Theorem exp3val 10615

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 8037	. . 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 9631	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
4		1zzd 9347	. . . . . . . 8 |- ( A e. CC -> 1 e. ZZ )
5		fvconst2g 5773	. . . . . . . . 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 2270	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
8		mulcl 8001	. . . . . . . . 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 10538	. . . . . . 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 9330	. . . . . 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 5695	. . . 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 9444	. . . . . . 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 2195	. . . . . . . . . . . 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 9332	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 e. ZZ )
29		zleloe 9367	. . . . . . . . . . 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 9366	. . . . . . . . . 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 9442	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
36	35	lt0neg1d 8536	. . . . . . . 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 9330	. . . . . . 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 5695	. . . . 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 1360	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A # 0 )
44	41, 43, 39	exp3vallem 10614	. . . . 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 8802	. . . 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 9332	. . . . 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 9397	. . . . 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 3587	. . 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 9332	. . . 4 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> 0 e. ZZ )
52		zdceq 9395	. . . 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 3587	. 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 3630	. . . . . . . 8 |- ( x = A -> { x } = { A } )
56	55	xpeq2d 4684	. . . . . . 7 |- ( x = A -> ( NN X. { x } ) = ( NN X. { A } ) )
57	56	seqeq3d 10529	. . . . . 6 |- ( x = A -> seq 1 ( x. , ( NN X. { x } ) ) = seq 1 ( x. , ( NN X. { A } ) ) )
58	57	fveq1d 5557	. . . . 5 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) )
59	57	fveq1d 5557	. . . . . 6 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) )
60	59	oveq2d 5935	. . . . 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 3577	. . . 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 3576	. . 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 2200	. . . 4 |- ( y = N -> ( y = 0 <-> N = 0 ) )
64		breq2 4034	. . . . 5 |- ( y = N -> ( 0 < y <-> 0 < N ) )
65		fveq2 5555	. . . . 5 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
66		negeq 8214	. . . . . . 7 |- ( y = N -> -u y = -u N )
67	66	fveq2d 5559	. . . . . 6 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) )
68	67	oveq2d 5935	. . . . 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 3584	. . . 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 3582	. . 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 10613	. . 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 6054	. 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 1364   e. wcel 2164   ifcif 3558   {csn 3619   class class class wbr 4030   X. cxp 4658   -->wf 5251   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   x. cmul 7879   < clt 8056   <_ cle 8057   -ucneg 8193   # cap 8602   / cdiv 8693   NNcn 8984   ZZcz 9320   seqcseq 10521   ^cexp 10612

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522  df-exp 10613

This theorem is referenced by:  expnnval  10616  exp0  10617  expnegap0  10621