Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovolval5lem2 Structured version   Visualization version   GIF version

Theorem ovolval5lem2 39543
 Description: |- ( ( ph /\ n e. NN ) -> <. ( ( 1st (𝐹 n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd (𝐹 n ) ) >. e. ( RR X. RR ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovolval5lem2.q 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
ovolval5lem2.y (𝜑𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
ovolval5lem2.z 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))
ovolval5lem2.f (𝜑𝐹:ℕ⟶(ℝ × ℝ))
ovolval5lem2.s (𝜑𝐴 ran ([,) ∘ 𝐹))
ovolval5lem2.w (𝜑𝑊 ∈ ℝ+)
ovolval5lem2.g 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)
Assertion
Ref Expression
ovolval5lem2 (𝜑 → ∃𝑧𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
Distinct variable groups:   𝐴,𝑓,𝑧   𝑛,𝐹   𝑓,𝐺   𝑛,𝐺   𝑧,𝑄   𝑛,𝑊   𝑧,𝑊   𝑧,𝑌   𝑓,𝑍,𝑧   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑧,𝑓)   𝐴(𝑛)   𝑄(𝑓,𝑛)   𝐹(𝑧,𝑓)   𝐺(𝑧)   𝑊(𝑓)   𝑌(𝑓,𝑛)   𝑍(𝑛)

Proof of Theorem ovolval5lem2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 ovolval5lem2.z . . . . . 6 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))
21a1i 11 . . . . 5 (𝜑𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
3 nnex 10903 . . . . . . 7 ℕ ∈ V
43a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
5 volioof 38880 . . . . . . . 8 (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
65a1i 11 . . . . . . 7 (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞))
7 rexpssxrxp 9963 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
87a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
9 ovolval5lem2.f . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
109ffvelrnda 6267 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ × ℝ))
11 xp1st 7089 . . . . . . . . . . 11 ((𝐹𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
1210, 11syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
13 ovolval5lem2.w . . . . . . . . . . . . 13 (𝜑𝑊 ∈ ℝ+)
1413rpred 11748 . . . . . . . . . . . 12 (𝜑𝑊 ∈ ℝ)
1514adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
16 2nn 11062 . . . . . . . . . . . . . . 15 2 ∈ ℕ
1716a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 2 ∈ ℕ)
18 nnnn0 11176 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1917, 18nnexpcld 12892 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
2019nnred 10912 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
2120adantl 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
2219nnne0d 10942 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
2322adantl 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
2415, 21, 23redivcld 10732 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
2512, 24resubcld 10337 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) ∈ ℝ)
26 xp2nd 7090 . . . . . . . . . 10 ((𝐹𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2710, 26syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2825, 27opelxpd 5073 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩ ∈ (ℝ × ℝ))
29 ovolval5lem2.g . . . . . . . 8 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)
3028, 29fmptd 6292 . . . . . . 7 (𝜑𝐺:ℕ⟶(ℝ × ℝ))
316, 8, 30fcoss 38397 . . . . . 6 (𝜑 → ((vol ∘ (,)) ∘ 𝐺):ℕ⟶(0[,]+∞))
324, 31sge0xrcl 39278 . . . . 5 (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) ∈ ℝ*)
332, 32eqeltrd 2688 . . . 4 (𝜑𝑍 ∈ ℝ*)
34 reex 9906 . . . . . . . . 9 ℝ ∈ V
3534, 34xpex 6860 . . . . . . . 8 (ℝ × ℝ) ∈ V
3635a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ∈ V)
3736, 4elmapd 7758 . . . . . 6 (𝜑 → (𝐺 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ↔ 𝐺:ℕ⟶(ℝ × ℝ)))
3830, 37mpbird 246 . . . . 5 (𝜑𝐺 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
39 ovolval5lem2.s . . . . . . 7 (𝜑𝐴 ran ([,) ∘ 𝐹))
4030ffvelrnda 6267 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ (ℝ × ℝ))
41 xp1st 7089 . . . . . . . . . . . . . 14 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
4240, 41syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
4342rexrd 9968 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ℝ*)
44 xp2nd 7090 . . . . . . . . . . . . . 14 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4540, 44syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4645rexrd 9968 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) ∈ ℝ*)
4713adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑊 ∈ ℝ+)
4819nnrpd 11746 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
4948adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
5047, 49rpdivcld 11765 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ+)
5112, 50ltsubrpd 11780 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹𝑛)))
52 id 22 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
53 opex 4859 . . . . . . . . . . . . . . . . . . 19 ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩ ∈ V
5453a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩ ∈ V)
5529fvmpt2 6200 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)
5652, 54, 55syl2anc 691 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)
5756fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩))
58 ovex 6577 . . . . . . . . . . . . . . . . . 18 ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) ∈ V
59 fvex 6113 . . . . . . . . . . . . . . . . . 18 (2nd ‘(𝐹𝑛)) ∈ V
60 op1stg 7071 . . . . . . . . . . . . . . . . . 18 ((((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) ∈ V ∧ (2nd ‘(𝐹𝑛)) ∈ V) → (1st ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩) = ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))))
