Theorem exp3val 10999

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 8348	. . 3 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ N = 0 ) -> 1 e. CC )
2		simp1 1028	. . . . . . 7 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> A e. CC )
3		nnuz 9974	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
4		1zzd 9682	. . . . . . . 8 |- ( A e. CC -> 1 e. ZZ )
5		fvconst2g 5932	. . . . . . . . 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 2315	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
8		mulcl 8312	. . . . . . . . 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 10922	. . . . . . 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 492	. . . . 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 1029	. . . . . . 7 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> N e. ZZ )
14	13	ad2antrr 492	. . . . . 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 9665	. . . . . 6 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
17	14, 15, 16	sylanbrc 421	. . . . 5 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ 0 < N ) -> N e. NN )
18	12, 17	ffvelcdmd 5847	. . . 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 492	. . . . . 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 492	. . . . . . . 8 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. ZZ )
21	20	znegcld 9781	. . . . . . 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 533	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. N = 0 )
24		eqcom 2240	. . . . . . . . . . . 12 |- ( N = 0 <-> 0 = N )
25	23, 24	sylnib 687	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 = N )
26		ioran 764	. . . . . . . . . . 11 |- ( -. ( 0 < N \/ 0 = N ) <-> ( -. 0 < N /\ -. 0 = N ) )
27	22, 25, 26	sylanbrc 421	. . . . . . . . . 10 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. ( 0 < N \/ 0 = N ) )
28		0zd 9667	. . . . . . . . . . 11 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> 0 e. ZZ )
29		zleloe 9702	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> ( 0 < N \/ 0 = N ) ) )
30	28, 20, 29	syl2anc 415	. . . . . . . . . 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 684	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -. 0 <_ N )
32		zltnle 9701	. . . . . . . . . 10 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N < 0 <-> -. 0 <_ N ) )
33	20, 28, 32	syl2anc 415	. . . . . . . . 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 9779	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> N e. RR )
36	35	lt0neg1d 8850	. . . . . . . 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 9665	. . . . . . 7 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
39	21, 37, 38	sylanbrc 421	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> -u N e. NN )
40	19, 39	ffvelcdmd 5847	. . . . 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 492	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A e. CC )
42		simpll3 1069	. . . . . . 7 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( A # 0 \/ 0 <_ N ) )
43	31, 42	ecased 1390	. . . . . 6 |- ( ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) /\ -. 0 < N ) -> A # 0 )
44	41, 43, 39	exp3vallem 10998	. . . . 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 9119	. . . 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 9667	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> 0 e. ZZ )
47		simpl2 1032	. . . . 5 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> N e. ZZ )
48		zdclt 9733	. . . . 5 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
49	46, 47, 48	syl2anc 415	. . . 4 |- ( ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) /\ -. N = 0 ) -> DECID 0 < N )
50	18, 45, 49	ifcldadc 3670	. . 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 9667	. . . 4 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> 0 e. ZZ )
52		zdceq 9731	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
53	13, 51, 52	syl2anc 415	. . 3 |- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> DECID N = 0 )
54	1, 50, 53	ifcldadc 3670	. 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 3720	. . . . . . . 8 |- ( x = A -> { x } = { A } )
56	55	xpeq2d 4800	. . . . . . 7 |- ( x = A -> ( NN X. { x } ) = ( NN X. { A } ) )
57	56	seqeq3d 10913	. . . . . 6 |- ( x = A -> seq 1 ( x. , ( NN X. { x } ) ) = seq 1 ( x. , ( NN X. { A } ) ) )
58	57	fveq1d 5700	. . . . 5 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) )
59	57	fveq1d 5700	. . . . . 6 |- ( x = A -> ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) )
60	59	oveq2d 6104	. . . . 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 3660	. . . 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 3659	. . 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 2245	. . . 4 |- ( y = N -> ( y = 0 <-> N = 0 ) )
64		breq2 4135	. . . . 5 |- ( y = N -> ( 0 < y <-> 0 < N ) )
65		fveq2 5698	. . . . 5 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
66		negeq 8526	. . . . . . 7 |- ( y = N -> -u y = -u N )
67	66	fveq2d 5702	. . . . . 6 |- ( y = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` -u y ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) )
68	67	oveq2d 6104	. . . . 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 3667	. . . 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 3665	. . 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 10997	. . 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 6226	. 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 1323	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 ) ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   ifcif 3638   {csn 3709   class class class wbr 4131   X. cxp 4774   -->wf 5376   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   x. cmul 8190   < clt 8366   <_ cle 8367   -ucneg 8505   # cap 8917   / cdiv 9010   NNcn 9313   ZZcz 9655   seqcseq 10905   ^cexp 10996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-exp 10997

This theorem is used by:  expnnval  11000  exp0  11001  expnegap0  11005