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Theorem ovolunlem1 23558
Description: Lemma for ovolun 23560. (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 488 . . . 4 (𝜑𝐴 ⊆ ℝ)
3 ovolun.b . . . . 5 (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
43simpld 488 . . . 4 (𝜑𝐵 ⊆ ℝ)
52, 4unssd 3953 . . 3 (𝜑 → (𝐴𝐵) ⊆ ℝ)
6 ovolun.g1 . . . . . . . . . . . 12 (𝜑𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
7 elovolmlem 23535 . . . . . . . . . . . 12 (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
86, 7sylib 209 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
98adantr 472 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
109ffvelrnda 6551 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ (𝑛 / 2) ∈ ℕ) → (𝐺‘(𝑛 / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
11 nneo 11711 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1211adantl 473 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℕ ↔ ¬ ((𝑛 + 1) / 2) ∈ ℕ))
1312con2bid 345 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (((𝑛 + 1) / 2) ∈ ℕ ↔ ¬ (𝑛 / 2) ∈ ℕ))
1413biimpar 469 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → ((𝑛 + 1) / 2) ∈ ℕ)
15 ovolun.f1 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
16 elovolmlem 23535 . . . . . . . . . . . . 13 (𝐹 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1715, 16sylib 209 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1817adantr 472 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
1918ffvelrnda 6551 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ((𝑛 + 1) / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2014, 19syldan 585 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (𝑛 / 2) ∈ ℕ) → (𝐹‘((𝑛 + 1) / 2)) ∈ ( ≤ ∩ (ℝ × ℝ)))
2110, 20ifclda 4279 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))) ∈ ( ≤ ∩ (ℝ × ℝ)))
22 ovolun.h . . . . . . . 8 𝐻 = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (𝐺‘(𝑛 / 2)), (𝐹‘((𝑛 + 1) / 2))))
2321, 22fmptd 6576 . . . . . . 7 (𝜑𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
24 eqid 2765 . . . . . . . 8 ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
25 ovolun.u . . . . . . . 8 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
2624, 25ovolsf 23533 . . . . . . 7 (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
2723, 26syl 17 . . . . . 6 (𝜑𝑈:ℕ⟶(0[,)+∞))
28 rge0ssre 12487 . . . . . 6 (0[,)+∞) ⊆ ℝ
29 fss 6238 . . . . . 6 ((𝑈:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑈:ℕ⟶ℝ)
3027, 28, 29sylancl 580 . . . . 5 (𝜑𝑈:ℕ⟶ℝ)
3130frnd 6232 . . . 4 (𝜑 → ran 𝑈 ⊆ ℝ)
32 1nn 11289 . . . . . . 7 1 ∈ ℕ
33 1z 11657 . . . . . . . . . 10 1 ∈ ℤ
34 seqfn 13023 . . . . . . . . . 10 (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3533, 34mp1i 13 . . . . . . . . 9 (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3625fneq1i 6165 . . . . . . . . . 10 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ)
37 nnuz 11926 . . . . . . . . . . 11 ℕ = (ℤ‘1)
3837fneq2i 6166 . . . . . . . . . 10 (seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
3936, 38bitri 266 . . . . . . . . 9 (𝑈 Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝐻)) Fn (ℤ‘1))
4035, 39sylibr 225 . . . . . . . 8 (𝜑𝑈 Fn ℕ)
41 fndm 6170 . . . . . . . 8 (𝑈 Fn ℕ → dom 𝑈 = ℕ)
4240, 41syl 17 . . . . . . 7 (𝜑 → dom 𝑈 = ℕ)
4332, 42syl5eleqr 2851 . . . . . 6 (𝜑 → 1 ∈ dom 𝑈)
4443ne0d 4088 . . . . 5 (𝜑 → dom 𝑈 ≠ ∅)
45 dm0rn0 5512 . . . . . 6 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
4645necon3bii 2989 . . . . 5 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
4744, 46sylib 209 . . . 4 (𝜑 → ran 𝑈 ≠ ∅)
