Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovolval4lem1 Structured version   Visualization version   GIF version

Theorem ovolval4lem1 46604
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 13483 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
21a1i 11 . . . . . . 7 (𝜑 → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
3 ovolval4lem1.f . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
4 fco 6760 . . . . . . 7 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
52, 3, 4syl2anc 584 . . . . . 6 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
65ffnd 6737 . . . . 5 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
7 fniunfv 7266 . . . . 5 (((,) ∘ 𝐹) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
86, 7syl 17 . . . 4 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
98eqcomd 2740 . . 3 (𝜑 ran ((,) ∘ 𝐹) = 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
10 ovolval4lem1.a . . . . . . . . 9 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))}
11 ssrab2 4089 . . . . . . . . 9 {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))} ⊆ ℕ
1210, 11eqsstri 4029 . . . . . . . 8 𝐴 ⊆ ℕ
13 undif 4487 . . . . . . . 8 (𝐴 ⊆ ℕ ↔ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ)
1412, 13mpbi 230 . . . . . . 7 (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ
1514eqcomi 2743 . . . . . 6 ℕ = (𝐴 ∪ (ℕ ∖ 𝐴))
1615iuneq1i 45024 . . . . 5 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛)
17 iunxun 5098 . . . . 5 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
1816, 17eqtri 2762 . . . 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)))
203ffvelcdmda 7103 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ* × ℝ*))
21 xp1st 8044 . . . . . . . . . . 11 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
2220, 21syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
23 xp2nd 8045 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
2420, 23syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
2524, 22ifcld 4576 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) ∈ ℝ*)
2622, 25opelxpd 5727 . . . . . . . . 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 7133 . . . . . . . 8 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
29 fco 6760 . . . . . . . 8 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
302, 28, 29syl2anc 584 . . . . . . 7 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
3130ffnd 6737 . . . . . 6 (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
32 fniunfv 7266 . . . . . 6 (((,) ∘ 𝐺) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ran ((,) ∘ 𝐺))
3331, 32syl 17 . . . . 5 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ran ((,) ∘ 𝐺))
3433eqcomd 2740 . . . 4 (𝜑 ran ((,) ∘ 𝐺) = 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛))
3515iuneq1i 45024 . . . . . 6 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛)
36 iunxun 5098 . . . . . 6 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
3735, 36eqtri 2762 . . . . 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 480 . . . . . . . 8 ((𝜑𝑛𝐴) → 𝐺:ℕ⟶(ℝ* × ℝ*))
4012sseli 3990 . . . . . . . . 9 (𝑛𝐴𝑛 ∈ ℕ)
4140adantl 481 . . . . . . . 8 ((𝜑𝑛𝐴) → 𝑛 ∈ ℕ)
42 fvco3 7007 . . . . . . . 8 ((𝐺:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
4339, 41, 42syl2anc 584 . . . . . . 7 ((𝜑𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
443adantr 480 . . . . . . . . 9 ((𝜑𝑛𝐴) → 𝐹:ℕ⟶(ℝ* × ℝ*))
45 fvco3 7007 . . . . . . . . 9 ((𝐹:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
4644, 41, 45syl2anc 584 . . . . . . . 8 ((𝜑𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
47 simpl 482 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → 𝜑)
48 1st2nd2 8051 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
4920, 48syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
5047, 41, 49syl2anc 584 . . . . . . . . . 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 3501 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ ∈ V)
5351, 52fvmpt2d 7028 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
5447, 41, 53syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
5510eleq2i 2830 . . . . . . . . . . . . . . . . 17 (𝑛𝐴𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
5655biimpi 216 . . . . . . . . . . . . . . . 16 (𝑛𝐴𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
57 rabid 3454 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))} ↔ (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
5856, 57sylib 218 . . . . . . . . . . . . . . 15 (𝑛𝐴 → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
5958simprd 495 . . . . . . . . . . . . . 14 (𝑛𝐴 → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
6059adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑛𝐴) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
6160iftrued 4538 . . . . . . . . . . . 12 ((𝜑𝑛𝐴) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) = (2nd ‘(𝐹𝑛)))
6261opeq2d 4884 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
63 eqidd 2735 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩ = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
6454, 62, 633eqtrd 2778 . . . . . . . . . 10 ((𝜑𝑛𝐴) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
6550, 64eqtr4d 2777 . . . . . . . . 9 ((𝜑𝑛𝐴) → (𝐹𝑛) = (𝐺𝑛))
6665fveq2d 6910 . . . . . . . 8 ((𝜑𝑛𝐴) → ((,)‘(𝐹𝑛)) = ((,)‘(𝐺𝑛)))
