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Theorem ovolval4lem2 46571
Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 46568, 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 4554 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (2nd ‘(𝑓𝑛)))
43opeq2d 4904 . . . . . . . . . . . . . 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 5167 . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . 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 4557 . . . . . . . . . . . . . . 15 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (1st ‘(𝑓𝑛)))
1110opeq2d 4904 . . . . . . . . . . . . . 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 8907 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1413ffvelcdmda 7118 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
15 xp1st 8062 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1716leidd 11856 . . . . . . . . . . . . . . 15 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)))
18 df-br 5167 . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . 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 812 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ≤ )
23 xp2nd 8063 . . . . . . . . . . . . . 14 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2414, 23syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2524, 16ifcld 4594 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) ∈ ℝ)
26 opelxpi 5737 . . . . . . . . . . . 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 583 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ (ℝ × ℝ))
2822, 27elind 4223 . . . . . . . . . 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 7148 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31 reex 11275 . . . . . . . . . . . . 13 ℝ ∈ V
3231, 31xpex 7788 . . . . . . . . . . . 12 (ℝ × ℝ) ∈ V
3332inex2 5336 . . . . . . . . . . 11 ( ≤ ∩ (ℝ × ℝ)) ∈ V
3433a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
35 nnex 12299 . . . . . . . . . . 11 ℕ ∈ V
3635a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ℕ ∈ V)
3734, 36elmapd 8898 . . . . . . . . 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 11335 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4241a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
4313, 42fssd 6764 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
44 2fveq3 6925 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (1st ‘(𝑓𝑘)) = (1st ‘(𝑓𝑛)))
45 2fveq3 6925 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (2nd ‘(𝑓𝑘)) = (2nd ‘(𝑓𝑛)))
4644, 45breq12d 5179 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)) ↔ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
4746cbvrabv 3454 . . . . . . . . . . . . 13 {𝑘 ∈ ℕ ∣ (1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))}
4843, 29, 47ovolval4lem1 46570 . . . . . . . . . . . 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 4049 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝐺))
5251adantrr 716 . . . . . . . 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 6296 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
5655a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
57 coass 6296 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
5857a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
5954, 56, 583eqtr4d 2790 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
6059fveq2d 6924 . . . . . . . . . . 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 2780 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6362adantrl 715 . . . . . . . 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 5883 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
6665rneqd 5963 . . . . . . . . . . 11 (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6766unieqd 4944 . . . . . . . . . 10 (𝑔 = 𝐺 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6867sseq2d 4041 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝐺)))
69 coeq2 5883 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
7069fveq2d 6924 . . . . . . . . . 10 (𝑔 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
7170eqeq2d 2751 . . . . . . . . 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 3635 . . . . . . 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 583 . . . . . 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 3153 . . . . 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 4259 . . . . . . . . . 10 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
77 mapss 8947 . . . . . . . . . 10 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ))
7832, 76, 77mp2an 691 . . . . . . . . 9 (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ⊆ ((ℝ × ℝ) ↑m ℕ)
7978sseli 4004 . . . . . . . 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 5883 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
8382rneqd 5963 . . . . . . . . . . 11 (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8483unieqd 4944 . . . . . . . . . 10 (𝑓 = 𝑔 ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8584sseq2d 4041 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ 𝑔)))
86 coeq2 5883 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
8786fveq2d 6924 . . . . . . . . . 10 (𝑓 = 𝑔 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))
8887eqeq2d 2751 . . . . . . . . 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 3635 . . . . . . 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 583 . . . . . 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 3153 . . . . 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 3449 . . 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 2768 . 2 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
961, 95ovolval3 46568 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2108  wrex 3076  {crab 3443  Vcvv 3488  cin 3975  wss 3976  ifcif 4548  cop 4654   cuni 4931   class class class wbr 5166  cmpt 5249   × cxp 5698  ran crn 5701  ccom 5704  wf 6569  cfv 6573  (class class class)co 7448  1st c1st 8028  2nd c2nd 8029  m cmap 8884  infcinf 9510  cr 11183  *cxr 11323   < clt 11324  cle 11325  cn 12293  (,)cioo 13407  vol*covol 25516  volcvol 25517  Σ^csumge0 46283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cmp 23416  df-ovol 25518  df-vol 25519  df-sumge0 46284
This theorem is referenced by:  ovolval4  46572
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