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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  cr 9973  0cc0 9974  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cn 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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