Theorem deceq2

Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015) (Revised by AV, 6-Sep-2021)

	Ref	Expression
Assertion	deceq2	⊢ ( 𝐴 = 𝐵 → ; 𝐶 𝐴 = ; 𝐶 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝐴 = 𝐵 → ( ( ( 9 + 1 ) · 𝐶 ) + 𝐴 ) = ( ( ( 9 + 1 ) · 𝐶 ) + 𝐵 ) )
2		df-dec	⊢ ; 𝐶 𝐴 = ( ( ( 9 + 1 ) · 𝐶 ) + 𝐴 )
3		df-dec	⊢ ; 𝐶 𝐵 = ( ( ( 9 + 1 ) · 𝐶 ) + 𝐵 )
4	1 2 3	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ; 𝐶 𝐴 = ; 𝐶 𝐵 )