6158, 59, 60mp2an 704 . . . . . . . . . . . . . . . . 17 (1st ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩) = ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))
6261a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1st ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩) = ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))))
6357, 62eqtrd 2644 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))))
6463adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))))
6564breq1d 4593 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < (1st ‘(𝐹𝑛)) ↔ ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹𝑛))))
6651, 65mpbird 246 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) < (1st ‘(𝐹𝑛)))
6756fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩))
6858, 59op2nd 7068 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩) = (2nd ‘(𝐹𝑛))
6968a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (2nd ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩) = (2nd ‘(𝐹𝑛)))
7067, 69eqtrd 2644 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘(𝐹𝑛)))
7170adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘(𝐹𝑛)))
7271eqcomd 2616 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) = (2nd ‘(𝐺𝑛)))
7327, 72eqled 10019 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ≤ (2nd ‘(𝐺𝑛)))
74 icossioo 12135 . . . . . . . . . . . 12 ((((1st ‘(𝐺𝑛)) ∈ ℝ* ∧ (2nd ‘(𝐺𝑛)) ∈ ℝ*) ∧ ((1st ‘(𝐺𝑛)) < (1st ‘(𝐹𝑛)) ∧ (2nd ‘(𝐹𝑛)) ≤ (2nd ‘(𝐺𝑛)))) → ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))
7543, 46, 66, 73, 74syl22anc 1319 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))
76 1st2nd2 7096 . . . . . . . . . . . . . . 15 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
7710, 76syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
7877fveq2d 6107 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ([,)‘(𝐹𝑛)) = ([,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
79 df-ov 6552 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) = ([,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
8079a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) = ([,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
8178, 80eqtr4d 2647 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ([,)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))))
82 1st2nd2 7096 . . . . . . . . . . . . . . 15 ((𝐺𝑛) ∈ (ℝ × ℝ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
8340, 82syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
8483fveq2d 6107 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((,)‘(𝐺𝑛)) = ((,)‘⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩))
85 df-ov 6552 . . . . . . . . . . . . . 14 ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))) = ((,)‘⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
8685a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))) = ((,)‘⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩))
8784, 86eqtr4d 2647 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((,)‘(𝐺𝑛)) = ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))
8881, 87sseq12d 3597 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)) ↔ ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛)))))
8975, 88mpbird 246 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)))
9089ralrimiva 2949 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)))
91 ss2iun 4472 . . . . . . . . 9 (∀𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)) → 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)))
9290, 91syl 17 . . . . . . . 8 (𝜑 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)))
93 fvex 6113 . . . . . . . . . . . . 13 ([,)‘(𝐹𝑛)) ∈ V
9493rgenw 2908 . . . . . . . . . . . 12 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ∈ V
9594a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ∈ V)
96 dfiun3g 5299 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ∈ V → 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) = ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))))
9795, 96syl 17 . . . . . . . . . 10 (𝜑 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) = ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))))
98 icof 38406 . . . . . . . . . . . . . . 15 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
9998a1i 11 . . . . . . . . . . . . . 14 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
1009, 8, 99fcomptss 38390 . . . . . . . . . . . . 13 (𝜑 → ([,) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))))
101100eqcomd 2616 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))) = ([,) ∘ 𝐹))
102101rneqd 5274 . . . . . . . . . . 11 (𝜑 → ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))) = ran ([,) ∘ 𝐹))
103102unieqd 4382 . . . . . . . . . 10 (𝜑 ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))) = ran ([,) ∘ 𝐹))
10497, 103eqtr2d 2645 . . . . . . . . 9 (𝜑 ran ([,) ∘ 𝐹) = 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)))
105 fvex 6113 . . . . . . . . . . . . 13 ((,)‘(𝐺𝑛)) ∈ V
106105rgenw 2908 . . . . . . . . . . . 12 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)) ∈ V
107106a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ ((,)‘(𝐺𝑛)) ∈ V)
108 dfiun3g 5299 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ ((,)‘(𝐺𝑛)) ∈ V → 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)) = ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))))
109107, 108syl 17 . . . . . . . . . 10 (𝜑 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)) = ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))))
110 ioof 12142 . . . . . . . . . . . . . . 15 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
111110a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
