Definition df-dec 12722

Description: Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 30812. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)

Assertion

Ref	Expression
df-dec	⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)

Detailed syntax breakdown of Definition df-dec

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	cdc 12721	. 2 class ;𝐴𝐵
4		c9 12312	. . . . 5 class 9
5		c1 11111	. . . . 5 class 1
6		caddc 11113	. . . . 5 class +
7	4, 5, 6	co 7416	. . . 4 class (9 + 1)
8		cmul 11115	. . . 4 class ·
9	7, 1, 8	co 7416	. . 3 class ((9 + 1) · 𝐴)
10	9, 2, 6	co 7416	. 2 class (((9 + 1) · 𝐴) + 𝐵)
11	3, 10	wceq 1570	1 wff ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)

This definition is used by:  9p1e10  12723  dfdec10  12724  decex  12725  deceq1  12726  deceq2  12727  deccl  12736  10re  12744  25or6to4  43005  deccarry  48080