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Theorem ovolval3 45442
Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Ξ£^ and vol ∘ (,). See ovolval 24997 and ovolval2 45439 for alternative expressions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovolval3.a (πœ‘ β†’ 𝐴 βŠ† ℝ)
ovolval3.m 𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓)))}
Assertion
Ref Expression
ovolval3 (πœ‘ β†’ (vol*β€˜π΄) = inf(𝑀, ℝ*, < ))
Distinct variable groups:   𝐴,𝑓,𝑦   πœ‘,𝑓,𝑦
Allowed substitution hints:   𝑀(𝑦,𝑓)

Proof of Theorem ovolval3
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 ovolval3.a . . 3 (πœ‘ β†’ 𝐴 βŠ† ℝ)
2 eqid 2732 . . 3 {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))} = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))}
31, 2ovolval2 45439 . 2 (πœ‘ β†’ (vol*β€˜π΄) = inf({𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))}, ℝ*, < ))
4 ovolval3.m . . . . 5 𝑀 = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓)))}
5 reex 11203 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ ∈ V
65, 5xpex 7742 . . . . . . . . . . . . . . . . . . . . . 22 (ℝ Γ— ℝ) ∈ V
7 inss2 4229 . . . . . . . . . . . . . . . . . . . . . 22 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
8 mapss 8885 . . . . . . . . . . . . . . . . . . . . . 22 (((ℝ Γ— ℝ) ∈ V ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)) β†’ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) βŠ† ((ℝ Γ— ℝ) ↑m β„•))
96, 7, 8mp2an 690 . . . . . . . . . . . . . . . . . . . . 21 (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) βŠ† ((ℝ Γ— ℝ) ↑m β„•)
109sseli 3978 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑓 ∈ ((ℝ Γ— ℝ) ↑m β„•))
11 elmapi 8845 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ ((ℝ Γ— ℝ) ↑m β„•) β†’ 𝑓:β„•βŸΆ(ℝ Γ— ℝ))
1210, 11syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑓:β„•βŸΆ(ℝ Γ— ℝ))
1312ffvelcdmda 7086 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (π‘“β€˜π‘›) ∈ (ℝ Γ— ℝ))
14 1st2nd2 8016 . . . . . . . . . . . . . . . . . 18 ((π‘“β€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (π‘“β€˜π‘›) = ⟨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩)
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (π‘“β€˜π‘›) = ⟨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩)
1615fveq2d 6895 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((,)β€˜(π‘“β€˜π‘›)) = ((,)β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩))
17 df-ov 7414 . . . . . . . . . . . . . . . . . 18 ((1st β€˜(π‘“β€˜π‘›))(,)(2nd β€˜(π‘“β€˜π‘›))) = ((,)β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩)
1817eqcomi 2741 . . . . . . . . . . . . . . . . 17 ((,)β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩) = ((1st β€˜(π‘“β€˜π‘›))(,)(2nd β€˜(π‘“β€˜π‘›)))
1918a1i 11 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((,)β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩) = ((1st β€˜(π‘“β€˜π‘›))(,)(2nd β€˜(π‘“β€˜π‘›))))
2016, 19eqtrd 2772 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((,)β€˜(π‘“β€˜π‘›)) = ((1st β€˜(π‘“β€˜π‘›))(,)(2nd β€˜(π‘“β€˜π‘›))))
2120fveq2d 6895 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (volβ€˜((,)β€˜(π‘“β€˜π‘›))) = (volβ€˜((1st β€˜(π‘“β€˜π‘›))(,)(2nd β€˜(π‘“β€˜π‘›)))))
22 xp1st 8009 . . . . . . . . . . . . . . . 16 ((π‘“β€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(π‘“β€˜π‘›)) ∈ ℝ)
