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Theorem ovolunlem1 23017
Description: Lemma for ovolun 23019. (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovolun.a (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
ovolun.b (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
ovolun.c (𝜑𝐶 ∈ ℝ+)
ovolun.s 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolun.t 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
ovolun.u 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ovolun.f1 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
ovolun.f2 (𝜑𝐴 ran ((,) ∘ 𝐹))
ovolun.f3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
ovolun.g1 (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
ovolun.g2 (𝜑𝐵 ran ((,) ∘ 𝐺))
ovolun.g3 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
ovolun.h 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))
Assertion
Ref Expression
ovolunlem1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Distinct variable groups:   𝐶,𝑛   𝑛,𝐹   𝐴,𝑛   𝐵,𝑛   𝑛,𝐺   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑈(𝑛)   𝐻(𝑛)

Proof of Theorem ovolunlem1
Dummy variables 𝑘 𝑧 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolun.a . . . . 5 (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
21simpld 474 . . . 4 (𝜑𝐴 ⊆ ℝ)
3 ovolun.b . . . . 5 (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
43simpld 474 . . . 4 (𝜑𝐵 ⊆ ℝ)
52, 4unssd 3751 . . 3 (𝜑 → (𝐴𝐵) ⊆ ℝ)
6 ovolun.g1 . . . . . . . . . . . 12 (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
7 reex 9884 . . . . . . . . . . . . . . 15 ℝ ∈ V
87, 7xpex 6838 . . . . . . . . . . . . . 14 (ℝ × ℝ) ∈ V
98inex2 4723 . . . . . . . . . . . . 13 ( ≤ ∩ (ℝ × ℝ)) ∈ V
10 nnex 10876 . . . . . . . . . . . . 13 ℕ ∈ V
119, 10elmap 7750 . . . . . . . . . . . 12 (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
126, 11sylib 207 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1312adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1413ffvelrnda 6252 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
15 nneo 11296 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1615adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1716con2bid 343 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝑛 + 1) / 2) ∈ ℕ ↔ ¬ (𝑛 / 2) ∈ ℕ))
1817biimpar 501 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((𝑛 + 1) / 2) ∈ ℕ)
19 ovolun.f1 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
209, 10elmap 7750 . . . . . . . . . . . . 13 (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2119, 20sylib 207 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2221adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2322ffvelrnda 6252 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2418, 23syldan 486 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2514, 24ifclda 4070 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) ∈ ( ≤ ∩ (ℝ × ℝ)))
26 ovolun.h . . . . . . . 8 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))
2725, 26fmptd 6277 . . . . . . 7 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28 eqid 2610 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
29 ovolun.u . . . . . . . 8 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
3028, 29ovolsf 22993 . . . . . . 7 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
3127, 30syl 17 . . . . . 6 (𝜑𝑈:ℕ⟶(0[,)+∞))
32 rge0ssre 12110 . . . . . 6 (0[,)+∞) ⊆ ℝ
33 fss 5955 . . . . . 6 ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
3431, 32, 33sylancl 693 . . . . 5 (𝜑𝑈:ℕ⟶ℝ)
35 frn 5952 . . . . 5 (𝑈:ℕ⟶ℝ → ran 𝑈 ⊆ ℝ)
3634, 35syl 17 . . . 4 (𝜑 → ran 𝑈 ⊆ ℝ)
37 1nn 10881 . . . . . . 7 1 ∈ ℕ
38 1z 11243 . . . . . . . . . 10 1 ∈ ℤ
39 seqfn 12633 . . . . . . . . . 10 (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
4038, 39mp1i 13 . . . . . . . . 9 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
4129fneq1i 5885 . . . . . . . . . 10 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
42 nnuz 11558 . . . . . . . . . . 11 ℕ = (ℤ‘1)
4342fneq2i 5886 . . . . . . . . . 10 (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
4441, 43bitri 263 . . . . . . . . 9 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
4540, 44sylibr 223 . . . . . . . 8 (𝜑𝑈 Fn ℕ)
46 fndm 5890 . . . . . . . 8 (𝑈 Fn ℕ → dom 𝑈 = ℕ)
4745, 46syl 17 . . . . . . 7 (𝜑 → dom 𝑈 = ℕ)
4837, 47syl5eleqr 2695 . . . . . 6 (𝜑 → 1 ∈ dom 𝑈)
49 ne0i 3880 . . . . . 6 (1 ∈ dom 𝑈 → dom 𝑈 ≠ ∅)
5048, 49syl 17 . . . . 5 (𝜑 → dom 𝑈 ≠ ∅)
51 dm0rn0 5250 . . . . . 6 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
5251necon3bii 2834 . . . . 5 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
5350, 52sylib 207 . . . 4 (𝜑 → ran 𝑈 ≠ ∅)
541simprd 478 . . . . . . . 8 (𝜑 → (vol*‘𝐴) ∈ ℝ)
553simprd 478 . . . . . . . 8 (𝜑 → (vol*‘𝐵) ∈ ℝ)
5654, 55readdcld 9926 . . . . . . 7 (𝜑 → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
57 ovolun.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ+)
5857rpred 11707 . . . . . . 7 (𝜑𝐶 ∈ ℝ)
5956, 58readdcld 9926 . . . . . 6 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ)
60 ovolun.s . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
61 ovolun.t . . . . . . . . 9 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
62 ovolun.f2 . . . . . . . . 9 (𝜑𝐴 ran ((,) ∘ 𝐹))
63 ovolun.f3 . . . . . . . . 9 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
64 ovolun.g2 . . . . . . . . 9 (𝜑𝐵 ran ((,) ∘ 𝐺))
65 ovolun.g3 . . . . . . . . 9 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
661, 3, 57, 60, 61, 29, 19, 62, 63, 6, 64, 65, 26ovolunlem1a 23016 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
6766ralrimiva 2949 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
68 breq1 4581 . . . . . . . . 9 (𝑧 = (𝑈𝑘) → (𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6968ralrn 6255 . . . . . . . 8 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
7045, 69syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
