Definition df-concat 13301

Description: Define the concatenation operator which combines two words. Definition in section 9.1 of [AhoHopUll] p. 318. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
df-concat	|- ++ = ( s e. _V , t e. _V |-> ( x e. ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) |-> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) ) )

Distinct variable group:   t, s, x

Detailed syntax breakdown of Definition df-concat

Step	Hyp	Ref	Expression
1		cconcat 13293	. 2 class ++
2		vs	. . 3 setvar s
3		vt	. . 3 setvar t
4		cvv 3200	. . 3 class _V
5		vx	. . . 4 setvar x
6		cc0 9936	. . . . 5 class 0
7	2	cv 1482	. . . . . . 7 class s
8		chash 13117	. . . . . . 7 class #
9	7, 8	cfv 5888	. . . . . 6 class ( # ` s )
10	3	cv 1482	. . . . . . 7 class t
11	10, 8	cfv 5888	. . . . . 6 class ( # ` t )
12		caddc 9939	. . . . . 6 class +
13	9, 11, 12	co 6650	. . . . 5 class ( ( # ` s ) + ( # ` t ) )
14		cfzo 12465	. . . . 5 class ..^
15	6, 13, 14	co 6650	. . . 4 class ( 0..^ ( ( # ` s ) + ( # ` t ) ) )
16	5	cv 1482	. . . . . 6 class x
17	6, 9, 14	co 6650	. . . . . 6 class ( 0..^ ( # ` s ) )
18	16, 17	wcel 1990	. . . . 5 wff x e. ( 0..^ ( # ` s ) )
19	16, 7	cfv 5888	. . . . 5 class ( s ` x )
20		cmin 10266	. . . . . . 7 class -
21	16, 9, 20	co 6650	. . . . . 6 class ( x - ( # ` s ) )
22	21, 10	cfv 5888	. . . . 5 class ( t ` ( x - ( # ` s ) ) )
23	18, 19, 22	cif 4086	. . . 4 class if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) )
24	5, 15, 23	cmpt 4729	. . 3 class ( x e. ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) |-> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) )
25	2, 3, 4, 4, 24	cmpt2 6652	. 2 class ( s e. _V , t e. _V |-> ( x e. ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) |-> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) ) )
26	1, 25	wceq 1483	1 wff ++ = ( s e. _V , t e. _V |-> ( x e. ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) |-> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) ) )

This definition is referenced by:  ccatfn  13357  ccatfval  13358