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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovolval4lem1 Structured version   Visualization version   GIF version

Theorem ovolval4lem1 41435
Description: |- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) 𝑛) = (((,) ∘ 𝐹) n ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovolval4lem1.f (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
ovolval4lem1.g 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
ovolval4lem1.a 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))}
Assertion
Ref Expression
ovolval4lem1 (𝜑 → ( ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))
Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝑛,𝐺   𝜑,𝑛

Proof of Theorem ovolval4lem1
StepHypRef Expression
1 ioof 12474 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
21a1i 11 . . . . . . 7 (𝜑 → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
3 ovolval4lem1.f . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
4 fco 6240 . . . . . . 7 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
52, 3, 4syl2anc 579 . . . . . 6 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
65ffnd 6224 . . . . 5 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
7 fniunfv 6697 . . . . 5 (((,) ∘ 𝐹) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
86, 7syl 17 . . . 4 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
98eqcomd 2771 . . 3 (𝜑 ran ((,) ∘ 𝐹) = 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
10 ovolval4lem1.a . . . . . . . . 9 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))}
11 ssrab2 3847 . . . . . . . . 9 {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))} ⊆ ℕ
1210, 11eqsstri 3795 . . . . . . . 8 𝐴 ⊆ ℕ
13 undif 4209 . . . . . . . 8 (𝐴 ⊆ ℕ ↔ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ)
1412, 13mpbi 221 . . . . . . 7 (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ
1514eqcomi 2774 . . . . . 6 ℕ = (𝐴 ∪ (ℕ ∖ 𝐴))
1615iuneq1i 39842 . . . . 5 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛)
17 iunxun 4762 . . . . 5 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
1816, 17eqtri 2787 . . . 4 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
1918a1i 11 . . 3 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
203ffvelrnda 6549 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ* × ℝ*))
21 xp1st 7398 . . . . . . . . . . 11 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
2220, 21syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
23 xp2nd 7399 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
2420, 23syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
2524, 22ifcld 4288 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) ∈ ℝ*)
2622, 25opelxpd 5315 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ ∈ (ℝ* × ℝ*))
27 ovolval4lem1.g . . . . . . . . 9 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
2826, 27fmptd 6574 . . . . . . . 8 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
29 fco 6240 . . . . . . . 8 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
302, 28, 29syl2anc 579 . . . . . . 7 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
3130ffnd 6224 . . . . . 6 (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
32 fniunfv 6697 . . . . . 6 (((,) ∘ 𝐺) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ran ((,) ∘ 𝐺))
3331, 32syl 17 . . . . 5 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ran ((,) ∘ 𝐺))
3433eqcomd 2771 . . . 4 (𝜑 ran ((,) ∘ 𝐺) = 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛))
3515iuneq1i 39842 . . . . . 6 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛)
36 iunxun 4762 . . . . . 6 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
3735, 36eqtri 2787 . . . . 5 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
3837a1i 11 . . . 4 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)))
3928adantr 472 . . . . . . . 8 ((𝜑𝑛𝐴) → 𝐺:ℕ⟶(ℝ* × ℝ*))
4012sseli 3757 . . . . . . . . 9 (𝑛𝐴𝑛 ∈ ℕ)
4140adantl 473 . . . . . . . 8 ((𝜑𝑛𝐴) → 𝑛 ∈ ℕ)
42 fvco3 6464 . . . . . . . 8 ((𝐺:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
4339, 41, 42syl2anc 579 . . . . . . 7 ((𝜑𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
443adantr 472 . . . . . . . . 9 ((𝜑𝑛𝐴) → 𝐹:ℕ⟶(ℝ* × ℝ*))
45 fvco3 6464 . . . . . . . . 9 ((𝐹:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
4644, 41, 45syl2anc 579 . . . . . . . 8 ((𝜑𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
47 simpl 474 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → 𝜑)
48 1st2nd2 7405 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
4920, 48syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
5047, 41, 49syl2anc 579 . . . . . . . . . 10 ((𝜑𝑛𝐴) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
5127a1i 11 . . . . . . . . . . . . 13 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩))
5226elexd 3367 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ ∈ V)
5351, 52fvmpt2d 6482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
5447, 41, 53syl2anc 579 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
5510eleq2i 2836 . . . . . . . . . . . . . . . . 17 (𝑛𝐴𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
5655biimpi 207 . . . . . . . . . . . . . . . 16 (𝑛𝐴𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
57 rabid 3263 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))} ↔ (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
5856, 57sylib 209 . . . . . . . . . . . . . . 15 (𝑛𝐴 → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
5958simprd 489 . . . . . . . . . . . . . 14 (𝑛𝐴 → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
6059adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑛𝐴) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
6160iftrued 4251 . . . . . . . . . . . 12 ((𝜑𝑛𝐴) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) = (2nd ‘(𝐹𝑛)))
