Theorem deceq1

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

	Ref	Expression
Assertion	deceq1	⊢ ( 𝐴 = 𝐵 → ; 𝐴 𝐶 = ; 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝐴 = 𝐵 → ( ( 9 + 1 ) · 𝐴 ) = ( ( 9 + 1 ) · 𝐵 ) )
2	1	oveq1d	⊢ ( 𝐴 = 𝐵 → ( ( ( 9 + 1 ) · 𝐴 ) + 𝐶 ) = ( ( ( 9 + 1 ) · 𝐵 ) + 𝐶 ) )
3		df-dec	⊢ ; 𝐴 𝐶 = ( ( ( 9 + 1 ) · 𝐴 ) + 𝐶 )
4		df-dec	⊢ ; 𝐵 𝐶 = ( ( ( 9 + 1 ) · 𝐵 ) + 𝐶 )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ; 𝐴 𝐶 = ; 𝐵 𝐶 )