Theorem isome 43133

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 5736	. . . . . . 7 ⊢ (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
3	1, 2	feq12d 6475	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
4	2	unieqd 4814	. . . . . . . 8 ⊢ (𝑥 = 𝑂 → ∪ dom 𝑥 = ∪ dom 𝑂)
5	4	pweqd 4516	. . . . . . 7 ⊢ (𝑥 = 𝑂 → 𝒫 ∪ dom 𝑥 = 𝒫 ∪ dom 𝑂)
6	2, 5	eqeq12d 2814	. . . . . 6 ⊢ (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 ∪ dom 𝑥 ↔ dom 𝑂 = 𝒫 ∪ dom 𝑂))
7	3, 6	anbi12d 633	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)))
8		fveq1 6644	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
9	8	eqeq1d 2800	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
10	7, 9	anbi12d 633	. . . 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 6644	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑧) = (𝑂‘𝑧))
12		fveq1 6644	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑦) = (𝑂‘𝑦))
13	11, 12	breq12d 5043	. . . . . 6 ⊢ (𝑥 = 𝑂 → ((𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝑦)))
14	13	ralbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
15	5, 14	raleqbidv 3354	. . . 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 633	. . 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 4516	. . . 4 ⊢ (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18		fveq1 6644	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∪ 𝑦) = (𝑂‘∪ 𝑦))
19		reseq1 5812	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥 ↾ 𝑦) = (𝑂 ↾ 𝑦))
20	19	fveq2d 6649	. . . . . 6 ⊢ (𝑥 = 𝑂 → (Σ^‘(𝑥 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑦)))
21	18, 20	breq12d 5043	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
22	21	imbi2d 344	. . . 4 ⊢ (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))) ↔ (𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
23	17, 22	raleqbidv 3354	. . 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 633	. 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 43129	. 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 3616	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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  dom cdm 5519   ↾ cres 5521  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ωcom 7560   ≼ cdom 8490  0cc0 10526  +∞cpnf 10661   ≤ cle 10665  [,]cicc 12729  Σ^{^}csumge0 43001  OutMeascome 43128

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

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-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ome 43129

This theorem is referenced by:  omef  43135  ome0  43136  omessle  43137  omedm  43138  omeunile  43144  0ome  43168  isomennd  43170