Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omessle Structured version   Visualization version   GIF version

Theorem omessle 39192
Description: The outer measure of a set is larger or equal to the measure of a subset, Definition 113A (ii) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
omessle.o (𝜑𝑂 ∈ OutMeas)
omessle.x 𝑋 = dom 𝑂
omessle.b (𝜑𝐵𝑋)
omessle.a (𝜑𝐴𝐵)
Assertion
Ref Expression
omessle (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))

Proof of Theorem omessle
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omessle.a . . 3 (𝜑𝐴𝐵)
2 omessle.o . . . . . . 7 (𝜑𝑂 ∈ OutMeas)
3 omessle.x . . . . . . 7 𝑋 = dom 𝑂
42, 3unidmex 38045 . . . . . 6 (𝜑𝑋 ∈ V)
5 omessle.b . . . . . 6 (𝜑𝐵𝑋)
64, 5ssexd 4728 . . . . 5 (𝜑𝐵 ∈ V)
76, 1ssexd 4728 . . . 4 (𝜑𝐴 ∈ V)
8 elpwg 4115 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
97, 8syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
101, 9mpbird 245 . 2 (𝜑𝐴 ∈ 𝒫 𝐵)
115, 3syl6sseq 3613 . . . 4 (𝜑𝐵 dom 𝑂)
12 elpwg 4115 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
136, 12syl 17 . . . 4 (𝜑 → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
1411, 13mpbird 245 . . 3 (𝜑𝐵 ∈ 𝒫 dom 𝑂)
15 isome 39188 . . . . . 6 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
162, 15syl 17 . . . . 5 (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
172, 16mpbid 220 . . . 4 (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
1817simplrd 788 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦))
19 pweq 4110 . . . . . 6 (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
2019raleqdv 3120 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦)))
21 fveq2 6088 . . . . . . 7 (𝑦 = 𝐵 → (𝑂𝑦) = (𝑂𝐵))
2221breq2d 4589 . . . . . 6 (𝑦 = 𝐵 → ((𝑂𝑧) ≤ (𝑂𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝐵)))
2322ralbidv 2968 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2420, 23bitrd 266 . . . 4 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2524rspcva 3279 . . 3 ((𝐵 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
2614, 18, 25syl2anc 690 . 2 (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
27 fveq2 6088 . . . 4 (𝑧 = 𝐴 → (𝑂𝑧) = (𝑂𝐴))
2827breq1d 4587 . . 3 (𝑧 = 𝐴 → ((𝑂𝑧) ≤ (𝑂𝐵) ↔ (𝑂𝐴) ≤ (𝑂𝐵)))
2928rspcva 3279 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)) → (𝑂𝐴) ≤ (𝑂𝐵))
3010, 26, 29syl2anc 690 1 (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  wss 3539  c0 3873  𝒫 cpw 4107   cuni 4366   class class class wbr 4577  dom cdm 5028  cres 5030  wf 5786  cfv 5790  (class class class)co 6527  ωcom 6934  cdom 7816  0cc0 9792  +∞cpnf 9927  cle 9931  [,]cicc 12005  Σ^csumge0 39059  OutMeascome 39183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ome 39184
This theorem is referenced by:  omessre  39204  omeiunltfirp  39213  carageniuncllem2  39216  caratheodorylem2  39221  omess0  39228  caragencmpl  39229
  Copyright terms: Public domain W3C validator