2313, 22syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(π‘“β€˜π‘›)) ∈ ℝ)
24 xp2nd 8010 . . . . . . . . . . . . . . . 16 ((π‘“β€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(π‘“β€˜π‘›)) ∈ ℝ)
2513, 24syl 17 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(π‘“β€˜π‘›)) ∈ ℝ)
26 elmapi 8845 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
2726adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
28 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•)
29 ovolfcl 24990 . . . . . . . . . . . . . . . . 17 ((𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ 𝑛 ∈ β„•) β†’ ((1st β€˜(π‘“β€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(π‘“β€˜π‘›)) ∈ ℝ ∧ (1st β€˜(π‘“β€˜π‘›)) ≀ (2nd β€˜(π‘“β€˜π‘›))))
3027, 28, 29syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((1st β€˜(π‘“β€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(π‘“β€˜π‘›)) ∈ ℝ ∧ (1st β€˜(π‘“β€˜π‘›)) ≀ (2nd β€˜(π‘“β€˜π‘›))))
3130simp3d 1144 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(π‘“β€˜π‘›)) ≀ (2nd β€˜(π‘“β€˜π‘›)))
32 volioo 25093 . . . . . . . . . . . . . . 15 (((1st β€˜(π‘“β€˜π‘›)) ∈ ℝ ∧ (2nd β€˜(π‘“β€˜π‘›)) ∈ ℝ ∧ (1st β€˜(π‘“β€˜π‘›)) ≀ (2nd β€˜(π‘“β€˜π‘›))) β†’ (volβ€˜((1st β€˜(π‘“β€˜π‘›))(,)(2nd β€˜(π‘“β€˜π‘›)))) = ((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›))))
3323, 25, 31, 32syl3anc 1371 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (volβ€˜((1st β€˜(π‘“β€˜π‘›))(,)(2nd β€˜(π‘“β€˜π‘›)))) = ((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›))))
3421, 33eqtrd 2772 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (volβ€˜((,)β€˜(π‘“β€˜π‘›))) = ((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›))))
35 ioof 13426 . . . . . . . . . . . . . . . 16 (,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ
36 ffun 6720 . . . . . . . . . . . . . . . 16 ((,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ β†’ Fun (,))
3735, 36ax-mp 5 . . . . . . . . . . . . . . 15 Fun (,)
3837a1i 11 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ Fun (,))
39 rexpssxrxp 11261 . . . . . . . . . . . . . . . 16 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
4039, 13sselid 3980 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (π‘“β€˜π‘›) ∈ (ℝ* Γ— ℝ*))
4135fdmi 6729 . . . . . . . . . . . . . . . . 17 dom (,) = (ℝ* Γ— ℝ*)
4241eqcomi 2741 . . . . . . . . . . . . . . . 16 (ℝ* Γ— ℝ*) = dom (,)
4342a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (ℝ* Γ— ℝ*) = dom (,))
4440, 43eleqtrd 2835 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (π‘“β€˜π‘›) ∈ dom (,))
45 fvco 6989 . . . . . . . . . . . . . 14 ((Fun (,) ∧ (π‘“β€˜π‘›) ∈ dom (,)) β†’ ((vol ∘ (,))β€˜(π‘“β€˜π‘›)) = (volβ€˜((,)β€˜(π‘“β€˜π‘›))))
4638, 44, 45syl2anc 584 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((vol ∘ (,))β€˜(π‘“β€˜π‘›)) = (volβ€˜((,)β€˜(π‘“β€˜π‘›))))
4715fveq2d 6895 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘›)) = ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩))
48 df-ov 7414 . . . . . . . . . . . . . . . 16 ((1st β€˜(π‘“β€˜π‘›))(abs ∘ βˆ’ )(2nd β€˜(π‘“β€˜π‘›))) = ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩)
4948eqcomi 2741 . . . . . . . . . . . . . . 15 ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩) = ((1st β€˜(π‘“β€˜π‘›))(abs ∘ βˆ’ )(2nd β€˜(π‘“β€˜π‘›)))
