¬ 𝑘 = 1)
105104iffalsed 4130 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → if(𝑘 = 1, ⟨𝐴, 𝐵⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
106100, 105eqtrd 2685 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (𝐺𝑘) = ⟨0, 0⟩)
107106fveq2d 6233 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = (2nd ‘⟨0, 0⟩))
108 c0ex 10072 . . . . . . . . . . . . 13 0 ∈ V
109108, 108op2nd 7219 . . . . . . . . . . . 12 (2nd ‘⟨0, 0⟩) = 0
110107, 109syl6eq 2701 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (2nd ‘(𝐺𝑘)) = 0)
111106fveq2d 6233 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = (1st ‘⟨0, 0⟩))
112108, 108op1st 7218 . . . . . . . . . . . 12 (1st ‘⟨0, 0⟩) = 0
113111, 112syl6eq 2701 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (1st ‘(𝐺𝑘)) = 0)
114110, 113oveq12d 6708 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = (0 − 0))
115 0m0e0 11168 . . . . . . . . . 10 (0 − 0) = 0
116114, 115syl6eq 2701 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → ((2nd ‘(𝐺𝑘)) − (1st ‘(𝐺𝑘))) = 0)
11794, 116eqtrd 2685 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ) ∧ 𝑘 ∈ ((1 + 1)...𝑥)) → (((abs ∘ − ) ∘ 𝐺)‘𝑘) = 0)
11873, 76, 78, 84, 117seqid2 12887 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥))
119 1z 11445 . . . . . . . 8 1 ∈ ℤ
12025adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
12126ovolfsval 23285 . . . . . . . . . 10 ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 1 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))))
122120, 38, 121sylancl 695 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))))
12357fveq2i 6232 . . . . . . . . . . 11 (2nd ‘(𝐺‘1)) = (2nd ‘⟨𝐴, 𝐵⟩)
12450adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
125123, 124syl5eq 2697 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (2nd ‘(𝐺‘1)) = 𝐵)
12657fveq2i 6232 . . . . . . . . . . 11 (1st ‘(𝐺‘1)) = (1st ‘⟨𝐴, 𝐵⟩)
12741adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℕ) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
128126, 127syl5eq 2697 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ) → (1st ‘(𝐺‘1)) = 𝐴)
129125, 128oveq12d 6708 . . . . . . . . 9 ((𝜑𝑥 ∈ ℕ) → ((2nd ‘(𝐺‘1)) − (1st ‘(𝐺‘1))) = (𝐵𝐴))
130122, 129eqtrd 2685 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘1) = (𝐵𝐴))
131119, 130seq1i 12855 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘1) = (𝐵𝐴))
132118, 131eqtr3d 2687 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) = (𝐵𝐴))
13336leidd 10632 . . . . . . 7 (𝜑 → (𝐵𝐴) ≤ (𝐵𝐴))
134133adantr 480 . . . . . 6 ((𝜑𝑥 ∈ ℕ) → (𝐵𝐴) ≤ (𝐵𝐴))
135132, 134eqbrtrd 4707 . . . . 5 ((𝜑𝑥 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
136135ralrimiva 2995 . . . 4 (𝜑 → ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴))
137 ffn 6083 . . . . . 6 (seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
13829, 137syl 17 . . . . 5 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
139 breq1 4688 . . . . . 6 (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) → (𝑧 ≤ (𝐵𝐴) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
140139ralrn 6402 . . . . 5 (seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴) ↔ ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
141138, 140syl 17 . . . 4 (𝜑 → (∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴) ↔ ∀𝑥 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑥) ≤ (𝐵𝐴)))
142136, 141mpbird 247 . . 3 (𝜑 → ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴))
143 supxrleub 12194 . . . 4 ((ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* ∧ (𝐵𝐴) ∈ ℝ*) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (𝐵𝐴) ↔ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴)))
14433, 37, 143syl2anc 694 . . 3 (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (𝐵𝐴) ↔ ∀𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑧 ≤ (𝐵𝐴)))
145142, 144mpbird 247 . 2 (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (𝐵𝐴))
1466, 35, 37, 71, 145xrletrd 12031 1 (𝜑 → (vol*‘(𝐴[,]𝐵)) ≤ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  ifcif 4119  cop 4216   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  ran crn 5144  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  2c2 11108  cuz 11725  [,)cico 12215  [,]cicc 12216  ...cfz 12364  seqcseq 12841  abscabs 14018  vol*covol 23277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-ovol 23279
This theorem is referenced by:  ovolicc  23337
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