Theorem omeunile 46426

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 44952	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
6	5	pwexd 5397	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
7		ssexg 5341	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
8	2, 6, 7	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ V)
9		elpwg 4625	. . . . . 6 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋))
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋))
11	2, 10	mpbird 257	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋)
12		omedm 46420	. . . . . . 7 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
13	3, 12	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
14	4	pweqi 4638	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
15	14	eqcomi 2749	. . . . . . 7 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
17	13, 16	eqtr2d 2781	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
18	17	pweqd 4639	. . . 4 ⊢ (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
19	11, 18	eleqtrd 2846	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝒫 dom 𝑂)
20		isome 46415	. . . . . 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 17	. . . . 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 232	. . . 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 495	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
24		breq1 5169	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
25		unieq 4942	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ∪ 𝑦 = ∪ 𝑌)
26	25	fveq2d 6924	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑂‘∪ 𝑦) = (𝑂‘∪ 𝑌))
27		reseq2 6004	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑂 ↾ 𝑦) = (𝑂 ↾ 𝑌))
28	27	fveq2d 6924	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Σ^‘(𝑂 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑌)))
29	26, 28	breq12d 5179	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
30	24, 29	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))) ↔ (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))))
31	30	rspcva 3633	. . 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 583	. 2 ⊢ (𝜑 → (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
33	1, 32	mpd 15	1 ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ωcom 7903   ≼ cdom 9001  0cc0 11184  +∞cpnf 11321   ≤ cle 11325  [,]cicc 13410  Σ^{^}csumge0 46283  OutMeascome 46410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ome 46411

This theorem is referenced by:  omeunle  46437  omeiunle  46438