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Theorem ovolval4lem2 41436
Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 41433, 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 4248 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (2nd ‘(𝑓𝑛)))
43opeq2d 4565 . . . . . . . . . . . . . 14 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩)
54adantl 473 . . . . . . . . . . . . 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 4809 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
76biimpi 207 . . . . . . . . . . . . . 14 ((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
87adantl 473 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛))⟩ ∈ ≤ )
95, 8eqeltrd 2843 . . . . . . . . . . . 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 4251 . . . . . . . . . . . . . . 15 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) = (1st ‘(𝑓𝑛)))
1110opeq2d 4565 . . . . . . . . . . . . . 14 (¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ = ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩)
1211adantl 473 . . . . . . . . . . . . 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 8081 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ × ℝ))
1413ffvelrnda 6548 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑓𝑛) ∈ (ℝ × ℝ))
15 xp1st 7397 . . . . . . . . . . . . . . . . 17 ((𝑓𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1614, 15syl 17 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ∈ ℝ)
1716leidd 10847 . . . . . . . . . . . . . . 15 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)))
18 df-br 4809 . . . . . . . . . . . . . . 15 ((1st ‘(𝑓𝑛)) ≤ (1st ‘(𝑓𝑛)) ↔ ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
1917, 18sylib 209 . . . . . . . . . . . . . 14 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
2019adantr 472 . . . . . . . . . . . . 13 (((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ ¬ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))) → ⟨(1st ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))⟩ ∈ ≤ )
2112, 20eqeltrd 2843 . . . . . . . . . . . 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 847 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ ≤ )
23 xp2nd 7398 . . . . . . . . . . . . . 14 ((𝑓𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2414, 23syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝑓𝑛)) ∈ ℝ)
2524, 16ifcld 4287 . . . . . . . . . . . 12 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛))) ∈ ℝ)
26 opelxpi 5313 . . . . . . . . . . . 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 579 . . . . . . . . . . 11 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝑓𝑛)), if((1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛)), (2nd ‘(𝑓𝑛)), (1st ‘(𝑓𝑛)))⟩ ∈ (ℝ × ℝ))
2822, 27elind 3959 . . . . . . . . . 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 6573 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31 reex 10279 . . . . . . . . . . . . 13 ℝ ∈ V
3231, 31xpex 7159 . . . . . . . . . . . 12 (ℝ × ℝ) ∈ V
3332inex2 4960 . . . . . . . . . . 11 ( ≤ ∩ (ℝ × ℝ)) ∈ V
3433a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ( ≤ ∩ (ℝ × ℝ)) ∈ V)
35 nnex 11280 . . . . . . . . . . 11 ℕ ∈ V
3635a1i 11 . . . . . . . . . 10 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ℕ ∈ V)
3734, 36elmapd 8073 . . . . . . . . 9 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ↔ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))))
3830, 37mpbird 248 . . . . . . . 8 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
3938adantr 472 . . . . . . 7 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐺 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ))
40 simpr 477 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝑓))
41 rexpssxrxp 10337 . . . . . . . . . . . . . . 15 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
4241a1i 11 . . . . . . . . . . . . . 14 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
4313, 42fssd 6236 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → 𝑓:ℕ⟶(ℝ* × ℝ*))
44 2fveq3 6379 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (1st ‘(𝑓𝑘)) = (1st ‘(𝑓𝑛)))
45 2fveq3 6379 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (2nd ‘(𝑓𝑘)) = (2nd ‘(𝑓𝑛)))
4644, 45breq12d 4821 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘)) ↔ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))))
4746cbvrabv 3347 . . . . . . . . . . . . 13 {𝑘 ∈ ℕ ∣ (1st ‘(𝑓𝑘)) ≤ (2nd ‘(𝑓𝑘))} = {𝑛 ∈ ℕ ∣ (1st ‘(𝑓𝑛)) ≤ (2nd ‘(𝑓𝑛))}
4843, 29, 47ovolval4lem1 41435 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ( ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺))))
4948simpld 488 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5049adantr 472 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝐺))
5140, 50sseqtrd 3800 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝐴 ran ((,) ∘ 𝑓)) → 𝐴 ran ((,) ∘ 𝐺))
5251adantrr 708 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝐴 ran ((,) ∘ 𝐺))
53 simpr 477 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))
5448simprd 489 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (vol ∘ ((,) ∘ 𝑓)) = (vol ∘ ((,) ∘ 𝐺)))
55 coass 5839 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓))
5655a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = (vol ∘ ((,) ∘ 𝑓)))
57 coass 5839 . . . . . . . . . . . . . 14 ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺))
