Theorem exp3val 10667

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 8070	. . 3 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ N = 0 ) -> 1 e. CC )
2		simp1 999	. . . . . . 7 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> A e. CC )
3		nnuz 9666	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
4		1zzd 9381	. . . . . . . 8 |- ( A e. CC -> 1 e. ZZ )
5		fvconst2g 5788	. . . . . . . . 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 2281	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
8		mulcl 8034	. . . . . . . . 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 10590	. . . . . . 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 1000	. . . . . . 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 9364	. . . . . 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 5710	. . . 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 9479	. . . . . . 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 528	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. N = 0 )
24		eqcom 2206	. . . . . . . . . . . 12 |- ( N = 0 <-> 0 = N )
25	23, 24	sylnib 677	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 = N )
26		ioran 753	. . . . . . . . . . 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 9366	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 e. ZZ )
29		zleloe 9401	. . . . . . . . . . 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 674	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 <_ N )
32		zltnle 9400	. . . . . . . . . 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 9477	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
36	35	lt0neg1d 8570	. . . . . . . 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 9364	. . . . . . 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 5710	. . . . 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 1040	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( A # 0 \/ 0 <_ N ) )
43	31, 42	ecased 1361	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A # 0 )
44	41, 43, 39	exp3vallem 10666	. . . . 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 8836	. . . 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 9366	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> 0 e. ZZ )
47		simpl2 1003	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> N e. ZZ )
48		zdclt 9432	. . . . 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 3599	. . 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 9366	. . . 4 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> 0 e. ZZ )
52		zdceq 9430	. . . 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 3599	. 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 3643	. . . . . . . 8 |- ( x = A -> { x } = { A } )
56	55	xpeq2d 4697	. . . . . . 7 |- ( x = A -> ( NN X. { x } ) = ( NN X. { A } ) )
57	56	seqeq3d 10581	. . . . . 6 |- ( x = A -> seq 1 ( x. , ( NN X. { x } ) ) = seq 1 ( x. , ( NN X. { A } ) ) )
58	57	fveq1d 5572	. . . . 5 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) )
59	57	fveq1d 5572	. . . . . 6 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) )
60	59	oveq2d 5950	. . . . 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 3589	. . . 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 3588	. . 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 2211	. . . 4 |- ( y = N -> ( y = 0 <-> N = 0 ) )
64		breq2 4047	. . . . 5 |- ( y = N -> ( 0 < y <-> 0 < N ) )
65		fveq2 5570	. . . . 5 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
66		negeq 8247	. . . . . . 7 |- ( y = N -> -u y = -u N )
67	66	fveq2d 5574	. . . . . 6 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) )
68	67	oveq2d 5950	. . . . 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 3596	. . . 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 3594	. . 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 10665	. . 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 6070	. 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 1294	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 709  DECID wdc 835   /\ w3a 980   = wceq 1372   e. wcel 2175   ifcif 3570   {csn 3632   class class class wbr 4043   X. cxp 4671   -->wf 5264   ` cfv 5268  (class class class)co 5934   CCcc 7905   0cc0 7907   1c1 7908   x. cmul 7912   < clt 8089   <_ cle 8090   -ucneg 8226   # cap 8636   / cdiv 8727   NNcn 9018   ZZcz 9354   seqcseq 10573   ^cexp 10664

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-frec 6467  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-n0 9278  df-z 9355  df-uz 9631  df-seqfrec 10574  df-exp 10665

This theorem is referenced by:  expnnval  10668  exp0  10669  expnegap0  10673