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

Theorem ovolval4lem1 41184
 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 12309 . . . . . . . 8 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
21a1i 11 . . . . . . 7 (𝜑 → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
3 ovolval4lem1.f . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ* × ℝ*))
4 fco 6096 . . . . . . 7 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
52, 3, 4syl2anc 694 . . . . . 6 (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
6 ffn 6083 . . . . . 6 (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
75, 6syl 17 . . . . 5 (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
8 fniunfv 6545 . . . . 5 (((,) ∘ 𝐹) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
97, 8syl 17 . . . 4 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ran ((,) ∘ 𝐹))
109eqcomd 2657 . . 3 (𝜑 ran ((,) ∘ 𝐹) = 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
11 ovolval4lem1.a . . . . . . . . 9 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))}
12 ssrab2 3720 . . . . . . . . 9 {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))} ⊆ ℕ
1311, 12eqsstri 3668 . . . . . . . 8 𝐴 ⊆ ℕ
14 undif 4082 . . . . . . . 8 (𝐴 ⊆ ℕ ↔ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ)
1513, 14mpbi 220 . . . . . . 7 (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ
1615eqcomi 2660 . . . . . 6 ℕ = (𝐴 ∪ (ℕ ∖ 𝐴))
1716iuneq1i 39573 . . . . 5 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛)
18 iunxun 4637 . . . . 5 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
1917, 18eqtri 2673 . . . 4 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
2019a1i 11 . . 3 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
213ffvelrnda 6399 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ* × ℝ*))
22 xp1st 7242 . . . . . . . . . . 11 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
2321, 22syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
24 xp2nd 7243 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
2521, 24syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
2625, 23ifcld 4164 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) ∈ ℝ*)
2723, 26opelxpd 5183 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ ∈ (ℝ* × ℝ*))
28 ovolval4lem1.g . . . . . . . . 9 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
2927, 28fmptd 6425 . . . . . . . 8 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
30 fco 6096 . . . . . . . 8 (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
312, 29, 30syl2anc 694 . . . . . . 7 (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
32 ffn 6083 . . . . . . 7 (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐺) Fn ℕ)
3331, 32syl 17 . . . . . 6 (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
34 fniunfv 6545 . . . . . 6 (((,) ∘ 𝐺) Fn ℕ → 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ran ((,) ∘ 𝐺))
3533, 34syl 17 . . . . 5 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ran ((,) ∘ 𝐺))
3635eqcomd 2657 . . . 4 (𝜑 ran ((,) ∘ 𝐺) = 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛))
3716iuneq1i 39573 . . . . . 6 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛)
38 iunxun 4637 . . . . . 6 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
3937, 38eqtri 2673 . . . . 5 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
4039a1i 11 . . . 4 (𝜑 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)))
4129adantr 480 . . . . . . . 8 ((𝜑𝑛𝐴) → 𝐺:ℕ⟶(ℝ* × ℝ*))
4213sseli 3632 . . . . . . . . 9 (𝑛𝐴𝑛 ∈ ℕ)
4342adantl 481 . . . . . . . 8 ((𝜑𝑛𝐴) → 𝑛 ∈ ℕ)
44 fvco3 6314 . . . . . . . 8 ((𝐺:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
4541, 43, 44syl2anc 694 . . . . . . 7 ((𝜑𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
463adantr 480 . . . . . . . . 9 ((𝜑𝑛𝐴) → 𝐹:ℕ⟶(ℝ* × ℝ*))
47 fvco3 6314 . . . . . . . . 9 ((𝐹:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
4846, 43, 47syl2anc 694 . . . . . . . 8 ((𝜑𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
49 simpl 472 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → 𝜑)
50 1st2nd2 7249 . . . . . . . . . . . 12 ((𝐹𝑛) ∈ (ℝ* × ℝ*) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
5121, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
5249, 43, 51syl2anc 694 . . . . . . . . . 10 ((𝜑𝑛𝐴) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
5328a1i 11 . . . . . . . . . . . . 13 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩))
5427elexd 3245 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ ∈ V)
5553, 54fvmpt2d 6332 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
5649, 43, 55syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
5711eleq2i 2722 . . . . . . . . . . . . . . . . 17 (𝑛𝐴𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
5857biimpi 206 . . . . . . . . . . . . . . . 16 (𝑛𝐴𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
59 rabid 3145 . . . . . . . . . . . . . . . 16 (𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))} ↔ (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
6058, 59sylib 208 . . . . . . . . . . . . . . 15 (𝑛𝐴 → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
6160simprd 478 . . . . . . . . . . . . . 14 (𝑛𝐴 → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
6261adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑛𝐴) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
6362iftrued 4127 . . . . . . . . . . . 12 ((𝜑𝑛𝐴) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) = (2nd ‘(𝐹𝑛)))
6463opeq2d 4440 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
65 eqidd 2652 . . . . . . . . . . 11 ((𝜑𝑛𝐴) → ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩ = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
6656, 64, 653eqtrd 2689 . . . . . . . . . 10 ((𝜑𝑛𝐴) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
6752, 66eqtr4d 2688 . . . . . . . . 9 ((𝜑𝑛𝐴) → (𝐹𝑛) = (𝐺𝑛))
6867fveq2d 6233 . . . . . . . 8 ((𝜑𝑛𝐴) → ((,)‘(𝐹𝑛)) = ((,)‘(𝐺𝑛)))
6948, 68eqtrd 2685 . . . . . . 7 ((𝜑𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐺𝑛)))
