Theorem isome 41502

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 5556	. . . . . . 7 ⊢ (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
3	1, 2	feq12d 6266	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
4	2	unieqd 4668	. . . . . . . 8 ⊢ (𝑥 = 𝑂 → ∪ dom 𝑥 = ∪ dom 𝑂)
5	4	pweqd 4383	. . . . . . 7 ⊢ (𝑥 = 𝑂 → 𝒫 ∪ dom 𝑥 = 𝒫 ∪ dom 𝑂)
6	2, 5	eqeq12d 2840	. . . . . 6 ⊢ (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 ∪ dom 𝑥 ↔ dom 𝑂 = 𝒫 ∪ dom 𝑂))
7	3, 6	anbi12d 626	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)))
8		fveq1 6432	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
9	8	eqeq1d 2827	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
10	7, 9	anbi12d 626	. . . 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 6432	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑧) = (𝑂‘𝑧))
12		fveq1 6432	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑦) = (𝑂‘𝑦))
13	11, 12	breq12d 4886	. . . . . 6 ⊢ (𝑥 = 𝑂 → ((𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝑦)))
14	13	ralbidv 3195	. . . . 5 ⊢ (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
15	5, 14	raleqbidv 3364	. . . 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 626	. . 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 4383	. . . 4 ⊢ (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18		fveq1 6432	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∪ 𝑦) = (𝑂‘∪ 𝑦))
19		reseq1 5623	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥 ↾ 𝑦) = (𝑂 ↾ 𝑦))
20	19	fveq2d 6437	. . . . . 6 ⊢ (𝑥 = 𝑂 → (Σ^‘(𝑥 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑦)))
21	18, 20	breq12d 4886	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
22	21	imbi2d 332	. . . 4 ⊢ (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))) ↔ (𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
23	17, 22	raleqbidv 3364	. . 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 626	. 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 41498	. 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 3574	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 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ∅c0 4144  𝒫 cpw 4378  ∪ cuni 4658   class class class wbr 4873  dom cdm 5342   ↾ cres 5344  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  ωcom 7326   ≼ cdom 8220  0cc0 10252  +∞cpnf 10388   ≤ cle 10392  [,]cicc 12466  Σ^{^}csumge0 41370  OutMeascome 41497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ome 41498

This theorem is referenced by:  omef  41504  ome0  41505  omessle  41506  omedm  41507  omeunile  41513  0ome  41537  isomennd  41539