Theorem ome0 46947

Description: The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
ome0.1	⊢ (𝜑 → 𝑂 ∈ OutMeas)

Assertion

Ref	Expression
ome0	⊢ (𝜑 → (𝑂‘∅) = 0)

Proof of Theorem ome0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ome0.1	. . . 4 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		isome 46944	. . . . 5 ⊢ (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^‘(𝑂 ↾ 𝑥))))))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^‘(𝑂 ↾ 𝑥))))))
4	1, 3	mpbid 233	. . 3 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^‘(𝑂 ↾ 𝑥)))))
5	4	simplld 773	. 2 ⊢ (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0))
6	5	simprd 496	1 ⊢ (𝜑 → (𝑂‘∅) = 0)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∅c0 4268  𝒫 cpw 4536  ∪ cuni 4845   class class class wbr 5079  dom cdm 5625   ↾ cres 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  ωcom 7813   ≼ cdom 8888  0cc0 11036  +∞cpnf 11174   ≤ cle 11178  [,]cicc 13299  Σ^{^}csumge0 46812  OutMeascome 46939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ome 46940

This theorem is referenced by:  caragen0  46956  caragenunidm  46958  caratheodory  46978