Definition df-exp 10684

Description: Define exponentiation to nonnegative integer powers. For example, ( 5 ^ 2 ) = 2 5 (see ex-exp 15663).

This definition is not meant to be used directly; instead, exp0 10688 and expp1 10691 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts.

10-Jun-2005: The definition was extended to include zero exponents, so that 0 ^ 0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134 (see 0exp0e1 10689).

4-Jun-2014: The definition was extended to include negative integer exponents. For example, ( -u 3 ^ -u 2 ) = ( 1 / 9 ) (ex-exp 15663). The case x = 0 , y < 0 gives the value ( 1 / 0 ), so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)

Assertion

Ref	Expression
df-exp	|- ^ = ( x e. CC , y e. ZZ |-> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-exp

Step	Hyp	Ref	Expression
1		cexp 10683	. 2 class ^
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 7923	. . 3 class CC
5		cz 9372	. . 3 class ZZ
6	3	cv 1372	. . . . 5 class y
7		cc0 7925	. . . . 5 class 0
8	6, 7	wceq 1373	. . . 4 wff y = 0
9		c1 7926	. . . 4 class 1
10		clt 8107	. . . . . 6 class <
11	7, 6, 10	wbr 4044	. . . . 5 wff 0 < y
12		cmul 7930	. . . . . . 7 class x.
13		cn 9036	. . . . . . . 8 class NN
14	2	cv 1372	. . . . . . . . 9 class x
15	14	csn 3633	. . . . . . . 8 class { x }
16	13, 15	cxp 4673	. . . . . . 7 class ( NN X. { x } )
17	12, 16, 9	cseq 10592	. . . . . 6 class seq 1 ( x. , ( NN X. { x } ) )
18	6, 17	cfv 5271	. . . . 5 class ( seq 1 ( x. , ( NN X. { x } ) ) ` y )
19	6	cneg 8244	. . . . . . 7 class -u y
20	19, 17	cfv 5271	. . . . . 6 class ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y )
21		cdiv 8745	. . . . . 6 class /
22	9, 20, 21	co 5944	. . . . 5 class ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) )
23	11, 18, 22	cif 3571	. . . 4 class if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) )
24	8, 9, 23	cif 3571	. . 3 class if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) )
25	2, 3, 4, 5, 24	cmpo 5946	. 2 class ( x e. CC , y e. ZZ |-> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) ) )
26	1, 25	wceq 1373	1 wff ^ = ( x e. CC , y e. ZZ |-> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) ) )

This definition is referenced by:  exp3val  10686