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Theorem ovolval5lem2 41188
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 11064 . . . . . . 7 ℕ ∈ V
43a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
5 volioof 40522 . . . . . . . 8 (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
65a1i 11 . . . . . . 7 (𝜑 → (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞))
7 rexpssxrxp 10122 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
87a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
9 ovolval5lem2.f . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
109ffvelrnda 6399 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (ℝ × ℝ))
11 xp1st 7242 . . . . . . . . . . 11 ((𝐹𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
1210, 11syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐹𝑛)) ∈ ℝ)
13 ovolval5lem2.w . . . . . . . . . . . . 13 (𝜑𝑊 ∈ ℝ+)
1413rpred 11910 . . . . . . . . . . . 12 (𝜑𝑊 ∈ ℝ)
1514adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑊 ∈ ℝ)
16 2nn 11223 . . . . . . . . . . . . . . 15 2 ∈ ℕ
1716a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 2 ∈ ℕ)
18 nnnn0 11337 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1917, 18nnexpcld 13070 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
2019nnred 11073 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ)
2120adantl 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ)
2219nnne0d 11103 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (2↑𝑛) ≠ 0)
2322adantl 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0)
2415, 21, 23redivcld 10891 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ)
2512, 24resubcld 10496 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) ∈ ℝ)
26 xp2nd 7243 . . . . . . . . . 10 ((𝐹𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2710, 26syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ∈ ℝ)
2825, 27opelxpd 5183 . . . . . . . 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 6425 . . . . . . 7 (𝜑𝐺:ℕ⟶(ℝ × ℝ))
316, 8, 30fcoss 39716 . . . . . 6 (𝜑 → ((vol ∘ (,)) ∘ 𝐺):ℕ⟶(0[,]+∞))
324, 31sge0xrcl 40920 . . . . 5 (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) ∈ ℝ*)
332, 32eqeltrd 2730 . . . 4 (𝜑𝑍 ∈ ℝ*)
34 reex 10065 . . . . . . . . 9 ℝ ∈ V
3534, 34xpex 7004 . . . . . . . 8 (ℝ × ℝ) ∈ V
3635a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ∈ V)
3736, 4elmapd 7913 . . . . . 6 (𝜑 → (𝐺 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ↔ 𝐺:ℕ⟶(ℝ × ℝ)))
3830, 37mpbird 247 . . . . 5 (𝜑𝐺 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
39 ovolval5lem2.s . . . . . . 7 (𝜑𝐴 ran ([,) ∘ 𝐹))
4030ffvelrnda 6399 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ (ℝ × ℝ))
41 xp1st 7242 . . . . . . . . . . . . . 14 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
4240, 41syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
4342rexrd 10127 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ℝ*)
44 xp2nd 7243 . . . . . . . . . . . . . 14 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4540, 44syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4645rexrd 10127 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) ∈ ℝ*)
4713adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → 𝑊 ∈ ℝ+)
4819nnrpd 11908 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ+)
4948adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+)
5047, 49rpdivcld 11927 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝑊 / (2↑𝑛)) ∈ ℝ+)
5112, 50ltsubrpd 11942 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹𝑛)))
52 id 22 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
53 opex 4962 . . . . . . . . . . . . . . . . . . 19 ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩ ∈ V
5453a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩ ∈ V)
5529fvmpt2 6330 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩ ∈ V) → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)
5652, 54, 55syl2anc 694 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (𝐺𝑛) = ⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩)
5756fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = (1st ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩))
58 ovex 6718 . . . . . . . . . . . . . . . . . 18 ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) ∈ V
59 fvex 6239 . . . . . . . . . . . . . . . . . 18 (2nd ‘(𝐹𝑛)) ∈ V
60 op1stg 7222 . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))))
6463adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))))
6564breq1d 4695 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛)) < (1st ‘(𝐹𝑛)) ↔ ((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))) < (1st ‘(𝐹𝑛))))
6651, 65mpbird 247 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) < (1st ‘(𝐹𝑛)))
