Theorem omsfval

Description: Value of the outer measure evaluated for a given set A . (Contributed by Thierry Arnoux, 15-Sep-2019) (Revised by AV, 4-Oct-2020)

	Ref	Expression
Assertion	omsfval	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → ( ( toOMeas ‘ 𝑅 ) ‘ 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) )

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) )
2		simp1	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → 𝑄 ∈ 𝑉 )
3	1 2	fexd	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → 𝑅 ∈ V )
4		omsval	⊢ ( 𝑅 ∈ V → ( toOMeas ‘ 𝑅 ) = ( 𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) )
5	3 4	syl	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → ( toOMeas ‘ 𝑅 ) = ( 𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) )
6		simpr	⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) ∧ 𝑎 = 𝐴 ) → 𝑎 = 𝐴 )
7	6	sseq1d	⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) ∧ 𝑎 = 𝐴 ) → ( 𝑎 ⊆ ∪ 𝑧 ↔ 𝐴 ⊆ ∪ 𝑧 ) )
8	7	anbi1d	⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) ∧ 𝑎 = 𝐴 ) → ( ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) ↔ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) ) )
9	8	rabbidv	⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) ∧ 𝑎 = 𝐴 ) → { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } = { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } )
10	9	mpteq1d	⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) ∧ 𝑎 = 𝐴 ) → ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) = ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) )
11	10	rneqd	⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) ∧ 𝑎 = 𝐴 ) → ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) = ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) )
12	11	infeq1d	⊢ ( ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) ∧ 𝑎 = 𝐴 ) → inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) )
13		simp3	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → 𝐴 ⊆ ∪ 𝑄 )
14		fdm	⊢ ( 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) → dom 𝑅 = 𝑄 )
15	14	3ad2ant2	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → dom 𝑅 = 𝑄 )
16	15	unieqd	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → ∪ dom 𝑅 = ∪ 𝑄 )
17	13 16	sseqtrrd	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → 𝐴 ⊆ ∪ dom 𝑅 )
18	2	uniexd	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → ∪ 𝑄 ∈ V )
19		ssexg	⊢ ( ( 𝐴 ⊆ ∪ 𝑄 ∧ ∪ 𝑄 ∈ V ) → 𝐴 ∈ V )
20	13 18 19	syl2anc	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → 𝐴 ∈ V )
21		elpwg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom 𝑅 ) )
22	20 21	syl	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → ( 𝐴 ∈ 𝒫 ∪ dom 𝑅 ↔ 𝐴 ⊆ ∪ dom 𝑅 ) )
23	17 22	mpbird	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → 𝐴 ∈ 𝒫 ∪ dom 𝑅 )
24		xrltso	⊢ < Or ℝ*
25		iccssxr	⊢ ( 0 [,] +∞ ) ⊆ ℝ*
26		soss	⊢ ( ( 0 [,] +∞ ) ⊆ ℝ* → ( < Or ℝ* → < Or ( 0 [,] +∞ ) ) )
27	25 26	ax-mp	⊢ ( < Or ℝ* → < Or ( 0 [,] +∞ ) )
28	24 27	mp1i	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → < Or ( 0 [,] +∞ ) )
29	28	infexd	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ∈ V )
30	5 12 23 29	fvmptd	⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ 𝐴 ⊆ ∪ 𝑄 ) → ( ( toOMeas ‘ 𝑅 ) ‘ 𝐴 ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( 𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) )