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Theorem ovolval 24001
Description: The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)
Hypothesis
Ref Expression
ovolval.1 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
Assertion
Ref Expression
ovolval (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Distinct variable group:   𝐴,𝑓,𝑦
Allowed substitution hints:   𝑀(𝑦,𝑓)

Proof of Theorem ovolval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 reex 10616 . . 3 ℝ ∈ V
21elpw2 5239 . 2 (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
3 cleq1lem 14330 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
43rexbidv 3294 . . . . . 6 (𝑥 = 𝐴 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
54rabbidv 3478 . . . . 5 (𝑥 = 𝐴 → {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))})
6 ovolval.1 . . . . 5 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}
75, 6syl6eqr 2871 . . . 4 (𝑥 = 𝐴 → {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} = 𝑀)
87infeq1d 8929 . . 3 (𝑥 = 𝐴 → inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) = inf(𝑀, ℝ*, < ))
9 df-ovol 23992 . . 3 vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
10 xrltso 12522 . . . 4 < Or ℝ*
1110infex 8945 . . 3 inf(𝑀, ℝ*, < ) ∈ V
128, 9, 11fvmpt 6761 . 2 (𝐴 ∈ 𝒫 ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
132, 12sylbir 236 1 (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  {crab 3139  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830   × cxp 5546  ran crn 5549  ccom 5552  cfv 6348  (class class class)co 7145  m cmap 8395  supcsup 8892  infcinf 8893  cr 10524  1c1 10526   + caddc 10528  *cxr 10662   < clt 10663  cle 10664  cmin 10858  cn 11626  (,)cioo 12726  seqcseq 13357  abscabs 14581  vol*covol 23990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-ovol 23992
This theorem is referenced by:  ovolcl  24006  ovollb  24007  ovolgelb  24008  ovolge0  24009  ovolsslem  24012  ovolshft  24039  ovolicc2  24050  ismblfin  34814  ovolval2  42803
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