Theorem isome 47598

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 23	. . . . . . 7 ⊢ (𝑥 = 𝑂 → 𝑥 = 𝑂)
2		dmeq 5884	. . . . . . 7 ⊢ (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
3	1, 2	feq12d 6698	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
4	2	unieqd 4879	. . . . . . . 8 ⊢ (𝑥 = 𝑂 → ∪ dom 𝑥 = ∪ dom 𝑂)
5	4	pweqd 4573	. . . . . . 7 ⊢ (𝑥 = 𝑂 → 𝒫 ∪ dom 𝑥 = 𝒫 ∪ dom 𝑂)
6	2, 5	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 ∪ dom 𝑥 ↔ dom 𝑂 = 𝒫 ∪ dom 𝑂))
7	3, 6	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)))
8		fveq1 6885	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
9	8	eqeq1d 2762	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
10	7, 9	anbi12d 644	. . . 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 6885	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑧) = (𝑂‘𝑧))
12		fveq1 6885	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥‘𝑦) = (𝑂‘𝑦))
13	11, 12	breq12d 5115	. . . . . 6 ⊢ (𝑥 = 𝑂 → ((𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ (𝑂‘𝑧) ≤ (𝑂‘𝑦)))
14	13	ralbidv 3185	. . . . 5 ⊢ (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)))
15	5, 14	raleqbidv 3334	. . . 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 644	. . 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 4573	. . . 4 ⊢ (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18		fveq1 6885	. . . . . 6 ⊢ (𝑥 = 𝑂 → (𝑥‘∪ 𝑦) = (𝑂‘∪ 𝑦))
19		reseq1 5963	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑥 ↾ 𝑦) = (𝑂 ↾ 𝑦))
20	19	fveq2d 6890	. . . . . 6 ⊢ (𝑥 = 𝑂 → (Σ^‘(𝑥 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑦)))
21	18, 20	breq12d 5115	. . . . 5 ⊢ (𝑥 = 𝑂 → ((𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
22	21	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^‘(𝑥 ↾ 𝑦))) ↔ (𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
23	17, 22	raleqbidv 3334	. . 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 644	. 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 47594	. 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 3633	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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102  dom cdm 5650   ↾ cres 5652  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421  ωcom 7878   ≼ cdom 8975  0cc0 11204  +∞cpnf 11344   ≤ cle 11348  [,]cicc 13483  Σ^{^}csumge0 47466  OutMeascome 47593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ome 47594

This theorem is used by:  omef  47600  ome0  47601  omessle  47602  omedm  47603  omeunile  47609  0ome  47633  isomennd  47635