5049a1i 11 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜βŸ¨(1st β€˜(π‘“β€˜π‘›)), (2nd β€˜(π‘“β€˜π‘›))⟩) = ((1st β€˜(π‘“β€˜π‘›))(abs ∘ βˆ’ )(2nd β€˜(π‘“β€˜π‘›))))
5123recnd 11244 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(π‘“β€˜π‘›)) ∈ β„‚)
5225recnd 11244 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(π‘“β€˜π‘›)) ∈ β„‚)
53 eqid 2732 . . . . . . . . . . . . . . . . 17 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
5453cnmetdval 24294 . . . . . . . . . . . . . . . 16 (((1st β€˜(π‘“β€˜π‘›)) ∈ β„‚ ∧ (2nd β€˜(π‘“β€˜π‘›)) ∈ β„‚) β†’ ((1st β€˜(π‘“β€˜π‘›))(abs ∘ βˆ’ )(2nd β€˜(π‘“β€˜π‘›))) = (absβ€˜((1st β€˜(π‘“β€˜π‘›)) βˆ’ (2nd β€˜(π‘“β€˜π‘›)))))
5551, 52, 54syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((1st β€˜(π‘“β€˜π‘›))(abs ∘ βˆ’ )(2nd β€˜(π‘“β€˜π‘›))) = (absβ€˜((1st β€˜(π‘“β€˜π‘›)) βˆ’ (2nd β€˜(π‘“β€˜π‘›)))))
56 abssub 15275 . . . . . . . . . . . . . . . 16 (((1st β€˜(π‘“β€˜π‘›)) ∈ β„‚ ∧ (2nd β€˜(π‘“β€˜π‘›)) ∈ β„‚) β†’ (absβ€˜((1st β€˜(π‘“β€˜π‘›)) βˆ’ (2nd β€˜(π‘“β€˜π‘›)))) = (absβ€˜((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›)))))
5751, 52, 56syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (absβ€˜((1st β€˜(π‘“β€˜π‘›)) βˆ’ (2nd β€˜(π‘“β€˜π‘›)))) = (absβ€˜((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›)))))
5823, 25, 31abssubge0d 15380 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ (absβ€˜((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›)))) = ((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›))))
5955, 57, 583eqtrd 2776 . . . . . . . . . . . . . 14 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((1st β€˜(π‘“β€˜π‘›))(abs ∘ βˆ’ )(2nd β€˜(π‘“β€˜π‘›))) = ((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›))))
6047, 50, 593eqtrd 2776 . . . . . . . . . . . . 13 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘›)) = ((2nd β€˜(π‘“β€˜π‘›)) βˆ’ (1st β€˜(π‘“β€˜π‘›))))
6134, 46, 603eqtr4d 2782 . . . . . . . . . . . 12 ((𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) ∧ 𝑛 ∈ β„•) β†’ ((vol ∘ (,))β€˜(π‘“β€˜π‘›)) = ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘›)))
6261mpteq2dva 5248 . . . . . . . . . . 11 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ (𝑛 ∈ β„• ↦ ((vol ∘ (,))β€˜(π‘“β€˜π‘›))) = (𝑛 ∈ β„• ↦ ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘›))))
63 volioof 44782 . . . . . . . . . . . . 13 (vol ∘ (,)):(ℝ* Γ— ℝ*)⟢(0[,]+∞)
6463a1i 11 . . . . . . . . . . . 12 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ (vol ∘ (,)):(ℝ* Γ— ℝ*)⟢(0[,]+∞))
6539a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*))
6612, 65fssd 6735 . . . . . . . . . . . 12 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑓:β„•βŸΆ(ℝ* Γ— ℝ*))
67 fcompt 7133 . . . . . . . . . . . 12 (((vol ∘ (,)):(ℝ* Γ— ℝ*)⟢(0[,]+∞) ∧ 𝑓:β„•βŸΆ(ℝ* Γ— ℝ*)) β†’ ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ β„• ↦ ((vol ∘ (,))β€˜(π‘“β€˜π‘›))))
6864, 66, 67syl2anc 584 . . . . . . . . . . 11 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ ((vol ∘ (,)) ∘ 𝑓) = (𝑛 ∈ β„• ↦ ((vol ∘ (,))β€˜(π‘“β€˜π‘›))))
69 absf 15286 . . . . . . . . . . . . . 14 abs:β„‚βŸΆβ„
70 subf 11464 . . . . . . . . . . . . . 14 βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚
71 fco 6741 . . . . . . . . . . . . . 14 ((abs:β„‚βŸΆβ„ ∧ βˆ’ :(β„‚ Γ— β„‚)βŸΆβ„‚) β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