5857a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝐺) = (vol ∘ ((,) ∘ 𝐺)))
5954, 56, 583eqtr4d 2808 . . . . . . . . . . . 12 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝐺))
6059fveq2d 6378 . . . . . . . . . . 11 (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6160adantr 472 . . . . . . . . . 10 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6253, 61eqtrd 2798 . . . . . . . . 9 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6362adantrl 707 . . . . . . . 8 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
6452, 63jca 507 . . . . . . 7 ((𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))) → (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
65 coeq2 5448 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ((,) ∘ 𝑔) = ((,) ∘ 𝐺))
6665rneqd 5520 . . . . . . . . . . 11 (𝑔 = 𝐺 → ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6766unieqd 4603 . . . . . . . . . 10 (𝑔 = 𝐺 ran ((,) ∘ 𝑔) = ran ((,) ∘ 𝐺))
6867sseq2d 3792 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐴 ran ((,) ∘ 𝑔) ↔ 𝐴 ran ((,) ∘ 𝐺)))
69 coeq2 5448 . . . . . . . . . . 11 (𝑔 = 𝐺 → ((vol ∘ (,)) ∘ 𝑔) = ((vol ∘ (,)) ∘ 𝐺))
7069fveq2d 6378 . . . . . . . . . 10 (𝑔 = 𝐺 → (Σ^‘((vol ∘ (,)) ∘ 𝑔)) = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))
7170eqeq2d 2774 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))))
7268, 71anbi12d 624 . . . . . . . 8 (𝑔 = 𝐺 → ((𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))) ↔ (𝐴 ran ((,) ∘ 𝐺) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝐺)))))
7372rspcev 3460 . . . . . . 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 579 . . . . . 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 3174 . . . . 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 3992 . . . . . . . . . 10 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
77 mapss 8104 . . . . . . . . . 10 (((ℝ × ℝ) ∈ V ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ))
7832, 76, 77mp2an 683 . . . . . . . . 9 (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ⊆ ((ℝ × ℝ) ↑𝑚 ℕ)
7978sseli 3756 . . . . . . . 8 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
8079adantr 472 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → 𝑔 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
81 simpr 477 . . . . . . 7 ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) ∧ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))) → (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
82 coeq2 5448 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((,) ∘ 𝑓) = ((,) ∘ 𝑔))
8382rneqd 5520 . . . . . . . . . . 11 (𝑓 = 𝑔 → ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8483unieqd 4603 . . . . . . . . . 10 (𝑓 = 𝑔 ran ((,) ∘ 𝑓) = ran ((,) ∘ 𝑔))
8584sseq2d 3792 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐴 ran ((,) ∘ 𝑓) ↔ 𝐴 ran ((,) ∘ 𝑔)))
86 coeq2 5448 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((vol ∘ (,)) ∘ 𝑓) = ((vol ∘ (,)) ∘ 𝑔))
8786fveq2d 6378 . . . . . . . . . 10 (𝑓 = 𝑔 → (Σ^‘((vol ∘ (,)) ∘ 𝑓)) = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))
8887eqeq2d 2774 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔))))
8985, 88anbi12d 624 . . . . . . . 8 (𝑓 = 𝑔 → ((𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))))
9089rspcev 3460 . . . . . . 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 579 . . . . . 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 3174 . . . . 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 200 . . . 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 3333 . . 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 2786 . 2 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑔) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑔)))}
961, 95ovolval3 41433 1 (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wcel 2155  wrex 3055  {crab 3058  Vcvv 3349  cin 3730  wss 3731  ifcif 4242  cop 4339   cuni 4593   class class class wbr 4808  cmpt 4887   × cxp 5274  ran crn 5277  ccom 5280  wf 6063  cfv 6067  (class class class)co 6841  1st c1st 7363  2nd c2nd 7364  𝑚 cmap 8059  infcinf 8553  cr 10187  *cxr 10326   < clt 10327  cle 10328  cn 11273  (,)cioo 12376  vol*covol 23519  volcvol 23520  Σ^csumge0 41148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-se 5236  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-isom 6076  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-of 7094  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-2o 7764  df-oadd 7767  df-er 7946  df-map 8061  df-pm 8062  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-fi 8523  df-sup 8554  df-inf 8555  df-oi 8621  df-card 9015  df-cda 9242  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-2 11334  df-3 11335  df-n0 11538  df-z 11624  df-uz 11886  df-q 11989  df-rp 12028  df-xneg 12145  df-xadd 12146  df-xmul 12147  df-ioo 12380  df-ico 12382  df-icc 12383  df-fz 12533  df-fzo 12673  df-fl 12800  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14125  df-re 14126  df-im 14127  df-sqrt 14261  df-abs 14262  df-clim 14505  df-rlim 14506  df-sum 14703  df-rest 16350  df-topgen 16371  df-psmet 20010  df-xmet 20011  df-met 20012  df-bl 20013  df-mopn 20014  df-top 20977  df-topon 20994  df-bases 21029  df-cmp 21469  df-ovol 23521  df-vol 23522  df-sumge0 41149
This theorem is referenced by:  ovolval4  41437
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