Theorem exp3val 9957

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 7504	. . 3 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ N = 0 ) -> 1 e. CC )
2		simp1 943	. . . . . . 7 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> A e. CC )
3		nnuz 9054	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
4		1zzd 8777	. . . . . . . 8 |- ( A e. CC -> 1 e. ZZ )
5		fvconst2g 5511	. . . . . . . . 9 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
6		simpl 107	. . . . . . . . 9 |- ( ( A e. CC /\ x e. NN ) -> A e. CC )
7	5, 6	eqeltrd 2164	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
8		mulcl 7469	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
9	8	adantl 271	. . . . . . . 8 |- ( ( A e. CC /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
10	3, 4, 7, 9	seqf 9880	. . . . . . 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 472	. . . . 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 944	. . . . . . 7 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> N e. ZZ )
14	13	ad2antrr 472	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> N e. ZZ )
15		simpr 108	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> 0 < N )
16		elnnz 8760	. . . . . 6 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
17	14, 15, 16	sylanbrc 408	. . . . 5 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> N e. NN )
18	12, 17	ffvelrnd 5435	. . . 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 472	. . . . . 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 472	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. ZZ )
21	20	znegcld 8870	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. ZZ )
22		simpr 108	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 < N )
23		simplr 497	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. N = 0 )
24		eqcom 2090	. . . . . . . . . . . 12 |- ( N = 0 <-> 0 = N )
25	23, 24	sylnib 636	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 = N )
26		ioran 704	. . . . . . . . . . 11 |- ( -. ( 0 < N \/ 0 = N ) <-> ( -. 0 < N /\ -. 0 = N ) )
27	22, 25, 26	sylanbrc 408	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. ( 0 < N \/ 0 = N ) )
28		0zd 8762	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 e. ZZ )
29		zleloe 8797	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> ( 0 < N \/ 0 = N ) ) )
30	28, 20, 29	syl2anc 403	. . . . . . . . . 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 633	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 <_ N )
32		zltnle 8796	. . . . . . . . . 10 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N < 0 <-> -. 0 <_ N ) )
33	20, 28, 32	syl2anc 403	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( N < 0 <-> -. 0 <_ N ) )
34	31, 33	mpbird 165	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N < 0 )
35	20	zred 8868	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
36	35	lt0neg1d 7993	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( N < 0 <-> 0 < -u N ) )
37	34, 36	mpbid 145	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 < -u N )
38		elnnz 8760	. . . . . . 7 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
39	21, 37, 38	sylanbrc 408	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. NN )
40	19, 39	ffvelrnd 5435	. . . . 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 472	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A e. CC )
42		simpll3 984	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( A # 0 \/ 0 <_ N ) )
43	31, 42	ecased 1285	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A # 0 )
44	41, 43, 39	exp3vallem 9956	. . . . 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 8248	. . . 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 8762	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> 0 e. ZZ )
47		simpl2 947	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> N e. ZZ )
48		zdclt 8824	. . . . 5 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
49	46, 47, 48	syl2anc 403	. . . 4 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> DECID 0 < N )
50	18, 45, 49	ifcldadc 3420	. . 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 8762	. . . 4 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> 0 e. ZZ )
52		zdceq 8822	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
53	13, 51, 52	syl2anc 403	. . 3 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> DECID N = 0 )
54	1, 50, 53	ifcldadc 3420	. 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 3457	. . . . . . . 8 |- ( x = A -> { x } = { A } )
56	55	xpeq2d 4462	. . . . . . 7 |- ( x = A -> ( NN X. { x } ) = ( NN X. { A } ) )
57	56	seqeq3d 9866	. . . . . 6 |- ( x = A -> seq 1 ( x. , ( NN X. { x } ) ) = seq 1 ( x. , ( NN X. { A } ) ) )
58	57	fveq1d 5307	. . . . 5 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) )
59	57	fveq1d 5307	. . . . . 6 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) )
60	59	oveq2d 5668	. . . . 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 3410	. . . 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 3409	. . 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 2094	. . . 4 |- ( y = N -> ( y = 0 <-> N = 0 ) )
64		breq2 3849	. . . . 5 |- ( y = N -> ( 0 < y <-> 0 < N ) )
65		fveq2 5305	. . . . 5 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
66		negeq 7675	. . . . . . 7 |- ( y = N -> -u y = -u N )
67	66	fveq2d 5309	. . . . . 6 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) )
68	67	oveq2d 5668	. . . . 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 3417	. . . 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 3415	. . 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 9955	. . 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	ovmpt2g 5779	. 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 1219	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 102   <-> wb 103   \/ wo 664  DECID wdc 780   /\ w3a 924   = wceq 1289   e. wcel 1438   ifcif 3393   {csn 3446   class class class wbr 3845   X. cxp 4436   -->wf 5011   ` cfv 5015  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351   x. cmul 7355   < clt 7522   <_ cle 7523   -ucneg 7654   # cap 8058   / cdiv 8139   NNcn 8422   ZZcz 8750   seqcseq 9852   ^cexp 9954

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-iseq 9853  df-seq3 9854  df-exp 9955

This theorem is referenced by:  expnnval  9958  exp0  9959  expnegap0  9963