6756fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘⟨((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹𝑛))⟩))
6858, 59op2nd 7219 . . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (2nd ‘(𝐺𝑛)) = (2nd ‘(𝐹𝑛)))
7170adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘(𝐹𝑛)))
7271eqcomd 2657 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) = (2nd ‘(𝐺𝑛)))
7327, 72eqled 10178 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐹𝑛)) ≤ (2nd ‘(𝐺𝑛)))
74 icossioo 12302 . . . . . . . . . . . 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 1367 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))
76 1st2nd2 7249 . . . . . . . . . . . . . . 15 ((𝐹𝑛) ∈ (ℝ × ℝ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
7710, 76syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
7877fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ([,)‘(𝐹𝑛)) = ([,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
79 df-ov 6693 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) = ([,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
8079a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) = ([,)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
8178, 80eqtr4d 2688 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ([,)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))))
82 1st2nd2 7249 . . . . . . . . . . . . . . 15 ((𝐺𝑛) ∈ (ℝ × ℝ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
8340, 82syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
8483fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((,)‘(𝐺𝑛)) = ((,)‘⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩))
85 df-ov 6693 . . . . . . . . . . . . . 14 ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))) = ((,)‘⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
8685a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))) = ((,)‘⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩))
8784, 86eqtr4d 2688 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((,)‘(𝐺𝑛)) = ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))
8881, 87sseq12d 3667 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)) ↔ ((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛)))))
8975, 88mpbird 247 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)))
9089ralrimiva 2995 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)))
91 ss2iun 4568 . . . . . . . . 9 (∀𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ ((,)‘(𝐺𝑛)) → 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)))
9290, 91syl 17 . . . . . . . 8 (𝜑 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)))
93 fvex 6239 . . . . . . . . . . . . 13 ([,)‘(𝐹𝑛)) ∈ V
9493rgenw 2953 . . . . . . . . . . . 12 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ∈ V
9594a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ∈ V)
96 dfiun3g 5410 . . . . . . . . . . 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 39725 . . . . . . . . . . . . . . 15 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
9998a1i 11 . . . . . . . . . . . . . 14 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
1009, 8, 99fcomptss 39709 . . . . . . . . . . . . 13 (𝜑 → ([,) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))))
101100eqcomd 2657 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))) = ([,) ∘ 𝐹))
102101rneqd 5385 . . . . . . . . . . 11 (𝜑 → ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))) = ran ([,) ∘ 𝐹))
103102unieqd 4478 . . . . . . . . . 10 (𝜑 ran (𝑛 ∈ ℕ ↦ ([,)‘(𝐹𝑛))) = ran ([,) ∘ 𝐹))
10497, 103eqtr2d 2686 . . . . . . . . 9 (𝜑 ran ([,) ∘ 𝐹) = 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)))
105 fvex 6239 . . . . . . . . . . . . 13 ((,)‘(𝐺𝑛)) ∈ V
106105rgenw 2953 . . . . . . . . . . . 12 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)) ∈ V
107106a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ ((,)‘(𝐺𝑛)) ∈ V)
108 dfiun3g 5410 . . . . . . . . . . 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 12309 . . . . . . . . . . . . . . 15 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
111110a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (,):(ℝ* × ℝ*)⟶𝒫 ℝ)
11230, 8, 111fcomptss 39709 . . . . . . . . . . . . 13 (𝜑 → ((,) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))))
113112eqcomd 2657 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))) = ((,) ∘ 𝐺))
114113rneqd 5385 . . . . . . . . . . 11 (𝜑 → ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))) = ran ((,) ∘ 𝐺))
115114unieqd 4478 . . . . . . . . . 10 (𝜑 ran (𝑛 ∈ ℕ ↦ ((,)‘(𝐺𝑛))) = ran ((,) ∘ 𝐺))
116109, 115eqtr2d 2686 . . . . . . . . 9 (𝜑 ran ((,) ∘ 𝐺) = 𝑛 ∈ ℕ ((,)‘(𝐺𝑛)))
117104, 116sseq12d 3667 . . . . . . . 8 (𝜑 → ( ran ([,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐺) ↔ 𝑛 ∈ ℕ ([,)‘(𝐹𝑛)) ⊆ 𝑛 ∈ ℕ ((,)‘(𝐺𝑛))))
11892, 117mpbird 247 . . . . . . 7 (𝜑 ran ([,) ∘ 𝐹) ⊆ ran ((,) ∘ 𝐺))
