Theorem omeunile 41511

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 40033	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
6	5	pwexd 5081	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
7		ssexg 5031	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
8	2, 6, 7	syl2anc 579	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ V)
9		elpwg 4388	. . . . . 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 249	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋)
12		omedm 41505	. . . . . . 7 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
13	3, 12	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
14	4	pweqi 4384	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
15	14	eqcomi 2834	. . . . . . 7 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
17	13, 16	eqtr2d 2862	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
18	17	pweqd 4385	. . . 4 ⊢ (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
19	11, 18	eleqtrd 2908	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝒫 dom 𝑂)
20		isome 41500	. . . . . 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 224	. . . 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 491	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
24		breq1 4878	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
25		unieq 4668	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ∪ 𝑦 = ∪ 𝑌)
26	25	fveq2d 6441	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑂‘∪ 𝑦) = (𝑂‘∪ 𝑌))
27		reseq2 5628	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑂 ↾ 𝑦) = (𝑂 ↾ 𝑌))
28	27	fveq2d 6441	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Σ^‘(𝑂 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑌)))
29	26, 28	breq12d 4888	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
30	24, 29	imbi12d 336	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))) ↔ (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))))
31	30	rspcva 3524	. . 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 579	. 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 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   ⊆ wss 3798  ∅c0 4146  𝒫 cpw 4380  ∪ cuni 4660   class class class wbr 4875  dom cdm 5346   ↾ cres 5348  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  ωcom 7331   ≼ cdom 8226  0cc0 10259  +∞cpnf 10395   ≤ cle 10399  [,]cicc 12473  Σ^{^}csumge0 41368  OutMeascome 41495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ome 41496

This theorem is referenced by:  omeunle  41522  omeiunle  41523