Theorem dp2eq1

Description: Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015)

	Ref	Expression
Assertion	dp2eq1	⊢ ( 𝐴 = 𝐵 → _ 𝐴 𝐶 = _ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 + ( 𝐶 / ; 1 0 ) ) = ( 𝐵 + ( 𝐶 / ; 1 0 ) ) )
2		df-dp2	⊢ _ 𝐴 𝐶 = ( 𝐴 + ( 𝐶 / ; 1 0 ) )
3		df-dp2	⊢ _ 𝐵 𝐶 = ( 𝐵 + ( 𝐶 / ; 1 0 ) )
4	1 2 3	3eqtr4g	⊢ ( 𝐴 = 𝐵 → _ 𝐴 𝐶 = _ 𝐵 𝐶 )