7167, 70mpbird 246 . . . . . 6 (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
72 breq2 4582 . . . . . . . 8 (𝑘 = (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) → (𝑧𝑘𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
7372ralbidv 2969 . . . . . . 7 (𝑘 = (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) → (∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
7473rspcev 3282 . . . . . 6 (((((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
7559, 71, 74syl2anc 691 . . . . 5 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
76 ressxr 9940 . . . . . . 7 ℝ ⊆ ℝ*
7736, 76syl6ss 3580 . . . . . 6 (𝜑 → ran 𝑈 ⊆ ℝ*)
78 supxrbnd2 11983 . . . . . 6 (ran 𝑈 ⊆ ℝ* → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
7977, 78syl 17 . . . . 5 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
8075, 79mpbid 221 . . . 4 (𝜑 → sup(ran 𝑈, ℝ*, < ) < +∞)
81 supxrbnd 11989 . . . 4 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ*, < ) < +∞) → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
8236, 53, 80, 81syl3anc 1318 . . 3 (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
83 nncn 10878 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
8483adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
85842timesd 11125 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
8685oveq1d 6542 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
87 1cnd 9913 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℂ)
8884, 84, 87addsubassd 10264 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑚 + 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
8986, 88eqtrd 2644 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
90 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
91 nnm1nn0 11184 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ0)
9291adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝑚 − 1) ∈ ℕ0)
93 nnnn0addcl 11173 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ0) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
9490, 92, 93syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
9589, 94eqeltrd 2688 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
96 oveq1 6534 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
9796eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
9896fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
99 oveq1 6534 . . . . . . . . . . . . . . . . 17 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
10099oveq1d 6542 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 + 1) / 2) = ((((2 · 𝑚) − 1) + 1) / 2))
101100fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
10297, 98, 101ifbieq12d 4063 . . . . . . . . . . . . . 14 (𝑛 = ((2 · 𝑚) − 1) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
103 fvex 6098 . . . . . . . . . . . . . . 15 (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
104 fvex 6098 . . . . . . . . . . . . . . 15 (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
105103, 104ifex 4106 . . . . . . . . . . . . . 14 if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
106102, 26, 105fvmpt 6176 . . . . . . . . . . . . 13 (((2 · 𝑚) − 1) ∈ ℕ → (𝐻‘((2 · 𝑚) − 1)) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
10795, 106syl 17 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))))
108 2nn 11035 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℕ
109 nnmulcl 10893 . . . . . . . . . . . . . . . . . . . 20 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
110108, 90, 109sylancr 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
111110nncnd 10886 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
112 ax-1cn 9851 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
113 npcan 10142 . . . . . . . . . . . . . . . . . 18 (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
114111, 112, 113sylancl 693 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
115114oveq1d 6542 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
116 2cn 10941 . . . . . . . . . . . . . . . . . 18 2 ∈ ℂ
117 2ne0 10963 . . . . . . . . . . . . . . . . . 18 2 ≠ 0
118 divcan3 10563 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
119116, 117, 118mp3an23 1408 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
12084, 119syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
121115, 120eqtrd 2644 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
122121, 90eqeltrd 2688 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
123 nneo 11296 . . . . . . . . . . . . . . . 16 (((2 · 𝑚) − 1) ∈ ℕ → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
12495, 123syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
125124con2bid 343 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
126122, 125mpbid 221 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
127126iffalsed 4047 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
128121fveq2d 6092 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹𝑚))
129107, 127, 1283eqtrd 2648 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚))
130 fveq2 6088 . . . . . . . . . . . . 13 (𝑘 = ((2 · 𝑚) − 1) → (𝐻𝑘) = (𝐻‘((2 · 𝑚) − 1)))
131130eqeq1d 2612 . . . . . . . . . . . 12 (𝑘 = ((2 · 𝑚) − 1) → ((𝐻𝑘) = (𝐹𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)))
132131rspcev 3282 . . . . . . . . . . 11 ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
13395, 129, 132syl2anc 691 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
134 fveq2 6088 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐹𝑚)))
135134breq1d 4588 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐹𝑚)) < 𝑧))
136 fveq2 6088 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐹𝑚)))
137136breq2d 4590 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐹𝑚))))
138135, 137anbi12d 743 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐹𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
