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Theorem ovolval4lem2 44881
Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 44878, 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 4492 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (2nd ‘(𝑓𝑛)))
43opeq2d 4837 . . . . . . . . . . . . . 14 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
54adantl 482 . . . . . . . . . . . . 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 5106 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
76biimpi 215 . . . . . . . . . . . . . 14 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
87adantl 482 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
95, 8eqeltrd 2838 . . . . . . . . . . . 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 4495 . . . . . . . . . . . . . . 15 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (1st ‘(𝑓𝑛)))
1110opeq2d 4837 . . . . . . . . . . . . . 14 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩)
1211adantl 482 . . . . . . . . . . . . 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 8787 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1413ffvelcdmda 7035 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
15 xp1st 7953 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1716leidd 11721 . . . . . . . . . . . . . . 15 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)))
18 df-br 5106 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
1917, 18sylib 217 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
2019adantr 481 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
2112, 20eqeltrd 2838 . . . . . . . . . . . 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 811 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ≤ )
23 xp2nd 7954 . . . . . . . . . . . . . 14 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2414, 23syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2524, 16ifcld 4532 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) ∈ ℝ)
26 opelxpi 5670 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ (ℝ × ℝ))
2822, 27elind 4154 . . . . . . . . . 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 7062 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31 reex 11142 . . . . . . . . . . . . 13 ℝ ∈ V
3231, 31xpex 7687 . . . . . . . . . . . 12 (ℝ × ℝ) ∈ V
3332inex2 5275 . . . . . . . . . . 11 ( ≤ ∩ (ℝ × ℝ)) ∈ V
3433a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
35 nnex 12159 . . . . . . . . . . 11 ℕ ∈ V
3635a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ℕ ∈ V)
3734, 36elmapd 8779 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
3830, 37mpbird 256 . . . . . . . 8 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
3938adantr 481 . . . . . . 7 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
40 simpr 485 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝑓))
41 rexpssxrxp 11200 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4241a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
4313, 42fssd 6686 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
44 2fveq3 6847 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (1st ‘(𝑓𝑘)) = (1st ‘(𝑓𝑛)))
45 2fveq3 6847 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (2nd ‘(𝑓𝑘)) = (2nd ‘(𝑓𝑛)))
4644, 45breq12d 5118 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)) ↔ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
4746cbvrabv 3417 . . . . . . . . . . . . 13 {𝑘 ∈ ℕ ∣ (1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))}
4843, 29, 47ovolval4lem1 44880 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ( ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺))))
4948simpld 495 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5049adantr 481 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5140, 50sseqtrd 3984 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝐺))
5251adantrr 715 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ran ((,) ∘ 𝐺))
53 simpr 485 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
5448simprd 496 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺)))
55 coass 6217 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
5655a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
57 coass 6217 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
5857a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
5954, 56, 583eqtr4d 2786 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
6059fveq2d 6846 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6160adantr 481 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6253, 61eqtrd 2776 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6362adantrl 714 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6452, 63jca 512 . . . . . . 7 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
65 coeq2 5814 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
6665rneqd 5893 . . . . . . . . . . 11 (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6766unieqd 4879 . . . . . . . . . 10 (𝑔 = 𝐺 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6867sseq2d 3976 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝐺)))
69 coeq2 5814 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
7069fveq2d 6846 . . . . . . . . . 10 (𝑔 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
7170eqeq2d 2747 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
7268, 71anbi12d 631 . . . . . . . 8 (𝑔 = 𝐺 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
7372rspcev 3581 . . . . . . 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))))
7439, 64, 73syl2anc 584 . . . . . 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))))
7574rexlimiva 3144 . . . . 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))))
76 inss2 4189 . . . . . . . . . 10 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
77 mapss 8827 . . . . . . . . . 10 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ))
7832, 76, 77mp2an 690 . . . . . . . . 9 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ)
7978sseli 3940 . . . . . . . 8 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑔 ∈ ((ℝ × ℝ) ↑m ℕ))
8079adantr 481 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → 𝑔 ∈ ((ℝ × ℝ) ↑m ℕ))
81 simpr 485 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
82 coeq2 5814 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
8382rneqd 5893 . . . . . . . . . . 11 (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8483unieqd 4879 . . . . . . . . . 10 (𝑓 = 𝑔 ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8584sseq2d 3976 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ 𝑔)))
86 coeq2 5814 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
8786fveq2d 6846 . . . . . . . . . 10 (𝑓 = 𝑔 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))
8887eqeq2d 2747 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
8985, 88anbi12d 631 . . . . . . . 8 (𝑓 = 𝑔 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))))
9089rspcev 3581 . . . . . . 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))))
9180, 81, 90syl2anc 584 . . . . . 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))))
9291rexlimiva 3144 . . . . 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))))
9375, 92impbii 208 . . . 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))))
9493rabbii 3413 . . 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)))}
952, 94eqtri 2764 . 2 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
961, 95ovolval3 44878 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  ifcif 4486  cop 4592   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  ran crn 5634  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  infcinf 9377  cr 11050  *cxr 11188   < clt 11189  cle 11190  cn 12153  (,)cioo 13264  vol*covol 24826  volcvol 24827  Σ^csumge0 44593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-sumge0 44594
This theorem is referenced by:  ovolval4  44882
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