481simprd 489 . . . . . . . 8 (𝜑 → (vol*‘𝐴) ∈ ℝ)
493simprd 489 . . . . . . . 8 (𝜑 → (vol*‘𝐵) ∈ ℝ)
5048, 49readdcld 10325 . . . . . . 7 (𝜑 → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
51 ovolun.c . . . . . . . 8 (𝜑𝐶 ∈ ℝ+)
5251rpred 12073 . . . . . . 7 (𝜑𝐶 ∈ ℝ)
5350, 52readdcld 10325 . . . . . 6 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ)
54 ovolun.s . . . . . . . . 9 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
55 ovolun.t . . . . . . . . 9 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
56 ovolun.f2 . . . . . . . . 9 (𝜑𝐴 ran ((,) ∘ 𝐹))
57 ovolun.f3 . . . . . . . . 9 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2)))
58 ovolun.g2 . . . . . . . . 9 (𝜑𝐵 ran ((,) ∘ 𝐺))
59 ovolun.g3 . . . . . . . . 9 (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2)))
601, 3, 51, 54, 55, 25, 15, 56, 57, 6, 58, 59, 22ovolunlem1a 23557 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
6160ralrimiva 3113 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
62 breq1 4814 . . . . . . . . 9 (𝑧 = (𝑈𝑘) → (𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6362ralrn 6554 . . . . . . . 8 (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6440, 63syl 17 . . . . . . 7 (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑘 ∈ ℕ (𝑈𝑘) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
6561, 64mpbird 248 . . . . . 6 (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
66 brralrspcev 4871 . . . . . 6 (((((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ ∧ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)) → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
6753, 65, 66syl2anc 579 . . . . 5 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘)
68 ressxr 10339 . . . . . . 7 ℝ ⊆ ℝ*
6931, 68syl6ss 3775 . . . . . 6 (𝜑 → ran 𝑈 ⊆ ℝ*)
70 supxrbnd2 12357 . . . . . 6 (ran 𝑈 ⊆ ℝ* → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
7169, 70syl 17 . . . . 5 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧𝑘 ↔ sup(ran 𝑈, ℝ*, < ) < +∞))
7267, 71mpbid 223 . . . 4 (𝜑 → sup(ran 𝑈, ℝ*, < ) < +∞)
73 supxrbnd 12363 . . . 4 ((ran 𝑈 ⊆ ℝ ∧ ran 𝑈 ≠ ∅ ∧ sup(ran 𝑈, ℝ*, < ) < +∞) → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
7431, 47, 72, 73syl3anc 1490 . . 3 (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ)
75 nncn 11285 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
7675adantl 473 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
77762timesd 11523 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) = (𝑚 + 𝑚))
7877oveq1d 6859 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = ((𝑚 + 𝑚) − 1))
79 1cnd 10290 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℂ)
8076, 76, 79addsubassd 10668 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((𝑚 + 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
8178, 80eqtrd 2799 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) = (𝑚 + (𝑚 − 1)))
82 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
83 nnm1nn0 11583 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ → (𝑚 − 1) ∈ ℕ0)
8483adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → (𝑚 − 1) ∈ ℕ0)
85 nnnn0addcl 11572 . . . . . . . . . . . . 13 ((𝑚 ∈ ℕ ∧ (𝑚 − 1) ∈ ℕ0) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
8682, 84, 85syl2anc 579 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝑚 + (𝑚 − 1)) ∈ ℕ)
8781, 86eqeltrd 2844 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) − 1) ∈ ℕ)
88 oveq1 6851 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 / 2) = (((2 · 𝑚) − 1) / 2))
8988eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → ((𝑛 / 2) ∈ ℕ ↔ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