7045, 69eqtr4d 2688 . . . . . 6 ((𝜑𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
7170iuneq2dv 4574 . . . . 5 (𝜑 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) = 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛))
7229adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
73 eldifi 3765 . . . . . . . . . . 11 (𝑛 ∈ (ℕ ∖ 𝐴) → 𝑛 ∈ ℕ)
7473adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝑛 ∈ ℕ)
7572, 74, 44syl2anc 694 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺𝑛)))
76 simpl 472 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝜑)
7776, 74, 55syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩)
7873anim1i 591 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))))
7978, 59sylibr 224 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))})
8079, 57sylibr 224 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛𝐴)
8180adantll 750 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → 𝑛𝐴)
82 eldifn 3766 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℕ ∖ 𝐴) → ¬ 𝑛𝐴)
8382ad2antlr 763 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛))) → ¬ 𝑛𝐴)
8481, 83pm2.65da 599 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ¬ (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
8584iffalsed 4130 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))) = (1st ‘(𝐹𝑛)))
8685opeq2d 4440 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ⟨(1st ‘(𝐹𝑛)), if((1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛)), (1st ‘(𝐹𝑛)))⟩ = ⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
8777, 86eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺𝑛) = ⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
8887fveq2d 6233 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐺𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩))
89 iooid 12241 . . . . . . . . . . . 12 ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛))) = ∅
9089eqcomi 2660 . . . . . . . . . . 11 ∅ = ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛)))
91 df-ov 6693 . . . . . . . . . . 11 ((1st ‘(𝐹𝑛))(,)(1st ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩)
9290, 91eqtr2i 2674 . . . . . . . . . 10 ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩) = ∅
9392a1i 11 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹𝑛)), (1st ‘(𝐹𝑛))⟩) = ∅)
9475, 88, 933eqtrd 2689 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ∅)
9594iuneq2dv 4574 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)∅)
96 iun0 4608 . . . . . . . 8 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅
9796a1i 11 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅)
9895, 97eqtrd 2685 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∅)
9976, 3syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
10099, 74, 47syl2anc 694 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
10176, 74, 51syl2anc 694 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
102101fveq2d 6233 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
103 df-ov 6693 . . . . . . . . . . 11 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
104103a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
105 simplr 807 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → 𝑛 ∈ (ℕ ∖ 𝐴))
10674, 23syldan 486 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
107106adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) ∈ ℝ*)
10874, 25syldan 486 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
109108adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
110 simpr 476 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛)))
111107, 109xrltnled 39892 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ((1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛)) ↔ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
112110, 111mpbird 247 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛)))
113107, 109, 112xrltled 39800 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → (1st ‘(𝐹𝑛)) ≤ (2nd ‘(𝐹𝑛)))
114105, 113, 80syl2anc 694 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → 𝑛𝐴)
11582ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))) → ¬ 𝑛𝐴)
116114, 115condan 852 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛)))
117 ioo0 12238 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑛)) ∈ ℝ* ∧ (2nd ‘(𝐹𝑛)) ∈ ℝ*) → (((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅ ↔ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
118106, 108, 117syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅ ↔ (2nd ‘(𝐹𝑛)) ≤ (1st ‘(𝐹𝑛))))
119116, 118mpbird 247 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) = ∅)
120104, 119eqtr3d 2687 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩) = ∅)
121100, 102, 1203eqtrd 2689 . . . . . . . 8 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ∅)
122121iuneq2dv 4574 . . . . . . 7 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)∅)
123122, 97eqtrd 2685 . . . . . 6 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∅)
12498, 123eqtr4d 2688 . . . . 5 (𝜑 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
12571, 124uneq12d 3801 . . . 4 (𝜑 → ( 𝑛𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)) = ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
12636, 40, 1253eqtrrd 2690 . . 3 (𝜑 → ( 𝑛𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)) = ran ((,) ∘ 𝐺))
12710, 20, 1263eqtrd 2689 . 2 (𝜑 ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺))
128 volf 23343 . . . . . 6 vol:dom vol⟶(0[,]+∞)
129128a1i 11 . . . . 5 (𝜑 → vol:dom vol⟶(0[,]+∞))
1303adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶(ℝ* × ℝ*))
131 simpr 476 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
132130, 131, 47syl2anc 694 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹𝑛)))
13351fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((,)‘(𝐹𝑛)) = ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
134103eqcomi 2660 . . . . . . . . . . 11 ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛)))
135134a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
