Theorem omeunile 46933

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 45481	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
6	5	pwexd 5321	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
7		ssexg 5264	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
8	2, 6, 7	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ V)
9		elpwg 4544	. . . . . 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 46927	. . . . . . 7 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
13	3, 12	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
14	4	pweqi 4557	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
15	14	eqcomi 2745	. . . . . . 7 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
17	13, 16	eqtr2d 2772	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
18	17	pweqd 4558	. . . 4 ⊢ (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
19	11, 18	eleqtrd 2838	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝒫 dom 𝑂)
20		isome 46922	. . . . . 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 5088	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
25		unieq 4861	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ∪ 𝑦 = ∪ 𝑌)
26	25	fveq2d 6844	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑂‘∪ 𝑦) = (𝑂‘∪ 𝑌))
27		reseq2 5939	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑂 ↾ 𝑦) = (𝑂 ↾ 𝑌))
28	27	fveq2d 6844	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Σ^‘(𝑂 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑌)))
29	26, 28	breq12d 5098	. . . . 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 3562	. . 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 585	. 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 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∪ cuni 4850   class class class wbr 5085  dom cdm 5631   ↾ cres 5633  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ωcom 7817   ≼ cdom 8891  0cc0 11038  +∞cpnf 11176   ≤ cle 11180  [,]cicc 13301  Σ^{^}csumge0 46790  OutMeascome 46917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ome 46918

This theorem is referenced by:  omeunle  46944  omeiunle  46945