11939, 118sstrd 3646 . . . . . 6 (𝜑𝐴 ran ((,) ∘ 𝐺))
120119, 2jca 553 . . . . 5 (𝜑 → (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
121 coeq2 5313 . . . . . . . . . 10 (𝑓 = 𝐺 → ((,) ∘ 𝑓) = ((,) ∘ 𝐺))
122121rneqd 5385 . . . . . . . . 9 (𝑓 = 𝐺 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
123122unieqd 4478 . . . . . . . 8 (𝑓 = 𝐺 ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
124123sseq2d 3666 . . . . . . 7 (𝑓 = 𝐺 → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ 𝐺)))
125 coeq2 5313 . . . . . . . . 9 (𝑓 = 𝐺 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
126125fveq2d 6233 . . . . . . . 8 (𝑓 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
127126eqeq2d 2661 . . . . . . 7 (𝑓 = 𝐺 → (𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
128124, 127anbi12d 747 . . . . . 6 (𝑓 = 𝐺 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
129128rspcev 3340 . . . . 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 694 . . . 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 2655 . . . . . 6 (𝑧 = 𝑍 → (𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
133132anbi2d 740 . . . . 5 (𝑧 = 𝑍 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
134133rexbidv 3081 . . . 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 3399 . . 3 (𝑍𝑄 ↔ (𝑍 ∈ ℝ* ∧ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))))
137131, 136sylibr 224 . 2 (𝜑𝑍𝑄)
138 fveq2 6229 . . . . . . 7 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
139138fveq2d 6233 . . . . . 6 (𝑚 = 𝑛 → (1st ‘(𝐹𝑚)) = (1st ‘(𝐹𝑛)))
140138fveq2d 6233 . . . . . 6 (𝑚 = 𝑛 → (2nd ‘(𝐹𝑚)) = (2nd ‘(𝐹𝑛)))
141139, 140breq12d 4698 . . . . 5 (𝑚 = 𝑛 → ((1st ‘(𝐹𝑚)) < (2nd ‘(𝐹𝑚)) ↔ (1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛))))
142141cbvrabv 3230 . . . 4 {𝑚 ∈ ℕ ∣ (1st ‘(𝐹𝑚)) < (2nd ‘(𝐹𝑚))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹𝑛)) < (2nd ‘(𝐹𝑛))}
14312, 27, 13, 142ovolval5lem1 41187 . . 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 2793 . . . . . . . 8 𝑛𝐺
14530, 8fssd 6095 . . . . . . . 8 (𝜑𝐺:ℕ⟶(ℝ* × ℝ*))
146144, 145volioofmpt 40529 . . . . . . 7 (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))))
14764, 71oveq12d 6708 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))) = (((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛))))
148147fveq2d 6233 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (vol‘((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛)))) = (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))
149148mpteq2dva 4777 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐺𝑛))(,)(2nd ‘(𝐺𝑛))))) = (𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛))))))
150146, 149eqtrd 2685 . . . . . 6 (𝜑 → ((vol ∘ (,)) ∘ 𝐺) = (𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛))))))
151150fveq2d 6233 . . . . 5 (𝜑 → (Σ^‘((vol ∘ (,)) ∘ 𝐺)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))))
1522, 151eqtrd 2685 . . . 4 (𝜑𝑍 = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(((1st ‘(𝐹𝑛)) − (𝑊 / (2↑𝑛)))(,)(2nd ‘(𝐹𝑛)))))))
153 ovolval5lem2.y . . . . . 6 (𝜑𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
154 nfcv 2793 . . . . . . . 8 𝑛𝐹
155 ressxr 10121 . . . . . . . . . . 11 ℝ ⊆ ℝ*
156 xpss2 5162 . . . . . . . . . . 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 6095 . . . . . . . 8 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
160154, 159volicofmpt 40532 . . . . . . 7 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛))))))
161160fveq2d 6233 . . . . . 6 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))))
162153, 161eqtrd 2685 . . . . 5 (𝜑𝑌 = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))))
163162oveq1d 6705 . . . 4 (𝜑 → (𝑌 +𝑒 𝑊) = ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑛))[,)(2nd ‘(𝐹𝑛)))))) +𝑒 𝑊))
164152, 163breq12d 4698 . . 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 247 . 2 (𝜑𝑍 ≤ (𝑌 +𝑒 𝑊))
166 breq1 4688 . . 3 (𝑧 = 𝑍 → (𝑧 ≤ (𝑌 +𝑒 𝑊) ↔ 𝑍 ≤ (𝑌 +𝑒 𝑊)))
167166rspcev 3340 . 2 ((𝑍𝑄𝑍 ≤ (𝑌 +𝑒 𝑊)) → ∃𝑧𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
168137, 165, 167syl2anc 694 1 (𝜑 → ∃𝑧𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  ran crn 5144  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  cr 9973  0cc0 9974  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  +crp 11870   +𝑒 cxad 11982  (,)cioo 12213  [,)cico 12215  [,]cicc 12216  cexp 12900  volcvol 23278  Σ^csumge0 40897
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-fi 8358  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-xneg 11984  df-xadd 11985  df-xmul 11986  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-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898
This theorem is referenced by:  ovolval5lem3  41189
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