6261opeq2d 4566 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
63 eqidd 2766 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩ = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
6454, 62, 633eqtrd 2803 . . . . . . . . . 10 ((𝜑𝑛𝐴) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
6550, 64eqtr4d 2802 . . . . . . . . 9 ((𝜑𝑛𝐴) → (𝐹𝑛) = (𝐺𝑛))
6665fveq2d 6379 . . . . . . . 8 ((𝜑𝑛𝐴) → ((,)‘(𝐹𝑛)) = ((,)‘(𝐺𝑛)))
6746, 66eqtrd 2799 . . . . . . 7 ((𝜑𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐺𝑛)))
6843, 67eqtr4d 2802 . . . . . 6 ((𝜑𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
6968iuneq2dv 4698 . . . . 5 (𝜑 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) = 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛))
7028adantr 472 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
71 eldifi 3894 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ 𝐴) → 𝑛 ∈ ℕ)
7271adantl 473 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝑛 ∈ ℕ)
7370, 72, 42syl2anc 579 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
74 simpl 474 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝜑)
7574, 72, 53syl2anc 579 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
7671anim1i 608 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
7776, 57sylibr 225 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
7877, 55sylibr 225 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛𝐴)
7978adantll 705 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛𝐴)
80 eldifn 3895 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℕ ∖ 𝐴) → ¬ 𝑛𝐴)
8180ad2antlr 718 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → ¬ 𝑛𝐴)
8279, 81pm2.65da 851 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ¬ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
8382iffalsed 4254 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) = (1st ‘(𝐹𝑛)))
8483opeq2d 4566 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ = ⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
8575, 84eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
8685fveq2d 6379 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐺𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩))
87 iooid 12405 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛))) = ∅
8887eqcomi 2774 . . . . . . . . . . 11 ∅ = ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛)))
89 df-ov 6845 . . . . . . . . . . 11 ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
9088, 89eqtr2i 2788 . . . . . . . . . 10 ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩) = ∅
9190a1i 11 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩) = ∅)
9273, 86, 913eqtrd 2803 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ∅)
9392iuneq2dv 4698 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)∅)
94 iun0 4732 . . . . . . . 8 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅
9594a1i 11 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅)
9693, 95eqtrd 2799 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∅)
9774, 3syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
9897, 72, 45syl2anc 579 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
9974, 72, 49syl2anc 579 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
10099fveq2d 6379 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
101 df-ov 6845 . . . . . . . . . . 11 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
102101a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
103 simplr 785 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → 𝑛 ∈ (ℕ ∖ 𝐴))
10472, 22syldan 585 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
105104adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
10672, 24syldan 585 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
107106adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
108 simpr 477 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛)))
109105, 107xrltnled 40149 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ((1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛)) ↔ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
110108, 109mpbird 248 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛)))
111105, 107, 110xrltled 12183 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
112103, 111, 78syl2anc 579 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → 𝑛𝐴)
11380ad2antlr 718 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ¬ 𝑛𝐴)
114112, 113condan 852 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛)))
115 ioo0 12402 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ* ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ*) → (((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅ ↔ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
116104, 106, 115syl2anc 579 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅ ↔ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
117114, 116mpbird 248 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅)
118102, 117eqtr3d 2801 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩) = ∅)
11998, 100, 1183eqtrd 2803 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ∅)
120119iuneq2dv 4698 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)∅)
121120, 95eqtrd 2799 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∅)
12296, 121eqtr4d 2802 . . . . 5 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
12369, 122uneq12d 3930 . . . 4 (𝜑 → ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
12434, 38, 1233eqtrrd 2804 . . 3 (𝜑 → ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)) = ran ((,) ∘ 𝐺))
1259, 19, 1243eqtrd 2803 . 2 (𝜑 ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺))
126 volf 23587 . . . . . 6 vol:dom vol⟶(0[,]+∞)
127126a1i 11 . . . . 5 (𝜑 → vol:dom vol⟶(0[,]+∞))
1283adantr 472 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶(ℝ* × ℝ*))
129 simpr 477 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
