Theorem dp2eq2

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

	Ref	Expression
Assertion	dp2eq2	⊢ ( 𝐴 = 𝐵 → _ 𝐶 𝐴 = _ 𝐶 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 / ; 1 0 ) = ( 𝐵 / ; 1 0 ) )
2	1	oveq2d	⊢ ( 𝐴 = 𝐵 → ( 𝐶 + ( 𝐴 / ; 1 0 ) ) = ( 𝐶 + ( 𝐵 / ; 1 0 ) ) )
3		df-dp2	⊢ _ 𝐶 𝐴 = ( 𝐶 + ( 𝐴 / ; 1 0 ) )
4		df-dp2	⊢ _ 𝐶 𝐵 = ( 𝐶 + ( 𝐵 / ; 1 0 ) )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → _ 𝐶 𝐴 = _ 𝐶 𝐵 )