9088fveq2d 6381 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐺‘(𝑛 / 2)) = (𝐺‘(((2 · 𝑚) − 1) / 2)))
91 oveq1 6851 . . . . . . . . . . . . . . . 16 (𝑛 = ((2 · 𝑚) − 1) → (𝑛 + 1) = (((2 · 𝑚) − 1) + 1))
9291fvoveq1d 6866 . . . . . . . . . . . . . . 15 (𝑛 = ((2 · 𝑚) − 1) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)))
9389, 90, 92ifbieq12d 4272 . . . . . . . . . . . . . 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))))
94 fvex 6390 . . . . . . . . . . . . . . 15 (𝐺‘(((2 · 𝑚) − 1) / 2)) ∈ V
95 fvex 6390 . . . . . . . . . . . . . . 15 (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) ∈ V
9694, 95ifex 4293 . . . . . . . . . . . . . 14 if((((2 · 𝑚) − 1) / 2) ∈ ℕ, (𝐺‘(((2 · 𝑚) − 1) / 2)), (𝐹‘((((2 · 𝑚) − 1) + 1) / 2))) ∈ V
9793, 22, 96fvmpt 6473 . . . . . . . . . . . . 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))))
9887, 97syl 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))))
99 2nn 11347 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℕ
100 nnmulcl 11301 . . . . . . . . . . . . . . . . . . . 20 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
10199, 82, 100sylancr 581 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℕ)
102101nncnd 11294 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚 ∈ ℕ) → (2 · 𝑚) ∈ ℂ)
103 ax-1cn 10249 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
104 npcan 10546 . . . . . . . . . . . . . . . . . 18 (((2 · 𝑚) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
105102, 103, 104sylancl 580 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ) → (((2 · 𝑚) − 1) + 1) = (2 · 𝑚))
106105oveq1d 6859 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = ((2 · 𝑚) / 2))
107 2cn 11349 . . . . . . . . . . . . . . . . . 18 2 ∈ ℂ
108 2ne0 11385 . . . . . . . . . . . . . . . . . 18 2 ≠ 0
109 divcan3 10967 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · 𝑚) / 2) = 𝑚)
110107, 108, 109mp3an23 1577 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℂ → ((2 · 𝑚) / 2) = 𝑚)
11176, 110syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) = 𝑚)
112106, 111eqtrd 2799 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) = 𝑚)
113112, 82eqeltrd 2844 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ)
114 nneo 11711 . . . . . . . . . . . . . . . 16 (((2 · 𝑚) − 1) ∈ ℕ → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
11587, 114syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ) → ((((2 · 𝑚) − 1) / 2) ∈ ℕ ↔ ¬ ((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ))
116115con2bid 345 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ) → (((((2 · 𝑚) − 1) + 1) / 2) ∈ ℕ ↔ ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ))
117113, 116mpbid 223 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ¬ (((2 · 𝑚) − 1) / 2) ∈ ℕ)
118117iffalsed 4256 . . . . . . . . . . . 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)))
119112fveq2d 6381 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐹‘((((2 · 𝑚) − 1) + 1) / 2)) = (𝐹𝑚))
12098, 118, 1193eqtrd 2803 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚))
121 fveqeq2 6386 . . . . . . . . . . . 12 (𝑘 = ((2 · 𝑚) − 1) → ((𝐻𝑘) = (𝐹𝑚) ↔ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)))
122121rspcev 3462 . . . . . . . . . . 11 ((((2 · 𝑚) − 1) ∈ ℕ ∧ (𝐻‘((2 · 𝑚) − 1)) = (𝐹𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
12387, 120, 122syl2anc 579 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚))
124 fveq2 6377 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐹𝑚)))
125124breq1d 4821 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐹𝑚)) < 𝑧))
