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Metamath Proof Explorer

Theorem mbfi1fseq

Description: A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function G and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014) (Revised by Mario Carneiro, 23-Aug-2014)

Ref Expression
Hypotheses mbfi1fseq.1 ⊢ ( 𝜑 → 𝐹 ∈ MblFn )
mbfi1fseq.2 ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ( 0 [,) +∞ ) )
Assertion mbfi1fseq ( 𝜑 → ∃ 𝑔 ( 𝑔 : ℕ ⟶ dom ∫1 ∧ ∀ 𝑛 ∈ ℕ ( 0𝑝 ∘r ≤ ( 𝑔 ‘ 𝑛 ) ∧ ( 𝑔 ‘ 𝑛 ) ∘r ≤ ( 𝑔 ‘ ( 𝑛 + 1 ) ) ) ∧ ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 mbfi1fseq.1 ⊢ ( 𝜑 → 𝐹 ∈ MblFn )
2 mbfi1fseq.2 ⊢ ( 𝜑 → 𝐹 : ℝ ⟶ ( 0 [,) +∞ ) )
3 oveq2 ⊢ ( 𝑗 = 𝑘 → ( 2 ↑ 𝑗 ) = ( 2 ↑ 𝑘 ) )
4 3 oveq2d ⊢ ( 𝑗 = 𝑘 → ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) = ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑘 ) ) )
5 4 fveq2d ⊢ ( 𝑗 = 𝑘 → ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) = ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑘 ) ) ) )
6 5 3 oveq12d ⊢ ( 𝑗 = 𝑘 → ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) = ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑘 ) ) ) / ( 2 ↑ 𝑘 ) ) )
7 fveq2 ⊢ ( 𝑧 = 𝑦 → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑦 ) )
8 7 fvoveq1d ⊢ ( 𝑧 = 𝑦 → ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑘 ) ) ) = ( ⌊ ‘ ( ( 𝐹 ‘ 𝑦 ) · ( 2 ↑ 𝑘 ) ) ) )
9 8 oveq1d ⊢ ( 𝑧 = 𝑦 → ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑘 ) ) ) / ( 2 ↑ 𝑘 ) ) = ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑦 ) · ( 2 ↑ 𝑘 ) ) ) / ( 2 ↑ 𝑘 ) ) )
10 6 9 cbvmpov ⊢ ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) = ( 𝑘 ∈ ℕ , 𝑦 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑦 ) · ( 2 ↑ 𝑘 ) ) ) / ( 2 ↑ 𝑘 ) ) )
11 eleq1w ⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ ( - 𝑚 [,] 𝑚 ) ↔ 𝑥 ∈ ( - 𝑚 [,] 𝑚 ) ) )
12 oveq2 ⊢ ( 𝑦 = 𝑥 → ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) = ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) )
13 12 breq1d ⊢ ( 𝑦 = 𝑥 → ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) ≤ 𝑚 ↔ ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 ) )
14 13 12 ifbieq1d ⊢ ( 𝑦 = 𝑥 → if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) , 𝑚 ) = if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑚 ) )
15 11 14 ifbieq1d ⊢ ( 𝑦 = 𝑥 → if ( 𝑦 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) , 𝑚 ) , 0 ) = if ( 𝑥 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑚 ) , 0 ) )
16 15 cbvmptv ⊢ ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) , 𝑚 ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑚 ) , 0 ) )
17 negeq ⊢ ( 𝑚 = 𝑘 → - 𝑚 = - 𝑘 )
18 id ⊢ ( 𝑚 = 𝑘 → 𝑚 = 𝑘 )
19 17 18 oveq12d ⊢ ( 𝑚 = 𝑘 → ( - 𝑚 [,] 𝑚 ) = ( - 𝑘 [,] 𝑘 ) )
20 19 eleq2d ⊢ ( 𝑚 = 𝑘 → ( 𝑥 ∈ ( - 𝑚 [,] 𝑚 ) ↔ 𝑥 ∈ ( - 𝑘 [,] 𝑘 ) ) )
21 oveq1 ⊢ ( 𝑚 = 𝑘 → ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) = ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) )
22 21 18 breq12d ⊢ ( 𝑚 = 𝑘 → ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 ↔ ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑘 ) )
23 22 21 18 ifbieq12d ⊢ ( 𝑚 = 𝑘 → if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑚 ) = if ( ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑘 , ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑘 ) )
24 20 23 ifbieq1d ⊢ ( 𝑚 = 𝑘 → if ( 𝑥 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑚 ) , 0 ) = if ( 𝑥 ∈ ( - 𝑘 [,] 𝑘 ) , if ( ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑘 , ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑘 ) , 0 ) )
25 24 mpteq2dv ⊢ ( 𝑚 = 𝑘 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑚 ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( - 𝑘 [,] 𝑘 ) , if ( ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑘 , ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑘 ) , 0 ) ) )
26 16 25 eqtrid ⊢ ( 𝑚 = 𝑘 → ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) , 𝑚 ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( - 𝑘 [,] 𝑘 ) , if ( ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑘 , ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑘 ) , 0 ) ) )
27 26 cbvmptv ⊢ ( 𝑚 ∈ ℕ ↦ ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( - 𝑚 [,] 𝑚 ) , if ( ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) ≤ 𝑚 , ( 𝑚 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑦 ) , 𝑚 ) , 0 ) ) ) = ( 𝑘 ∈ ℕ ↦ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( - 𝑘 [,] 𝑘 ) , if ( ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) ≤ 𝑘 , ( 𝑘 ( 𝑗 ∈ ℕ , 𝑧 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐹 ‘ 𝑧 ) · ( 2 ↑ 𝑗 ) ) ) / ( 2 ↑ 𝑗 ) ) ) 𝑥 ) , 𝑘 ) , 0 ) ) )
28 1 2 10 27 mbfi1fseqlem6 ⊢ ( 𝜑 → ∃ 𝑔 ( 𝑔 : ℕ ⟶ dom ∫1 ∧ ∀ 𝑛 ∈ ℕ ( 0𝑝 ∘r ≤ ( 𝑔 ‘ 𝑛 ) ∧ ( 𝑔 ‘ 𝑛 ) ∘r ≤ ( 𝑔 ‘ ( 𝑛 + 1 ) ) ) ∧ ∀ 𝑥 ∈ ℝ ( 𝑛 ∈ ℕ ↦ ( ( 𝑔 ‘ 𝑛 ) ‘ 𝑥 ) ) ⇝ ( 𝐹 ‘ 𝑥 ) ) )