139138biimprcd 239 . . . . . . . . . . 11 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
140139reximdv 2999 . . . . . . . . . 10 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
141133, 140syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
142141rexlimdva 3013 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
143142ralimdv 2946 . . . . . . 7 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
144 ovolfioo 22988 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
1452, 21, 144syl2anc 691 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
146 ovolfioo 22988 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1472, 27, 146syl2anc 691 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
148143, 145, 1473imtr4d 282 . . . . . 6 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐻)))
14962, 148mpd 15 . . . . 5 (𝜑𝐴 ran ((,) ∘ 𝐻))
150 oveq1 6534 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
151150eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
152150fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
153 oveq1 6534 . . . . . . . . . . . . . . . . 17 (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
154153oveq1d 6542 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → ((𝑛 + 1) / 2) = (((2 · 𝑚) + 1) / 2))
155154fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
156151, 152, 155ifbieq12d 4063 . . . . . . . . . . . . . 14 (𝑛 = (2 · 𝑚) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
157 fvex 6098 . . . . . . . . . . . . . . 15 (𝐺‘((2 · 𝑚) / 2)) ∈ V
158 fvex 6098 . . . . . . . . . . . . . . 15 (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
159157, 158ifex 4106 . . . . . . . . . . . . . 14 if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
160156, 26, 159fvmpt 6176 . . . . . . . . . . . . 13 ((2 · 𝑚) ∈ ℕ → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
161110, 160syl 17 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
162120, 90eqeltrd 2688 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
163162iftrued 4044 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) = (𝐺‘((2 · 𝑚) / 2)))
164120fveq2d 6092 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺𝑚))
165161, 163, 1643eqtrd 2648 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺𝑚))
166 fveq2 6088 . . . . . . . . . . . . 13 (𝑘 = (2 · 𝑚) → (𝐻𝑘) = (𝐻‘(2 · 𝑚)))
167166eqeq1d 2612 . . . . . . . . . . . 12 (𝑘 = (2 · 𝑚) → ((𝐻𝑘) = (𝐺𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)))
168167rspcev 3282 . . . . . . . . . . 11 (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
169110, 165, 168syl2anc 691 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
170 fveq2 6088 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐺𝑚)))
171170breq1d 4588 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐺𝑚)) < 𝑧))
172 fveq2 6088 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐺𝑚)))
173172breq2d 4590 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐺𝑚))))
174171, 173anbi12d 743 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐺𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
175174biimprcd 239 . . . . . . . . . . 11 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
176175reximdv 2999 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
177169, 176syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
178177rexlimdva 3013 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
179178ralimdv 2946 . . . . . . 7 (𝜑 → (∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
180 ovolfioo 22988 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1814, 12, 180syl2anc 691 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
182 ovolfioo 22988 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1834, 27, 182syl2anc 691 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
184179, 181, 1833imtr4d 282 . . . . . 6 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) → 𝐵 ran ((,) ∘ 𝐻)))
18564, 184mpd 15 . . . . 5 (𝜑𝐵 ran ((,) ∘ 𝐻))
186149, 185unssd 3751 . . . 4 (𝜑 → (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻))
18729ovollb 22999 . . . 4 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻)) → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
18827, 186, 187syl2anc 691 . . 3 (𝜑 → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
189 ovollecl 23003 . . 3 (((𝐴𝐵) ⊆ ℝ ∧ sup(ran 𝑈, ℝ*, < ) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < )) → (vol*‘(𝐴𝐵)) ∈ ℝ)
1905, 82, 188, 189syl3anc 1318 . 2 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ)
19159rexrd 9946 . . . 4 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*)
192 supxrleub 11987 . . . 4 ((ran 𝑈 ⊆ ℝ* ∧ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*) → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
19377, 191, 192syl2anc 691 . . 3 (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
19471, 193mpbird 246 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
195190, 82, 59, 188, 194letrd 10046 1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  cun 3538  cin 3539  wss 3540  c0 3874  ifcif 4036   cuni 4367   class class class wbr 4578  cmpt 4638   × cxp 5026  dom cdm 5028  ran crn 5029  ccom 5032   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  1st c1st 7035  2nd c2nd 7036  𝑚 cmap 7722  supcsup 8207  cc 9791  cr 9792  0cc0 9793  1c1 9794   + caddc 9796   · cmul 9798  +∞cpnf 9928  *cxr 9930   < clt 9931  cle 9932  cmin 10118   / cdiv 10536  cn 10870  2c2 10920  0cn0 11142  cz 11213  cuz 11522  +crp 11667  (,)cioo 12005  [,)cico 12007  seqcseq 12621  abscabs 13771  vol*covol 22983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-sup 8209  df-inf 8210  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-ioo 12009  df-ico 12011  df-fz 12156  df-fl 12413  df-seq 12622  df-exp 12681  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-ovol 22985
This theorem is referenced by:  ovolunlem2  23018
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