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Theorem ovolval4lem2 46606
Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 46603, 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 4537 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (2nd ‘(𝑓𝑛)))
43opeq2d 4885 . . . . . . . . . . . . . 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 5149 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
76biimpi 216 . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . 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 4540 . . . . . . . . . . . . . . 15 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (1st ‘(𝑓𝑛)))
1110opeq2d 4885 . . . . . . . . . . . . . 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 8888 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1413ffvelcdmda 7104 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
15 xp1st 8045 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1716leidd 11827 . . . . . . . . . . . . . . 15 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)))
18 df-br 5149 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
1917, 18sylib 218 . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . 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 813 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ≤ )
23 xp2nd 8046 . . . . . . . . . . . . . 14 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2414, 23syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2524, 16ifcld 4577 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) ∈ ℝ)
26 opelxpi 5726 . . . . . . . . . . . 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 4210 . . . . . . . . . 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 7134 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31 reex 11244 . . . . . . . . . . . . 13 ℝ ∈ V
3231, 31xpex 7772 . . . . . . . . . . . 12 (ℝ × ℝ) ∈ V
3332inex2 5324 . . . . . . . . . . 11 ( ≤ ∩ (ℝ × ℝ)) ∈ V
3433a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
35 nnex 12270 . . . . . . . . . . 11 ℕ ∈ V
3635a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ℕ ∈ V)
3734, 36elmapd 8879 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
3830, 37mpbird 257 . . . . . . . 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 484 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝑓))
41 rexpssxrxp 11304 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4241a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
4313, 42fssd 6754 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
44 2fveq3 6912 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (1st ‘(𝑓𝑘)) = (1st ‘(𝑓𝑛)))
45 2fveq3 6912 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (2nd ‘(𝑓𝑘)) = (2nd ‘(𝑓𝑛)))
4644, 45breq12d 5161 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)) ↔ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
4746cbvrabv 3444 . . . . . . . . . . . . 13 {𝑘 ∈ ℕ ∣ (1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))}
4843, 29, 47ovolval4lem1 46605 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ( ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺))))
4948simpld 494 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5049adantr 480 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5140, 50sseqtrd 4036 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝐺))
5251adantrr 717 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ran ((,) ∘ 𝐺))
53 simpr 484 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
5448simprd 495 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺)))
55 coass 6287 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
5655a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
57 coass 6287 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
5857a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
5954, 56, 583eqtr4d 2785 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
6059fveq2d 6911 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6160adantr 480 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6253, 61eqtrd 2775 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6362adantrl 716 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6452, 63jca 511 . . . . . . 7 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
65 coeq2 5872 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
6665rneqd 5952 . . . . . . . . . . 11 (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6766unieqd 4925 . . . . . . . . . 10 (𝑔 = 𝐺 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6867sseq2d 4028 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝐺)))
69 coeq2 5872 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
7069fveq2d 6911 . . . . . . . . . 10 (𝑔 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
7170eqeq2d 2746 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
7268, 71anbi12d 632 . . . . . . . 8 (𝑔 = 𝐺 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
7372rspcev 3622 . . . . . . 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 3145 . . . . 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 4246 . . . . . . . . . 10 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
77 mapss 8928 . . . . . . . . . 10 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ))
7832, 76, 77mp2an 692 . . . . . . . . 9 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ)
7978sseli 3991 . . . . . . . 8 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑔 ∈ ((ℝ × ℝ) ↑m ℕ))
8079adantr 480 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → 𝑔 ∈ ((ℝ × ℝ) ↑m ℕ))
81 simpr 484 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
82 coeq2 5872 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
8382rneqd 5952 . . . . . . . . . . 11 (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8483unieqd 4925 . . . . . . . . . 10 (𝑓 = 𝑔 ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8584sseq2d 4028 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ 𝑔)))
86 coeq2 5872 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
8786fveq2d 6911 . . . . . . . . . 10 (𝑓 = 𝑔 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))
8887eqeq2d 2746 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
8985, 88anbi12d 632 . . . . . . . 8 (𝑓 = 𝑔 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))))
9089rspcev 3622 . . . . . . 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 3145 . . . . 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 209 . . . 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 3439 . . 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 2763 . 2 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
961, 95ovolval3 46603 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  ifcif 4531  cop 4637   cuni 4912   class class class wbr 5148  cmpt 5231   × cxp 5687  ran crn 5690  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  infcinf 9479  cr 11152  *cxr 11292   < clt 11293  cle 11294  cn 12264  (,)cioo 13384  vol*covol 25511  volcvol 25512  Σ^csumge0 46318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cmp 23411  df-ovol 25513  df-vol 25514  df-sumge0 46319
This theorem is referenced by:  ovolval4  46607
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