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Metamath Proof Explorer

Theorem itg2ub

Description: The integral of a nonnegative real function F is an upper bound on the integrals of all simple functions G dominated by F . (Contributed by Mario Carneiro, 28-Jun-2014)

Ref Expression
Assertion itg2ub ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( ∫1 ‘ 𝐺 ) ≤ ( ∫2 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 eqid ⊢ { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } = { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) }
2 1 itg2lcl ⊢ { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } ⊆ ℝ*
3 1 itg2lr ⊢ ( ( 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( ∫1 ‘ 𝐺 ) ∈ { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } )
4 3 3adant1 ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( ∫1 ‘ 𝐺 ) ∈ { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } )
5 supxrub ⊢ ( ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } ⊆ ℝ* ∧ ( ∫1 ‘ 𝐺 ) ∈ { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } ) → ( ∫1 ‘ 𝐺 ) ≤ sup ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } , ℝ* , < ) )
6 2 4 5 sylancr ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( ∫1 ‘ 𝐺 ) ≤ sup ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } , ℝ* , < ) )
7 1 itg2val ⊢ ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → ( ∫2 ‘ 𝐹 ) = sup ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } , ℝ* , < ) )
8 7 3ad2ant1 ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( ∫2 ‘ 𝐹 ) = sup ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } , ℝ* , < ) )
9 6 8 breqtrrd ⊢ ( ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹 ) → ( ∫1 ‘ 𝐺 ) ≤ ( ∫2 ‘ 𝐹 ) )