6746, 66eqtrd 2774 . . . . . . 7 ((𝜑𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐺𝑛)))
6843, 67eqtr4d 2777 . . . . . 6 ((𝜑𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
6968iuneq2dv 5020 . . . . 5 (𝜑 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) = 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛))
7028adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
71 eldifi 4140 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ 𝐴) → 𝑛 ∈ ℕ)
7271adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝑛 ∈ ℕ)
7370, 72, 42syl2anc 584 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
74 simpl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝜑)
7574, 72, 53syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
7671anim1i 615 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
7776, 57sylibr 234 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
7877, 55sylibr 234 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛𝐴)
7978adantll 714 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛𝐴)
80 eldifn 4141 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℕ ∖ 𝐴) → ¬ 𝑛𝐴)
8180ad2antlr 727 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → ¬ 𝑛𝐴)
8279, 81pm2.65da 817 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ¬ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
8382iffalsed 4541 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) = (1st ‘(𝐹𝑛)))
8483opeq2d 4884 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ = ⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
8575, 84eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
8685fveq2d 6910 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐺𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩))
87 iooid 13411 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛))) = ∅
8887eqcomi 2743 . . . . . . . . . . 11 ∅ = ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛)))
89 df-ov 7433 . . . . . . . . . . 11 ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
9088, 89eqtr2i 2763 . . . . . . . . . 10 ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩) = ∅
9190a1i 11 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩) = ∅)
9273, 86, 913eqtrd 2778 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ∅)
9392iuneq2dv 5020 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)∅)
94 iun0 5066 . . . . . . . 8 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅
9594a1i 11 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅)
9693, 95eqtrd 2774 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∅)
9774, 3syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
9897, 72, 45syl2anc 584 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
9974, 72, 49syl2anc 584 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
10099fveq2d 6910 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
101 df-ov 7433 . . . . . . . . . . 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 769 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → 𝑛 ∈ (ℕ ∖ 𝐴))
10472, 22syldan 591 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
105104adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
10672, 24syldan 591 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
107106adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
108 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛)))
109105, 107xrltnled 45312 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ((1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛)) ↔ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
110108, 109mpbird 257 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛)))
111105, 107, 110xrltled 13188 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
112103, 111, 78syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → 𝑛𝐴)
11380ad2antlr 727 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ¬ 𝑛𝐴)
114112, 113condan 818 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛)))
115 ioo0 13408 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ* ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ*) → (((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅ ↔ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
116104, 106, 115syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅ ↔ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
117114, 116mpbird 257 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅)
118102, 117eqtr3d 2776 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩) = ∅)
11998, 100, 1183eqtrd 2778 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ∅)
120119iuneq2dv 5020 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)∅)
121120, 95eqtrd 2774 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∅)
12296, 121eqtr4d 2777 . . . . 5 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
12369, 122uneq12d 4178 . . . 4 (𝜑 → ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
12434, 38, 1233eqtrrd 2779 . . 3 (𝜑 → ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)) = ran ((,) ∘ 𝐺))
1259, 19, 1243eqtrd 2778 . 2 (𝜑 ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺))
126 volf 25577 . . . . . 6 vol:dom vol⟶(0[,]+∞)
127126a1i 11 . . . . 5 (𝜑 → vol:dom vol⟶(0[,]+∞))
1283adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶(ℝ* × ℝ*))
129 simpr 484 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
130128, 129, 45syl2anc 584 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