11230, 8, 111fcomptss 38390 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))))
113112eqcomd 2616 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))) = ((,) ∘ 𝐺))
114113rneqd 5274 . . . . . . . . . . 11 (𝜑 → ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))) = ran ((,) ∘ 𝐺))
115114unieqd 4382 . . . . . . . . . 10 (𝜑 ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))) = ran ((,) ∘ 𝐺))
116109, 115eqtr2d 2645 . . . . . . . . 9 (𝜑 ran ((,) ∘ 𝐺) = 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)))
117104, 116sseq12d 3597 . . . . . . . 8 (𝜑 → ( ran ([,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐺) ↔ 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ 𝑛 ∈ ℕ ((,)‘(𝐺𝑛))))
11892, 117mpbird 246 . . . . . . 7 (𝜑 ran ([,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐺))
11939, 118sstrd 3578 . . . . . 6 (𝜑𝐴 ran ((,) ∘ 𝐺))
120119, 2jca 553 . . . . 5 (𝜑 → (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
121 coeq2 5202 . . . . . . . . . 10 (𝑓 = 𝐺 → ((,) ∘ 𝑓) = ((,) ∘ 𝐺))
122121rneqd 5274 . . . . . . . . 9 (𝑓 = 𝐺 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
123122unieqd 4382 . . . . . . . 8 (𝑓 = 𝐺 ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
124123sseq2d 3596 . . . . . . 7 (𝑓 = 𝐺 → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ 𝐺)))
125 coeq2 5202 . . . . . . . . 9 (𝑓 = 𝐺 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
126125fveq2d 6107 . . . . . . . 8 (𝑓 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
127126eqeq2d 2620 . . . . . . 7 (𝑓 = 𝐺 → (𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
128124, 127anbi12d 743 . . . . . 6 (𝑓 = 𝐺 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
129128rspcev 3282 . . . . 5 ((𝐺 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
13038, 120, 129syl2anc 691 . . . 4 (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
13133, 130jca 553 . . 3 (𝜑 → (𝑍 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
132 eqeq1 2614 . . . . . 6 (𝑧 = 𝑍 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
133132anbi2d 736 . . . . 5 (𝑧 = 𝑍 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
134133rexbidv 3034 . . . 4 (𝑧 = 𝑍 → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
135 ovolval5lem2.q . . . 4 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
136134, 135elrab2 3333 . . 3 (𝑍𝑄 ↔ (𝑍 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
137131, 136sylibr 223 . 2 (𝜑𝑍𝑄)
138 fveq2 6103 . . . . . . 7 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
139138fveq2d 6107 . . . . . 6 (𝑚 = 𝑛 → (1st ‘(𝐹𝑚)) = (1st ‘(𝐹𝑛)))
140138fveq2d 6107 . . . . . 6 (𝑚 = 𝑛 → (2nd ‘(𝐹𝑚)) = (2nd ‘(𝐹𝑛)))
141139, 140breq12d 4596 . . . . 5 (𝑚 = 𝑛 → ((1st ‘(𝐹𝑚)) < (2nd ‘(𝐹𝑚)) ↔ (1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛))))
142141cbvrabv 3172 . . . 4 {𝑚 ∈ ℕ ∣ (1st ‘(𝐹𝑚)) < (2nd ‘(𝐹𝑚))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛))}
14312, 27, 13, 142ovolval5lem1 39542 . . 3 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))) +𝑒 𝑊))
144 nfcv 2751 . . . . . . . 8 𝑛𝐺
14530, 8fssd 5970 . . . . . . . 8 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
146144, 145volioofmpt 38887 . . . . . . 7 (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))))
14764, 71oveq12d 6567 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))) = (((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛))))
148147fveq2d 6107 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (vol‘((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛)))) = (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))
149148mpteq2dva 4672 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))) = (𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛))))))
150146, 149eqtrd 2644 . . . . . 6 (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛))))))
151150fveq2d 6107 . . . . 5 (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))))
1522, 151eqtrd 2644 . . . 4 (𝜑𝑍 = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))))
153 ovolval5lem2.y . . . . . 6 (𝜑𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
154 nfcv 2751 . . . . . . . 8 𝑛𝐹
155 ressxr 9962 . . . . . . . . . . 11 ℝ ⊆ ℝ*
156 xpss2 5152 . . . . . . . . . . 11 (ℝ ⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
157155, 156ax-mp 5 . . . . . . . . . 10 (ℝ × ℝ) ⊆ (ℝ × ℝ*)
158157a1i 11 . . . . . . . . 9 (𝜑 → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
1599, 158fssd 5970 . . . . . . . 8 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
160154, 159volicofmpt 38890 . . . . . . 7 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))))))
161160fveq2d 6107 . . . . . 6 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))))
162153, 161eqtrd 2644 . . . . 5 (𝜑𝑌 = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))))
163162oveq1d 6564 . . . 4 (𝜑 → (𝑌 +𝑒 𝑊) = ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))) +𝑒 𝑊))
164152, 163breq12d 4596 . . 3 (𝜑 → (𝑍 ≤ (𝑌 +𝑒 𝑊) ↔ (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))) +𝑒 𝑊)))
165143, 164mpbird 246 . 2 (𝜑𝑍 ≤ (𝑌 +𝑒 𝑊))
166 breq1 4586 . . 3 (𝑧 = 𝑍 → (𝑧 ≤ (𝑌 +𝑒 𝑊) ↔ 𝑍 ≤ (𝑌 +𝑒 𝑊)))
167166rspcev 3282 . 2 ((𝑍𝑄𝑍 ≤ (𝑌 +𝑒 𝑊)) → ∃𝑧𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
168137, 165, 167syl2anc 691 1 (𝜑 → ∃𝑧𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  ℝcr 9814  0cc0 9815  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  ℝ+crp 11708   +𝑒 cxad 11820  (,)cioo 12046  [,)cico 12048  [,]cicc 12049  ↑cexp 12722  volcvol 23039  Σ^csumge0 39255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256 This theorem is referenced by:  ovolval5lem3  39544
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