130128, 129, 45syl2anc 579 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
13149fveq2d 6379 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
132101eqcomi 2774 . . . . . . . . . . 11 ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛)))
133132a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
134130, 131, 1333eqtrd 2803 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
135 ioombl 23623 . . . . . . . . . 10 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol
136135a1i 11 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol)
137134, 136eqeltrd 2844 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
138137ralrimiva 3113 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
1396, 138jca 507 . . . . . 6 (𝜑 → (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
140 ffnfv 6578 . . . . . 6 (((,) ∘ 𝐹):ℕ⟶dom vol ↔ (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
141139, 140sylibr 225 . . . . 5 (𝜑 → ((,) ∘ 𝐹):ℕ⟶dom vol)
142 fco 6240 . . . . 5 ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐹):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
143127, 141, 142syl2anc 579 . . . 4 (𝜑 → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
144143ffnd 6224 . . 3 (𝜑 → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
14568adantlr 706 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
146137adantr 472 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
147145, 146eqeltrd 2844 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
148 simpll 783 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → 𝜑)
149 eldif 3742 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ 𝐴) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴))
150149bicomi 215 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴) ↔ 𝑛 ∈ (ℕ ∖ 𝐴))
151150biimpi 207 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
152151adantll 705 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
153117, 135syl6eqelr 2853 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ∅ ∈ dom vol)
15492, 153eqeltrd 2844 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
155148, 152, 154syl2anc 579 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
156147, 155pm2.61dan 847 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
157156ralrimiva 3113 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
15831, 157jca 507 . . . . . 6 (𝜑 → (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
159 ffnfv 6578 . . . . . 6 (((,) ∘ 𝐺):ℕ⟶dom vol ↔ (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
160158, 159sylibr 225 . . . . 5 (𝜑 → ((,) ∘ 𝐺):ℕ⟶dom vol)
161 fco 6240 . . . . 5 ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐺):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
162127, 160, 161syl2anc 579 . . . 4 (𝜑 → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
163162ffnd 6224 . . 3 (𝜑 → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
164145eqcomd 2771 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
165119, 92eqtr4d 2802 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
166148, 152, 165syl2anc 579 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
167164, 166pm2.61dan 847 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
168167fveq2d 6379 . . . 4 ((𝜑𝑛 ∈ ℕ) → (vol‘(((,) ∘ 𝐹)‘𝑛)) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
169 fnfun 6166 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → Fun ((,) ∘ 𝐹))
1706, 169syl 17 . . . . . 6 (𝜑 → Fun ((,) ∘ 𝐹))
171170adantr 472 . . . . 5 ((𝜑𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐹))
1725fdmd 6232 . . . . . . . 8 (𝜑 → dom ((,) ∘ 𝐹) = ℕ)
173172eqcomd 2771 . . . . . . 7 (𝜑 → ℕ = dom ((,) ∘ 𝐹))
174173adantr 472 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐹))
175129, 174eleqtrd 2846 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐹))
176 fvco 6463 . . . . 5 ((Fun ((,) ∘ 𝐹) ∧ 𝑛 ∈ dom ((,) ∘ 𝐹)) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
177171, 175, 176syl2anc 579 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
178 fnfun 6166 . . . . . . 7 (((,) ∘ 𝐺) Fn ℕ → Fun ((,) ∘ 𝐺))
17931, 178syl 17 . . . . . 6 (𝜑 → Fun ((,) ∘ 𝐺))
180179adantr 472 . . . . 5 ((𝜑𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐺))
18130fdmd 6232 . . . . . . . 8 (𝜑 → dom ((,) ∘ 𝐺) = ℕ)
182181eqcomd 2771 . . . . . . 7 (𝜑 → ℕ = dom ((,) ∘ 𝐺))
183182adantr 472 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐺))
184129, 183eleqtrd 2846 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐺))
185 fvco 6463 . . . . 5 ((Fun ((,) ∘ 𝐺) ∧ 𝑛 ∈ dom ((,) ∘ 𝐺)) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
186180, 184, 185syl2anc 579 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
187168, 177, 1863eqtr4d 2809 . . 3 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = ((vol ∘ ((,) ∘ 𝐺))‘𝑛))
188144, 163, 187eqfnfvd 6504 . 2 (𝜑 → (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺)))
189125, 188jca 507 1 (𝜑 → ( ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cdif 3729  cun 3730  wss 3732  c0 4079  ifcif 4243  𝒫 cpw 4315  cop 4340   cuni 4594   ciun 4676   class class class wbr 4809  cmpt 4888   × cxp 5275  dom cdm 5277  ran crn 5278  ccom 5281  Fun wfun 6062   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  cr 10188  0cc0 10189  +∞cpnf 10325  *cxr 10327   < clt 10328  cle 10329  cn 11274  (,)cioo 12377  [,]cicc 12380  volcvol 23521
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 2069  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-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  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 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xadd 12147  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-rlim 14507  df-sum 14704  df-xmet 20012  df-met 20013  df-ovol 23522  df-vol 23523
This theorem is referenced by:  ovolval4lem2  41436
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