Theorem omeunile 40719

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 39217	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
6		pwexg 4850	. . . . . . . 8 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
8		ssexg 4804	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝒫 𝑋 ∧ 𝒫 𝑋 ∈ V) → 𝑌 ∈ V)
9	2, 7, 8	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ V)
10		elpwg 4166	. . . . . 6 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝒫 𝑋 ↔ 𝑌 ⊆ 𝒫 𝑋))
12	2, 11	mpbird 247	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝒫 𝑋)
13		omedm 40713	. . . . . . 7 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
14	3, 13	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
15	4	pweqi 4162	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
16	15	eqcomi 2631	. . . . . . 7 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
18	14, 17	eqtr2d 2657	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
19	18	pweqd 4163	. . . 4 ⊢ (𝜑 → 𝒫 𝒫 𝑋 = 𝒫 dom 𝑂)
20	12, 19	eleqtrd 2703	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝒫 dom 𝑂)
21		isome 40708	. . . . . 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))))))
22	3, 21	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))))))
23	3, 22	mpbid 222	. . . 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)))))
24	23	simprd 479	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
25		breq1 4656	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ ω ↔ 𝑌 ≼ ω))
26		unieq 4444	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ∪ 𝑦 = ∪ 𝑌)
27	26	fveq2d 6195	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑂‘∪ 𝑦) = (𝑂‘∪ 𝑌))
28		reseq2 5391	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑂 ↾ 𝑦) = (𝑂 ↾ 𝑌))
29	28	fveq2d 6195	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Σ^‘(𝑂 ↾ 𝑦)) = (Σ^‘(𝑂 ↾ 𝑌)))
30	27, 29	breq12d 4666	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)) ↔ (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
31	25, 30	imbi12d 334	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))) ↔ (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))))
32	31	rspcva 3307	. . 3 ⊢ ((𝑌 ∈ 𝒫 dom 𝑂 ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))) → (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
33	20, 24, 32	syl2anc 693	. 2 ⊢ (𝜑 → (𝑌 ≼ ω → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌))))
34	1, 33	mpd 15	1 ⊢ (𝜑 → (𝑂‘∪ 𝑌) ≤ (Σ^‘(𝑂 ↾ 𝑌)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114   ↾ cres 5116  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ≼ cdom 7953  0cc0 9936  +∞cpnf 10071   ≤ cle 10075  [,]cicc 12178  Σ^{^}csumge0 40579  OutMeascome 40703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ome 40704

This theorem is referenced by:  omeunle  40730  omeiunle  40731