Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovolval4lem2 Structured version   Visualization version   GIF version

Theorem ovolval4lem2 41185
 Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 41182, but here 𝑓 is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovolval4lem2.a (𝜑𝐴 ⊆ ℝ)
ovolval4lem2.m 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
ovolval4lem2.g 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩)
Assertion
Ref Expression
ovolval4lem2 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Distinct variable groups:   𝐴,𝑓,𝑦   𝑛,𝐺   𝑓,𝑛   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝐴(𝑛)   𝐺(𝑦,𝑓)   𝑀(𝑦,𝑓,𝑛)

Proof of Theorem ovolval4lem2
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolval4lem2.a . 2 (𝜑𝐴 ⊆ ℝ)
2 ovolval4lem2.m . . 3 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}
3 iftrue 4125 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (2nd ‘(𝑓𝑛)))
43opeq2d 4440 . . . . . . . . . . . . . 14 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
54adantl 481 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
6 df-br 4686 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
76biimpi 206 . . . . . . . . . . . . . 14 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
87adantl 481 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
95, 8eqeltrd 2730 . . . . . . . . . . . 12 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ≤ )
10 iffalse 4128 . . . . . . . . . . . . . . 15 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (1st ‘(𝑓𝑛)))
1110opeq2d 4440 . . . . . . . . . . . . . 14 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩)
1211adantl 481 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩)
13 elmapi 7921 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1413ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
15 xp1st 7242 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1716leidd 10632 . . . . . . . . . . . . . . 15 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)))
18 df-br 4686 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
1917, 18sylib 208 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
2019adantr 480 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
2112, 20eqeltrd 2730 . . . . . . . . . . . 12 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ≤ )
229, 21pm2.61dan 849 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ≤ )
23 xp2nd 7243 . . . . . . . . . . . . . 14 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2414, 23syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2524, 16ifcld 4164 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) ∈ ℝ)
26 opelxpi 5182 . . . . . . . . . . . 12 (((1st ‘(𝑓𝑛)) ∈ ℝ ∧ if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) ∈ ℝ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ (ℝ × ℝ))
2716, 25, 26syl2anc 694 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ (ℝ × ℝ))
2822, 27elind 3831 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
29 ovolval4lem2.g . . . . . . . . . 10 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩)
3028, 29fmptd 6425 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31 reex 10065 . . . . . . . . . . . . 13 ℝ ∈ V
3231, 31xpex 7004 . . . . . . . . . . . 12 (ℝ × ℝ) ∈ V
3332inex2 4833 . . . . . . . . . . 11 ( ≤ ∩ (ℝ × ℝ)) ∈ V
3433a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
35 nnex 11064 . . . . . . . . . . 11 ℕ ∈ V
3635a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ℕ ∈ V)
3734, 36elmapd 7913 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
3830, 37mpbird 247 . . . . . . . 8 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
3938adantr 480 . . . . . . 7 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
40 simpr 476 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝑓))
41 rexpssxrxp 10122 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4241a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
4313, 42fssd 6095 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
44 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑓𝑘) = (𝑓𝑛))
4544fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (1st ‘(𝑓𝑘)) = (1st ‘(𝑓𝑛)))
4644fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (2nd ‘(𝑓𝑘)) = (2nd ‘(𝑓𝑛)))
4745, 46breq12d 4698 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)) ↔ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
4847cbvrabv 3230 . . . . . . . . . . . . 13 {𝑘 ∈ ℕ ∣ (1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))}
4943, 29, 48ovolval4lem1 41184 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ( ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺))))
5049simpld 474 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5150adantr 480 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5240, 51sseqtrd 3674 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝐺))
5352adantrr 753 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ran ((,) ∘ 𝐺))
54 simpr 476 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
5549simprd 478 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺)))
56 coass 5692 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
5756a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
58 coass 5692 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
5958a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
6055, 57, 593eqtr4d 2695 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
6160fveq2d 6233 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6261adantr 480 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6354, 62eqtrd 2685 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6463adantrl 752 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6553, 64jca 553 . . . . . . 7 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
66 coeq2 5313 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
6766rneqd 5385 . . . . . . . . . . 11 (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6867unieqd 4478 . . . . . . . . . 10 (𝑔 = 𝐺 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6968sseq2d 3666 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝐺)))
70 coeq2 5313 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
7170fveq2d 6233 . . . . . . . . . 10 (𝑔 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
7271eqeq2d 2661 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
7369, 72anbi12d 747 . . . . . . . 8 (𝑔 = 𝐺 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
7473rspcev 3340 . . . . . . 7 ((𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
7539, 65, 74syl2anc 694 . . . . . 6 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
7675rexlimiva 3057 . . . . 5 (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
77 inss2 3867 . . . . . . . . . 10 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
78 mapss 7942 . . . . . . . . . 10 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
7932, 77, 78mp2an 708 . . . . . . . . 9 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
8079sseli 3632 . . . . . . . 8 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
8180adantr 480 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → 𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
82 simpr 476 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
83 coeq2 5313 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
8483rneqd 5385 . . . . . . . . . . 11 (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8584unieqd 4478 . . . . . . . . . 10 (𝑓 = 𝑔 ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8685sseq2d 3666 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ 𝑔)))
87 coeq2 5313 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
8887fveq2d 6233 . . . . . . . . . 10 (𝑓 = 𝑔 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))
8988eqeq2d 2661 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
9086, 89anbi12d 747 . . . . . . . 8 (𝑓 = 𝑔 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))))
9190rspcev 3340 . . . . . . 7 ((𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
9281, 82, 91syl2anc 694 . . . . . 6 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
9392rexlimiva 3057 . . . . 5 (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))))
9476, 93impbii 199 . . . 4 (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
9594rabbii 3216 . . 3 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} = {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
962, 95eqtri 2673 . 2 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
971, 96ovolval3 41182 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  {crab 2945  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  ifcif 4119  ⟨cop 4216  ∪ cuni 4468   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  infcinf 8388  ℝcr 9973  ℝ*cxr 10111   < clt 10112   ≤ cle 10113  ℕcn 11058  (,)cioo 12213  vol*covol 23277  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:  ovolval4  41186
 Copyright terms: Public domain W3C validator