Theorem iccpart

Description: A special partition. Corresponds to fourierdlem2 in GS's mathbox. (Contributed by AV, 9-Jul-2020)

	Ref	Expression
Assertion	iccpart	⊢ ( 𝑀 ∈ ℕ → ( 𝑃 ∈ ( RePart ‘ 𝑀 ) ↔ ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		iccpval	⊢ ( 𝑀 ∈ ℕ → ( RePart ‘ 𝑀 ) = { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } )
2	1	eleq2d	⊢ ( 𝑀 ∈ ℕ → ( 𝑃 ∈ ( RePart ‘ 𝑀 ) ↔ 𝑃 ∈ { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } ) )
3		fveq1	⊢ ( 𝑝 = 𝑃 → ( 𝑝 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) )
4		fveq1	⊢ ( 𝑝 = 𝑃 → ( 𝑝 ‘ ( 𝑖 + 1 ) ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) )
5	3 4	breq12d	⊢ ( 𝑝 = 𝑃 → ( ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) )
6	5	ralbidv	⊢ ( 𝑝 = 𝑃 → ( ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ↔ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) )
7	6	elrab	⊢ ( 𝑃 ∈ { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } ↔ ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) )
8	2 7	bitrdi	⊢ ( 𝑀 ∈ ℕ → ( 𝑃 ∈ ( RePart ‘ 𝑀 ) ↔ ( 𝑃 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑃 ‘ 𝑖 ) < ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ) )