Theorem exp3val 10927

Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)

Assertion

Ref	Expression
exp3val	|- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> ( 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 exp3val

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

Step	Hyp	Ref	Expression
1		1cnd 8306	. . 3 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ N = 0 ) -> 1 e. CC )
2		simp1 1024	. . . . . . 7 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> A e. CC )
3		nnuz 9908	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
4		1zzd 9621	. . . . . . . 8 |- ( A e. CC -> 1 e. ZZ )
5		fvconst2g 5903	. . . . . . . . 9 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
6		simpl 109	. . . . . . . . 9 |- ( ( A e. CC /\ x e. NN ) -> A e. CC )
7	5, 6	eqeltrd 2311	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
8		mulcl 8270	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
9	8	adantl 277	. . . . . . . 8 |- ( ( A e. CC /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
10	3, 4, 7, 9	seqf 10850	. . . . . . 7 |- ( A e. CC -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
11	2, 10	syl 14	. . . . . 6 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
12	11	ad2antrr 488	. . . . 5 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
13		simp2 1025	. . . . . . 7 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> N e. ZZ )
14	13	ad2antrr 488	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> N e. ZZ )
15		simpr 110	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> 0 < N )
16		elnnz 9604	. . . . . 6 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
17	14, 15, 16	sylanbrc 417	. . . . 5 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> N e. NN )
18	12, 17	ffvelcdmd 5818	. . . 4 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) e. CC )
19	11	ad2antrr 488	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
20	13	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. ZZ )
21	20	znegcld 9720	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. ZZ )
22		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 < N )
23		simplr 529	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. N = 0 )
24		eqcom 2236	. . . . . . . . . . . 12 |- ( N = 0 <-> 0 = N )
25	23, 24	sylnib 683	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 = N )
26		ioran 760	. . . . . . . . . . 11 |- ( -. ( 0 < N \/ 0 = N ) <-> ( -. 0 < N /\ -. 0 = N ) )
27	22, 25, 26	sylanbrc 417	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. ( 0 < N \/ 0 = N ) )
28		0zd 9606	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 e. ZZ )
29		zleloe 9641	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> ( 0 < N \/ 0 = N ) ) )
30	28, 20, 29	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( 0 <_ N <-> ( 0 < N \/ 0 = N ) ) )
31	27, 30	mtbird 680	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 <_ N )
32		zltnle 9640	. . . . . . . . . 10 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N < 0 <-> -. 0 <_ N ) )
33	20, 28, 32	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( N < 0 <-> -. 0 <_ N ) )
34	31, 33	mpbird 167	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N < 0 )
35	20	zred 9718	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
36	35	lt0neg1d 8806	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( N < 0 <-> 0 < -u N ) )
37	34, 36	mpbid 147	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 < -u N )
38		elnnz 9604	. . . . . . 7 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
39	21, 37, 38	sylanbrc 417	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. NN )
40	19, 39	ffvelcdmd 5818	. . . . 5 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) e. CC )
41	2	ad2antrr 488	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A e. CC )
42		simpll3 1065	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( A # 0 \/ 0 <_ N ) )
43	31, 42	ecased 1386	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A # 0 )
44	41, 43, 39	exp3vallem 10926	. . . . 5 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) # 0 )
45	40, 44	recclapd 9072	. . . 4 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) e. CC )
46		0zd 9606	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> 0 e. ZZ )
47		simpl2 1028	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> N e. ZZ )
48		zdclt 9672	. . . . 5 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
49	46, 47, 48	syl2anc 411	. . . 4 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> DECID 0 < N )
50	18, 45, 49	ifcldadc 3656	. . 3 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) e. CC )
51		0zd 9606	. . . 4 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> 0 e. ZZ )
52		zdceq 9670	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
53	13, 51, 52	syl2anc 411	. . 3 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> DECID N = 0 )
54	1, 50, 53	ifcldadc 3656	. 2 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> if ( N = 0 , 1 , if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) e. CC )
55		sneq 3705	. . . . . . . 8 |- ( x = A -> { x } = { A } )
56	55	xpeq2d 4778	. . . . . . 7 |- ( x = A -> ( NN X. { x } ) = ( NN X. { A } ) )
57	56	seqeq3d 10841	. . . . . 6 |- ( x = A -> seq 1 ( x. , ( NN X. { x } ) ) = seq 1 ( x. , ( NN X. { A } ) ) )
58	57	fveq1d 5677	. . . . 5 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) )
59	57	fveq1d 5677	. . . . . 6 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) )
60	59	oveq2d 6074	. . . . 5 |- ( x = A -> ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) ) )
61	58, 60	ifeq12d 3646	. . . 4 |- ( x = A -> if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) = if ( 0 < y , ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) ) ) )
62	61	ifeq2d 3645	. . 3 |- ( x = A -> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) ) = if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) ) ) ) )
63		eqeq1 2241	. . . 4 |- ( y = N -> ( y = 0 <-> N = 0 ) )
64		breq2 4118	. . . . 5 |- ( y = N -> ( 0 < y <-> 0 < N ) )
65		fveq2 5675	. . . . 5 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
66		negeq 8482	. . . . . . 7 |- ( y = N -> -u y = -u N )
67	66	fveq2d 5679	. . . . . 6 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) )
68	67	oveq2d 6074	. . . . 5 |- ( y = N -> ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) ) = ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) )
69	64, 65, 68	ifbieq12d 3653	. . . 4 |- ( y = N -> if ( 0 < y , ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) ) ) = if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) )
70	63, 69	ifbieq2d 3651	. . 3 |- ( y = N -> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -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 ) ) ) ) )
71		df-exp 10925	. . 3 |- ^ = ( 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 ) ) ) ) )
72	62, 70, 71	ovmpog 6196	. 2 |- ( ( A e. CC /\ N e. ZZ /\ if ( N = 0 , 1 , if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) e. CC ) -> ( 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 ) ) ) ) )
73	54, 72	syld3an3 1319	1 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> ( 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:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   ifcif 3624   {csn 3694   class class class wbr 4114   X. cxp 4752   -->wf 5353   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   x. cmul 8148   < clt 8324   <_ cle 8325   -ucneg 8461   # cap 8872   / cdiv 8963   NNcn 9254   ZZcz 9594   seqcseq 10833   ^cexp 10924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  expnnval  10928  exp0  10929  expnegap0  10933