126 fveq2 6377 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐹𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐹𝑚)))
127126breq2d 4823 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐹𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐹𝑚))))
128125, 127anbi12d 624 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐹𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
129128biimprcd 241 . . . . . . . . . . 11 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ((𝐻𝑘) = (𝐹𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
130129reximdv 3162 . . . . . . . . . 10 (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐹𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
131123, 130syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
132131rexlimdva 3178 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
133132ralimdv 3110 . . . . . . 7 (𝜑 → (∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚))) → ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
134 ovolfioo 23528 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
1352, 17, 134syl2anc 579 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) ↔ ∀𝑧𝐴𝑚 ∈ ℕ ((1st ‘(𝐹𝑚)) < 𝑧𝑧 < (2nd ‘(𝐹𝑚)))))
136 ovolfioo 23528 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1372, 23, 136syl2anc 579 . . . . . . 7 (𝜑 → (𝐴 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐴𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
138133, 135, 1373imtr4d 285 . . . . . 6 (𝜑 → (𝐴 ran ((,) ∘ 𝐹) → 𝐴 ran ((,) ∘ 𝐻)))
13956, 138mpd 15 . . . . 5 (𝜑𝐴 ran ((,) ∘ 𝐻))
140 oveq1 6851 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → (𝑛 / 2) = ((2 · 𝑚) / 2))
141140eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → ((𝑛 / 2) ∈ ℕ ↔ ((2 · 𝑚) / 2) ∈ ℕ))
142140fveq2d 6381 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐺‘(𝑛 / 2)) = (𝐺‘((2 · 𝑚) / 2)))
143 oveq1 6851 . . . . . . . . . . . . . . . 16 (𝑛 = (2 · 𝑚) → (𝑛 + 1) = ((2 · 𝑚) + 1))
144143fvoveq1d 6866 . . . . . . . . . . . . . . 15 (𝑛 = (2 · 𝑚) → (𝐹‘((𝑛 + 1) / 2)) = (𝐹‘(((2 · 𝑚) + 1) / 2)))
145141, 142, 144ifbieq12d 4272 . . . . . . . . . . . . . 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))))
146 fvex 6390 . . . . . . . . . . . . . . 15 (𝐺‘((2 · 𝑚) / 2)) ∈ V
147 fvex 6390 . . . . . . . . . . . . . . 15 (𝐹‘(((2 · 𝑚) + 1) / 2)) ∈ V
148146, 147ifex 4293 . . . . . . . . . . . . . 14 if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) ∈ V
149145, 22, 148fvmpt 6473 . . . . . . . . . . . . 13 ((2 · 𝑚) ∈ ℕ → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
150101, 149syl 17 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))))
151111, 82eqeltrd 2844 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ) → ((2 · 𝑚) / 2) ∈ ℕ)
152151iftrued 4253 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → if(((2 · 𝑚) / 2) ∈ ℕ, (𝐺‘((2 · 𝑚) / 2)), (𝐹‘(((2 · 𝑚) + 1) / 2))) = (𝐺‘((2 · 𝑚) / 2)))
153111fveq2d 6381 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → (𝐺‘((2 · 𝑚) / 2)) = (𝐺𝑚))
154150, 152, 1533eqtrd 2803 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝐻‘(2 · 𝑚)) = (𝐺𝑚))
155 fveqeq2 6386 . . . . . . . . . . . 12 (𝑘 = (2 · 𝑚) → ((𝐻𝑘) = (𝐺𝑚) ↔ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)))
156155rspcev 3462 . . . . . . . . . . 11 (((2 · 𝑚) ∈ ℕ ∧ (𝐻‘(2 · 𝑚)) = (𝐺𝑚)) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
157101, 154, 156syl2anc 579 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚))
158 fveq2 6377 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (1st ‘(𝐻𝑘)) = (1st ‘(𝐺𝑚)))
