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

Theorem omeunile 46028
Description: The outer measure of the union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
omeunile.o (𝜑𝑂 ∈ OutMeas)
omeunile.x 𝑋 = dom 𝑂
omeunile.y (𝜑𝑌 ⊆ 𝒫 𝑋)
omeunile.ct (𝜑𝑌 ≼ ω)
Assertion
Ref Expression
omeunile (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))

Proof of Theorem omeunile
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omeunile.ct . 2 (𝜑𝑌 ≼ ω)
2 omeunile.y . . . . 5 (𝜑𝑌 ⊆ 𝒫 𝑋)
3 omeunile.o . . . . . . . . 9 (𝜑𝑂 ∈ OutMeas)
4 omeunile.x . . . . . . . . 9 𝑋 = dom 𝑂
53, 4unidmex 44553 . . . . . . . 8 (𝜑𝑋 ∈ V)
65pwexd 5379 . . . . . . 7 (𝜑 → 𝒫 𝑋 ∈ V)
7 ssexg 5324 . . . . . . 7 ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
82, 6, 7syl2anc 582 . . . . . 6 (𝜑𝑌 ∈ V)
9 elpwg 4607 . . . . . 6 (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
108, 9syl 17 . . . . 5 (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋𝑌 ⊆ 𝒫 𝑋))
112, 10mpbird 256 . . . 4 (𝜑𝑌 ∈ 𝒫 𝒫 𝑋)
12 omedm 46022 . . . . . . 7 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
133, 12syl 17 . . . . . 6 (𝜑 → dom 𝑂 = 𝒫 dom 𝑂)
144pweqi 4620 . . . . . . . 8 𝒫 𝑋 = 𝒫 dom 𝑂
1514eqcomi 2734 . . . . . . 7 𝒫 dom 𝑂 = 𝒫 𝑋
1615a1i 11 . . . . . 6 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
1713, 16eqtr2d 2766 . . . . 5 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1817pweqd 4621 . . . 4 (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
1911, 18eleqtrd 2827 . . 3 (𝜑𝑌 ∈ 𝒫 dom 𝑂)
20 isome 46017 . . . . . 6 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑥 ∈ 𝒫 𝑦(𝑂𝑥) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
213, 20syl 17 . . . . 5 (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑥 ∈ 𝒫 𝑦(𝑂𝑥) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
223, 21mpbid 231 . . . 4 (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑥 ∈ 𝒫 𝑦(𝑂𝑥) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
2322simprd 494 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))
24 breq1 5152 . . . . 5 (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
25 unieq 4920 . . . . . . 7 (𝑦 = 𝑌 𝑦 = 𝑌)
2625fveq2d 6900 . . . . . 6 (𝑦 = 𝑌 → (𝑂 𝑦) = (𝑂 𝑌))
27 reseq2 5980 . . . . . . 7 (𝑦 = 𝑌 → (𝑂𝑦) = (𝑂𝑌))
2827fveq2d 6900 . . . . . 6 (𝑦 = 𝑌 → (Σ^‘(𝑂𝑦)) = (Σ^‘(𝑂𝑌)))
2926, 28breq12d 5162 . . . . 5 (𝑦 = 𝑌 → ((𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)) ↔ (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3024, 29imbi12d 343 . . . 4 (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))) ↔ (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))))
3130rspcva 3604 . . 3 ((𝑌 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))) → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
3219, 23, 31syl2anc 582 . 2 (𝜑 → (𝑌 ≼ ω → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌))))
331, 32mpd 15 1 (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  wss 3944  c0 4322  𝒫 cpw 4604   cuni 4909   class class class wbr 5149  dom cdm 5678  cres 5680  wf 6545  cfv 6549  (class class class)co 7419  ωcom 7871  cdom 8962  0cc0 11140  +∞cpnf 11277  cle 11281  [,]cicc 13362  Σ^csumge0 45885  OutMeascome 46012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ome 46013
This theorem is referenced by:  omeunle  46039  omeiunle  46040
  Copyright terms: Public domain W3C validator