13149fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
132101eqcomi 2743 . . . . . . . . . . 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 2778 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
135 ioombl 25613 . . . . . . . . . 10 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol
136135a1i 11 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol)
137134, 136eqeltrd 2838 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
138137ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
1396, 138jca 511 . . . . . 6 (𝜑 → (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
140 ffnfv 7138 . . . . . 6 (((,) ∘ 𝐹):ℕ⟶dom vol ↔ (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
141139, 140sylibr 234 . . . . 5 (𝜑 → ((,) ∘ 𝐹):ℕ⟶dom vol)
142 fco 6760 . . . . 5 ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐹):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
143127, 141, 142syl2anc 584 . . . 4 (𝜑 → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
144143ffnd 6737 . . 3 (𝜑 → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
14568adantlr 715 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
146137adantr 480 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
147145, 146eqeltrd 2838 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
148 simpll 767 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → 𝜑)
149 eldif 3972 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ 𝐴) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴))
150149bicomi 224 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴) ↔ 𝑛 ∈ (ℕ ∖ 𝐴))
151150biimpi 216 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
152151adantll 714 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
153117, 135eqeltrrdi 2847 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ∅ ∈ dom vol)
15492, 153eqeltrd 2838 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
155148, 152, 154syl2anc 584 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
156147, 155pm2.61dan 813 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
157156ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
15831, 157jca 511 . . . . . 6 (𝜑 → (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
159 ffnfv 7138 . . . . . 6 (((,) ∘ 𝐺):ℕ⟶dom vol ↔ (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
160158, 159sylibr 234 . . . . 5 (𝜑 → ((,) ∘ 𝐺):ℕ⟶dom vol)
161 fco 6760 . . . . 5 ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐺):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
162127, 160, 161syl2anc 584 . . . 4 (𝜑 → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
163162ffnd 6737 . . 3 (𝜑 → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
164145eqcomd 2740 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
165119, 92eqtr4d 2777 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
166148, 152, 165syl2anc 584 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
167164, 166pm2.61dan 813 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
168167fveq2d 6910 . . . 4 ((𝜑𝑛 ∈ ℕ) → (vol‘(((,) ∘ 𝐹)‘𝑛)) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
169 fnfun 6668 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → Fun ((,) ∘ 𝐹))
1706, 169syl 17 . . . . . 6 (𝜑 → Fun ((,) ∘ 𝐹))
171170adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐹))
1725fdmd 6746 . . . . . . . 8 (𝜑 → dom ((,) ∘ 𝐹) = ℕ)
173172eqcomd 2740 . . . . . . 7 (𝜑 → ℕ = dom ((,) ∘ 𝐹))
174173adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐹))
175129, 174eleqtrd 2840 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐹))
176 fvco 7006 . . . . 5 ((Fun ((,) ∘ 𝐹) ∧ 𝑛 ∈ dom ((,) ∘ 𝐹)) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
177171, 175, 176syl2anc 584 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
178 fnfun 6668 . . . . . . 7 (((,) ∘ 𝐺) Fn ℕ → Fun ((,) ∘ 𝐺))
17931, 178syl 17 . . . . . 6 (𝜑 → Fun ((,) ∘ 𝐺))
180179adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐺))
18130fdmd 6746 . . . . . . . 8 (𝜑 → dom ((,) ∘ 𝐺) = ℕ)
182181eqcomd 2740 . . . . . . 7 (𝜑 → ℕ = dom ((,) ∘ 𝐺))
183182adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐺))
184129, 183eleqtrd 2840 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐺))
185 fvco 7006 . . . . 5 ((Fun ((,) ∘ 𝐺) ∧ 𝑛 ∈ dom ((,) ∘ 𝐺)) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
186180, 184, 185syl2anc 584 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
187168, 177, 1863eqtr4d 2784 . . 3 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = ((vol ∘ ((,) ∘ 𝐺))‘𝑛))
188144, 163, 187eqfnfvd 7053 . 2 (𝜑 → (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺)))
189125, 188jca 511 1 (𝜑 → ( ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  wss 3962  c0 4338  ifcif 4530  𝒫 cpw 4604  cop 4636   cuni 4911   ciun 4995   class class class wbr 5147  cmpt 5230   × cxp 5686  dom cdm 5688  ran crn 5689  ccom 5692  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  cr 11151  0cc0 11152  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  cn 12263  (,)cioo 13383  [,]cicc 13386  volcvol 25511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xadd 13152  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-xmet 21374  df-met 21375  df-ovol 25512  df-vol 25513
This theorem is referenced by:  ovolval4lem2  46605
  Copyright terms: Public domain W3C validator