Definition df-dec 12839

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 31074. (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 12838	. 2 class ;𝐴𝐵
4		c9 12428	. . . . 5 class 9
5		c1 11225	. . . . 5 class 1
6		caddc 11227	. . . . 5 class +
7	4, 5, 6	co 7440	. . . 4 class (9 + 1)
8		cmul 11229	. . . 4 class ·
9	7, 1, 8	co 7440	. . 3 class ((9 + 1) · 𝐴)
10	9, 2, 6	co 7440	. 2 class (((9 + 1) · 𝐴) + 𝐵)
11	3, 10	wceq 1570	1 wff ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)

This definition is used by:  9p1e10  12840  dfdec10  12841  decex  12842  deceq1  12843  deceq2  12844  deccl  12853  10re  12861  25or6to4  43447  deccarry  48584