159158breq1d 4821 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧 ↔ (1st ‘(𝐺𝑚)) < 𝑧))
160 fveq2 6377 . . . . . . . . . . . . . 14 ((𝐻𝑘) = (𝐺𝑚) → (2nd ‘(𝐻𝑘)) = (2nd ‘(𝐺𝑚)))
161160breq2d 4823 . . . . . . . . . . . . 13 ((𝐻𝑘) = (𝐺𝑚) → (𝑧 < (2nd ‘(𝐻𝑘)) ↔ 𝑧 < (2nd ‘(𝐺𝑚))))
162159, 161anbi12d 624 . . . . . . . . . . . 12 ((𝐻𝑘) = (𝐺𝑚) → (((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘))) ↔ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
163162biimprcd 241 . . . . . . . . . . 11 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ((𝐻𝑘) = (𝐺𝑚) → ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
164163reximdv 3162 . . . . . . . . . 10 (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → (∃𝑘 ∈ ℕ (𝐻𝑘) = (𝐺𝑚) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
165157, 164syl5com 31 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
166165rexlimdva 3178 . . . . . . . 8 (𝜑 → (∃𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∃𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
167166ralimdv 3110 . . . . . . 7 (𝜑 → (∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚))) → ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
168 ovolfioo 23528 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
1694, 8, 168syl2anc 579 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) ↔ ∀𝑧𝐵𝑚 ∈ ℕ ((1st ‘(𝐺𝑚)) < 𝑧𝑧 < (2nd ‘(𝐺𝑚)))))
170 ovolfioo 23528 . . . . . . . 8 ((𝐵 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
1714, 23, 170syl2anc 579 . . . . . . 7 (𝜑 → (𝐵 ran ((,) ∘ 𝐻) ↔ ∀𝑧𝐵𝑘 ∈ ℕ ((1st ‘(𝐻𝑘)) < 𝑧𝑧 < (2nd ‘(𝐻𝑘)))))
172167, 169, 1713imtr4d 285 . . . . . 6 (𝜑 → (𝐵 ran ((,) ∘ 𝐺) → 𝐵 ran ((,) ∘ 𝐻)))
17358, 172mpd 15 . . . . 5 (𝜑𝐵 ran ((,) ∘ 𝐻))
174139, 173unssd 3953 . . . 4 (𝜑 → (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻))
17525ovollb 23540 . . . 4 ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐴𝐵) ⊆ ran ((,) ∘ 𝐻)) → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
17623, 174, 175syl2anc 579 . . 3 (𝜑 → (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < ))
177 ovollecl 23544 . . 3 (((𝐴𝐵) ⊆ ℝ ∧ sup(ran 𝑈, ℝ*, < ) ∈ ℝ ∧ (vol*‘(𝐴𝐵)) ≤ sup(ran 𝑈, ℝ*, < )) → (vol*‘(𝐴𝐵)) ∈ ℝ)
1785, 74, 176, 177syl3anc 1490 . 2 (𝜑 → (vol*‘(𝐴𝐵)) ∈ ℝ)
17953rexrd 10345 . . . 4 (𝜑 → (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ∈ ℝ*)
180 supxrleub 12361 . . . 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)))
18169, 179, 180syl2anc 579 . . 3 (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶)))
18265, 181mpbird 248 . 2 (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
183178, 74, 53, 176, 182letrd 10450 1 (𝜑 → (vol*‘(𝐴𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  cun 3732  cin 3733  wss 3734  c0 4081  ifcif 4245   cuni 4596   class class class wbr 4811  cmpt 4890   × cxp 5277  dom cdm 5279  ran crn 5280  ccom 5283   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  1st c1st 7366  2nd c2nd 7367  𝑚 cmap 8062  supcsup 8555  cc 10189  cr 10190  0cc0 10191  1c1 10192   + caddc 10194   · cmul 10196  +∞cpnf 10327  *cxr 10329   < clt 10330  cle 10331  cmin 10522   / cdiv 10940  cn 11276  2c2 11329  0cn0 11540  cz 11626  cuz 11889  +crp 12031  (,)cioo 12380  [,)cico 12382  seqcseq 13011  abscabs 14262  vol*covol 23523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-sup 8557  df-inf 8558  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-rp 12032  df-ioo 12384  df-ico 12386  df-fz 12537  df-fl 12804  df-seq 13012  df-exp 13071  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-ovol 23525
This theorem is referenced by:  ovolunlem2  23559
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