Theorem omeunile 47111

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	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omeunile.x	⊢ 𝑋 = ∪ dom 𝑂
omeunile.y	⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋)
omeunile.ct	⊢ (𝜑 → 𝑌 ≼ ω)

Assertion

Ref	Expression
omeunile	⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))

Proof of Theorem omeunile

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omeunile.ct	. 2 ⊢ (𝜑 → 𝑌 ≼ ω)
2		omeunile.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝒫 𝑋)
3		omeunile.o	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
4		omeunile.x	. . . . . . . . 9 ⊢ 𝑋 = ∪ dom 𝑂
5	3, 4	unidmex 45662	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
6	5	pwexd 5351	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
7		ssexg 5294	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
8	2, 6, 7	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ V)
9		elpwg 4570	. . . . . 6 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋))
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋))
11	2, 10	mpbird 260	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋)
12		omedm 47105	. . . . . . 7 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
13	3, 12	syl 18	. . . . . 6 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
14	4	pweqi 4583	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
15	14	eqcomi 2778	. . . . . . 7 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
17	13, 16	eqtr2d 2805	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
18	17	pweqd 4584	. . . 4 ⊢ (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
19	11, 18	eleqtrd 2871	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝒫 dom 𝑂)
20		isome 47100	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑥 ∈ 𝒫 𝑦(𝑂‘𝑥) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
21	3, 20	syl 18	. . . . 5 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑥 ∈ 𝒫 𝑦(𝑂‘𝑥) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
22	3, 21	mpbid 235	. . . 4 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑥 ∈ 𝒫 𝑦(𝑂‘𝑥) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
23	22	simprd 500	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
24		breq1 5116	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
25		unieq 4887	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ∪ 𝑦 = ∪ 𝑌)
26	25	fveq2d 6886	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑂‘∪ 𝑦) = (𝑂‘∪ 𝑌))
27		reseq2 5974	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑂 ↾ 𝑦) = (𝑂 ↾ 𝑌))
28	27	fveq2d 6886	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Σ^‘(𝑂 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑌)))
29	26, 28	breq12d 5126	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
30	24, 29	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))) ↔ (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))))
31	30	rspcva 3588	. . 3 ⊢ ((𝑌 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))) → (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
32	19, 23, 31	syl2anc 595	. 2 ⊢ (𝜑 → (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
33	1, 32	mpd 16	1 ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876   class class class wbr 5113  dom cdm 5662   ↾ cres 5664  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ωcom 7862   ≼ cdom 8941  0cc0 11100  +∞cpnf 11240   ≤ cle 11244  [,]cicc 13375  Σ^{^}csumge0 46968  OutMeascome 47095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ome 47096

This theorem is referenced by:  omeunle  47122  omeiunle  47123