Theorem expval 12862

Description: Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)

Assertion

Ref	Expression
expval	|- ( ( A e. CC /\ N e. ZZ ) -> ( A ^ N ) = if ( N = 0 , 1 , if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) )

Proof of Theorem expval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( x = A /\ y = N ) -> y = N )
2	1	eqeq1d 2624	. . 3 |- ( ( x = A /\ y = N ) -> ( y = 0 <-> N = 0 ) )
3	1	breq2d 4665	. . . 4 |- ( ( x = A /\ y = N ) -> ( 0 < y <-> 0 < N ) )
4		simpl 473	. . . . . . . 8 |- ( ( x = A /\ y = N ) -> x = A )
5	4	sneqd 4189	. . . . . . 7 |- ( ( x = A /\ y = N ) -> { x } = { A } )
6	5	xpeq2d 5139	. . . . . 6 |- ( ( x = A /\ y = N ) -> ( NN X. { x } ) = ( NN X. { A } ) )
7	6	seqeq3d 12809	. . . . 5 |- ( ( x = A /\ y = N ) -> seq 1 ( x. , ( NN X. { x } ) ) = seq 1 ( x. , ( NN X. { A } ) ) )
8	7, 1	fveq12d 6197	. . . 4 |- ( ( x = A /\ y = N ) -> ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
9	1	negeqd 10275	. . . . . 6 |- ( ( x = A /\ y = N ) -> -u y = -u N )
10	7, 9	fveq12d 6197	. . . . 5 |- ( ( x = A /\ y = N ) -> ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) )
11	10	oveq2d 6666	. . . 4 |- ( ( x = A /\ y = N ) -> ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) )
12	3, 8, 11	ifbieq12d 4113	. . 3 |- ( ( x = A /\ y = N ) -> if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) = if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) )
13	2, 12	ifbieq2d 4111	. 2 |- ( ( x = A /\ y = N ) -> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) ) = if ( N = 0 , 1 , if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) )
14		df-exp 12861	. 2 |- ^ = ( 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 ) ) ) ) )
15		1ex 10035	. . 3 |- 1 e. _V
16		fvex 6201	. . . 4 |- ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) e. _V
17		ovex 6678	. . . 4 |- ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) e. _V
18	16, 17	ifex 4156	. . 3 |- if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) e. _V
19	15, 18	ifex 4156	. 2 |- if ( N = 0 , 1 , if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) e. _V
20	13, 14, 19	ovmpt2a 6791	1 |- ( ( A e. CC /\ N e. ZZ ) -> ( A ^ N ) = if ( N = 0 , 1 , if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   {csn 4177   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   -ucneg 10267   / cdiv 10684   NNcn 11020   ZZcz 11377   seqcseq 12801   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-neg 10269  df-seq 12802  df-exp 12861

This theorem is referenced by:  expnnval  12863  exp0  12864  expneg  12868