7269, 70, 71mp2an 690 . . . . . . . . . . . . 13 (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„
7372a1i 11 . . . . . . . . . . . 12 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ (abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„)
74 rr2sscn2 44155 . . . . . . . . . . . . . 14 (ℝ Γ— ℝ) βŠ† (β„‚ Γ— β„‚)
7574a1i 11 . . . . . . . . . . . . 13 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ (ℝ Γ— ℝ) βŠ† (β„‚ Γ— β„‚))
7612, 75fssd 6735 . . . . . . . . . . . 12 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑓:β„•βŸΆ(β„‚ Γ— β„‚))
77 fcompt 7133 . . . . . . . . . . . 12 (((abs ∘ βˆ’ ):(β„‚ Γ— β„‚)βŸΆβ„ ∧ 𝑓:β„•βŸΆ(β„‚ Γ— β„‚)) β†’ ((abs ∘ βˆ’ ) ∘ 𝑓) = (𝑛 ∈ β„• ↦ ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘›))))
7873, 76, 77syl2anc 584 . . . . . . . . . . 11 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ ((abs ∘ βˆ’ ) ∘ 𝑓) = (𝑛 ∈ β„• ↦ ((abs ∘ βˆ’ )β€˜(π‘“β€˜π‘›))))
7962, 68, 783eqtr4d 2782 . . . . . . . . . 10 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ ((vol ∘ (,)) ∘ 𝑓) = ((abs ∘ βˆ’ ) ∘ 𝑓))
8079fveq2d 6895 . . . . . . . . 9 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓)) = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))
8180eqeq2d 2743 . . . . . . . 8 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ (𝑦 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓)) ↔ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓))))
8281anbi2d 629 . . . . . . 7 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ ((𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓))) ↔ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))))
8382rexbiia 3092 . . . . . 6 (βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓))) ↔ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓))))
8483rabbii 3438 . . . . 5 {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((vol ∘ (,)) ∘ 𝑓)))} = {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))}
854, 84eqtr2i 2761 . . . 4 {𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))} = 𝑀
8685infeq1i 9475 . . 3 inf({𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))}, ℝ*, < ) = inf(𝑀, ℝ*, < )
8786a1i 11 . 2 (πœ‘ β†’ inf({𝑦 ∈ ℝ* ∣ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Ξ£^β€˜((abs ∘ βˆ’ ) ∘ 𝑓)))}, ℝ*, < ) = inf(𝑀, ℝ*, < ))
883, 87eqtrd 2772 1 (πœ‘ β†’ (vol*β€˜π΄) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βŸ¨cop 4634  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680  Fun wfun 6537  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976   ↑m cmap 8822  infcinf 9438  β„‚cc 11110  β„cr 11111  0cc0 11112  +∞cpnf 11247  β„*cxr 11249   < clt 11250   ≀ cle 11251   βˆ’ cmin 11446  β„•cn 12214  (,)cioo 13326  [,]cicc 13329  abscabs 15183  vol*covol 24986  volcvol 24987  Ξ£^csumge0 45157
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-ioo 13330  df-ico 13332  df-icc 13333  df-fz 13487  df-fzo 13630  df-fl 13759  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-rlim 15435  df-sum 15635  df-rest 17370  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-top 22403  df-topon 22420  df-bases 22456  df-cmp 22898  df-ovol 24988  df-vol 24989  df-sumge0 45158
This theorem is referenced by:  ovolval4lem2  45445
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