Theorem isome 46514

Description: Express the predicate "𝑂 is an outer measure." Definition 113A of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
isome	⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))

Distinct variable group:   𝑦,𝑂,𝑧

Allowed substitution hints:   𝑉(𝑦,𝑧)

Proof of Theorem isome

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑂 → 𝑥 = 𝑂)
2		dmeq 5913	. . . . . . 7 ⊢ (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
3	1, 2	feq12d 6723	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
4	2	unieqd 4919	. . . . . . . 8 ⊢ (𝑥 = 𝑂 → ∪ dom 𝑥 = ∪ dom 𝑂)
5	4	pweqd 4616	. . . . . . 7 ⊢ (𝑥 = 𝑂 → 𝒫 ∪ dom 𝑥 = 𝒫 ∪ dom 𝑂)
6	2, 5	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 ∪ dom 𝑥 ↔ dom 𝑂 = 𝒫 ∪ dom 𝑂))
7	3, 6	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)))
8		fveq1 6904	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
9	8	eqeq1d 2738	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
10	7, 9	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝑂 → (((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ↔ ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0)))
11		fveq1 6904	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑧) = (𝑂‘𝑧))
12		fveq1 6904	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑦) = (𝑂‘𝑦))
13	11, 12	breq12d 5155	. . . . . 6 ⊢ (𝑥 = 𝑂 → ((𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝑦)))
14	13	ralbidv 3177	. . . . 5 ⊢ (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
15	5, 14	raleqbidv 3345	. . . 4 ⊢ (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
16	10, 15	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑂 → ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦)) ↔ (((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))))
17	2	pweqd 4616	. . . 4 ⊢ (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18		fveq1 6904	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∪ 𝑦) = (𝑂‘∪ 𝑦))
19		reseq1 5990	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥 ↾ 𝑦) = (𝑂 ↾ 𝑦))
20	19	fveq2d 6909	. . . . . 6 ⊢ (𝑥 = 𝑂 → (Σ^‘(𝑥 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑦)))
21	18, 20	breq12d 5155	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
22	21	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))) ↔ (𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
23	17, 22	raleqbidv 3345	. . 3 ⊢ (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))) ↔ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
24	16, 23	anbi12d 632	. 2 ⊢ (𝑥 = 𝑂 → (((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦)))) ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
25		df-ome 46510	. 2 ⊢ OutMeas = {𝑥 ∣ ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))))}
26	24, 25	elab2g 3679	1 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∅c0 4332  𝒫 cpw 4599  ∪ cuni 4906   class class class wbr 5142  dom cdm 5684   ↾ cres 5686  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  ωcom 7888   ≼ cdom 8984  0cc0 11156  +∞cpnf 11293   ≤ cle 11297  [,]cicc 13391  Σ^{^}csumge0 46382  OutMeascome 46509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ome 46510

This theorem is referenced by:  omef  46516  ome0  46517  omessle  46518  omedm  46519  omeunile  46525  0ome  46549  isomennd  46551