136132, 133, 1353eqtrd 2689 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))))
137 ioombl 23379 . . . . . . . . . 10 ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol
138137a1i 11 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛))(,)(2nd ‘(𝐹𝑛))) ∈ dom vol)
139136, 138eqeltrd 2730 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
140139ralrimiva 2995 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
1417, 140jca 553 . . . . . 6 (𝜑 → (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
142 ffnfv 6428 . . . . . 6 (((,) ∘ 𝐹):ℕ⟶dom vol ↔ (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
143141, 142sylibr 224 . . . . 5 (𝜑 → ((,) ∘ 𝐹):ℕ⟶dom vol)
144 fco 6096 . . . . 5 ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐹):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
145129, 143, 144syl2anc 694 . . . 4 (𝜑 → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
146 ffn 6083 . . . 4 ((vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞) → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
147145, 146syl 17 . . 3 (𝜑 → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
14870adantlr 751 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
149139adantr 480 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
150148, 149eqeltrd 2730 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
151 simpll 805 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → 𝜑)
152 eldif 3617 . . . . . . . . . . . . 13 (𝑛 ∈ (ℕ ∖ 𝐴) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴))
153152bicomi 214 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴) ↔ 𝑛 ∈ (ℕ ∖ 𝐴))
154153biimpi 206 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
155154adantll 750 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
156119, 137syl6eqelr 2739 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → ∅ ∈ dom vol)
15794, 156eqeltrd 2730 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
158151, 155, 157syl2anc 694 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
159150, 158pm2.61dan 849 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
160159ralrimiva 2995 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
16133, 160jca 553 . . . . . 6 (𝜑 → (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
162 ffnfv 6428 . . . . . 6 (((,) ∘ 𝐺):ℕ⟶dom vol ↔ (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
163161, 162sylibr 224 . . . . 5 (𝜑 → ((,) ∘ 𝐺):ℕ⟶dom vol)
164 fco 6096 . . . . 5 ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐺):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
165129, 163, 164syl2anc 694 . . . 4 (𝜑 → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
166 ffn 6083 . . . 4 ((vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞) → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
167165, 166syl 17 . . 3 (𝜑 → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
168148eqcomd 2657 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
169121, 94eqtr4d 2688 . . . . . . 7 ((𝜑𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
170151, 155, 169syl2anc 694 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
171168, 170pm2.61dan 849 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
172171fveq2d 6233 . . . 4 ((𝜑𝑛 ∈ ℕ) → (vol‘(((,) ∘ 𝐹)‘𝑛)) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
173 fnfun 6026 . . . . . . 7 (((,) ∘ 𝐹) Fn ℕ → Fun ((,) ∘ 𝐹))
1747, 173syl 17 . . . . . 6 (𝜑 → Fun ((,) ∘ 𝐹))
175174adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐹))
176 fdm 6089 . . . . . . . . 9 (((,) ∘ 𝐹):ℕ⟶dom vol → dom ((,) ∘ 𝐹) = ℕ)
177143, 176syl 17 . . . . . . . 8 (𝜑 → dom ((,) ∘ 𝐹) = ℕ)
178177eqcomd 2657 . . . . . . 7 (𝜑 → ℕ = dom ((,) ∘ 𝐹))
179178adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐹))
180131, 179eleqtrd 2732 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐹))
181 fvco 6313 . . . . 5 ((Fun ((,) ∘ 𝐹) ∧ 𝑛 ∈ dom ((,) ∘ 𝐹)) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
182175, 180, 181syl2anc 694 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
183 fnfun 6026 . . . . . . 7 (((,) ∘ 𝐺) Fn ℕ → Fun ((,) ∘ 𝐺))
18433, 183syl 17 . . . . . 6 (𝜑 → Fun ((,) ∘ 𝐺))
185184adantr 480 . . . . 5 ((𝜑𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐺))
186 fdm 6089 . . . . . . . . 9 (((,) ∘ 𝐺):ℕ⟶dom vol → dom ((,) ∘ 𝐺) = ℕ)
187163, 186syl 17 . . . . . . . 8 (𝜑 → dom ((,) ∘ 𝐺) = ℕ)
188187eqcomd 2657 . . . . . . 7 (𝜑 → ℕ = dom ((,) ∘ 𝐺))
189188adantr 480 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐺))
190131, 189eleqtrd 2732 . . . . 5 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐺))
191 fvco 6313 . . . . 5 ((Fun ((,) ∘ 𝐺) ∧ 𝑛 ∈ dom ((,) ∘ 𝐺)) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
192185, 190, 191syl2anc 694 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
193172, 182, 1923eqtr4d 2695 . . 3 ((𝜑𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = ((vol ∘ ((,) ∘ 𝐺))‘𝑛))
194147, 167, 193eqfnfvd 6354 . 2 (𝜑 → (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺)))
195127, 194jca 553 1 (𝜑 → ( ran ((,) ∘ 𝐹) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {crab 2945  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ⊆ wss 3607  ∅c0 3948  ifcif 4119  𝒫 cpw 4191  ⟨cop 4216  ∪ cuni 4468  ∪ ciun 4552   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144   ∘ ccom 5147  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  ℝcr 9973  0cc0 9974  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112   ≤ cle 10113  ℕcn 11058  (,)cioo 12213  [,]cicc 12216  volcvol 23278 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-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  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-cda 9028  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-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  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-rlim 14264  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280 This theorem is referenced by:  ovolval4lem2  41185
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