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Theorem omessle 43076
 Description: The outer measure of a set is greater than 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 41618 . . . . . 6 (𝜑𝑋 ∈ V)
5 omessle.b . . . . . 6 (𝜑𝐵𝑋)
64, 5ssexd 5204 . . . . 5 (𝜑𝐵 ∈ V)
76, 1ssexd 5204 . . . 4 (𝜑𝐴 ∈ V)
8 elpwg 4514 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
97, 8syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
101, 9mpbird 260 . 2 (𝜑𝐴 ∈ 𝒫 𝐵)
115, 3sseqtrdi 3992 . . . 4 (𝜑𝐵 dom 𝑂)
12 elpwg 4514 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
136, 12syl 17 . . . 4 (𝜑 → (𝐵 ∈ 𝒫 dom 𝑂𝐵 dom 𝑂))
1411, 13mpbird 260 . . 3 (𝜑𝐵 ∈ 𝒫 dom 𝑂)
15 isome 43072 . . . . . 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 235 . . . 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 769 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦))
19 pweq 4527 . . . . . 6 (𝑦 = 𝐵 → 𝒫 𝑦 = 𝒫 𝐵)
2019raleqdv 3392 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦)))
21 fveq2 6652 . . . . . . 7 (𝑦 = 𝐵 → (𝑂𝑦) = (𝑂𝐵))
2221breq2d 5054 . . . . . 6 (𝑦 = 𝐵 → ((𝑂𝑧) ≤ (𝑂𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝐵)))
2322ralbidv 3187 . . . . 5 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2420, 23bitrd 282 . . . 4 (𝑦 = 𝐵 → (∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦) ↔ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)))
2524rspcva 3596 . . 3 ((𝐵 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
2614, 18, 25syl2anc 587 . 2 (𝜑 → ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵))
27 fveq2 6652 . . . 4 (𝑧 = 𝐴 → (𝑂𝑧) = (𝑂𝐴))
2827breq1d 5052 . . 3 (𝑧 = 𝐴 → ((𝑂𝑧) ≤ (𝑂𝐵) ↔ (𝑂𝐴) ≤ (𝑂𝐵)))
2928rspcva 3596 . 2 ((𝐴 ∈ 𝒫 𝐵 ∧ ∀𝑧 ∈ 𝒫 𝐵(𝑂𝑧) ≤ (𝑂𝐵)) → (𝑂𝐴) ≤ (𝑂𝐵))
3010, 26, 29syl2anc 587 1 (𝜑 → (𝑂𝐴) ≤ (𝑂𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  Vcvv 3469   ⊆ wss 3908  ∅c0 4265  𝒫 cpw 4511  ∪ cuni 4813   class class class wbr 5042  dom cdm 5532   ↾ cres 5534  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140  ωcom 7565   ≼ cdom 8494  0cc0 10526  +∞cpnf 10661   ≤ cle 10665  [,]cicc 12729  Σ^csumge0 42940  OutMeascome 43067 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ome 43068 This theorem is referenced by:  omessre  43088  omeiunltfirp  43097  carageniuncllem2  43100  caratheodorylem2  43105  omess0  43112  caragencmpl  43113
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