Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isome Structured version   Visualization version   GIF version

Theorem isome 41029
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 (𝑂𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
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.
StepHypRef Expression
1 id 22 . . . . . . 7 (𝑥 = 𝑂𝑥 = 𝑂)
2 dmeq 5356 . . . . . . 7 (𝑥 = 𝑂 → dom 𝑥 = dom 𝑂)
31, 2feq12d 6071 . . . . . 6 (𝑥 = 𝑂 → (𝑥:dom 𝑥⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
42unieqd 4478 . . . . . . . 8 (𝑥 = 𝑂 dom 𝑥 = dom 𝑂)
54pweqd 4196 . . . . . . 7 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
62, 5eqeq12d 2666 . . . . . 6 (𝑥 = 𝑂 → (dom 𝑥 = 𝒫 dom 𝑥 ↔ dom 𝑂 = 𝒫 dom 𝑂))
73, 6anbi12d 747 . . . . 5 (𝑥 = 𝑂 → ((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ↔ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂)))
8 fveq1 6228 . . . . . 6 (𝑥 = 𝑂 → (𝑥‘∅) = (𝑂‘∅))
98eqeq1d 2653 . . . . 5 (𝑥 = 𝑂 → ((𝑥‘∅) = 0 ↔ (𝑂‘∅) = 0))
107, 9anbi12d 747 . . . 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 6228 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑧) = (𝑂𝑧))
12 fveq1 6228 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
1311, 12breq12d 4698 . . . . . 6 (𝑥 = 𝑂 → ((𝑥𝑧) ≤ (𝑥𝑦) ↔ (𝑂𝑧) ≤ (𝑂𝑦)))
1413ralbidv 3015 . . . . 5 (𝑥 = 𝑂 → (∀𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
155, 14raleqbidv 3182 . . . 4 (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦) ↔ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)))
1610, 15anbi12d 747 . . 3 (𝑥 = 𝑂 → ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ↔ (((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦))))
172pweqd 4196 . . . 4 (𝑥 = 𝑂 → 𝒫 dom 𝑥 = 𝒫 dom 𝑂)
18 fveq1 6228 . . . . . 6 (𝑥 = 𝑂 → (𝑥 𝑦) = (𝑂 𝑦))
19 reseq1 5422 . . . . . . 7 (𝑥 = 𝑂 → (𝑥𝑦) = (𝑂𝑦))
2019fveq2d 6233 . . . . . 6 (𝑥 = 𝑂 → (Σ^‘(𝑥𝑦)) = (Σ^‘(𝑂𝑦)))
2118, 20breq12d 4698 . . . . 5 (𝑥 = 𝑂 → ((𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦)) ↔ (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
2221imbi2d 329 . . . 4 (𝑥 = 𝑂 → ((𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))) ↔ (𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2317, 22raleqbidv 3182 . . 3 (𝑥 = 𝑂 → (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))) ↔ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2416, 23anbi12d 747 . 2 (𝑥 = 𝑂 → (((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦)))) ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
25 df-ome 41025 . 2 OutMeas = {𝑥 ∣ ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥𝑧 ∈ 𝒫 𝑦(𝑥𝑧) ≤ (𝑥𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥 𝑦) ≤ (Σ^‘(𝑥𝑦))))}
2624, 25elab2g 3385 1 (𝑂𝑉 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  c0 3948  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  dom cdm 5143  cres 5145  wf 5922  cfv 5926  (class class class)co 6690  ωcom 7107  cdom 7995  0cc0 9974  +∞cpnf 10109  cle 10113  [,]cicc 12216  Σ^csumge0 40897  OutMeascome 41024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ome 41025
This theorem is referenced by:  omef  41031  ome0  41032  omessle  41033  omedm  41034  omeunile  41040  0ome  41064  isomennd  41066
  Copyright terms: Public domain W3C validator