Theorem expivallem 9256

Description: Lemma for expival 9257. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)

Assertion

Ref	Expression
expivallem	|- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 )

Proof of Theorem expivallem

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

Step	Hyp	Ref	Expression
1		fveq2 5178	. . . . . 6 |- ( n = 1 -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) )
2	1	breq1d 3774	. . . . 5 |- ( n = 1 -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) # 0 ) )
3	2	imbi2d 219	. . . 4 |- ( n = 1 -> ( ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 ) <-> ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) # 0 ) ) )
4		fveq2 5178	. . . . . 6 |- ( n = k -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) )
5	4	breq1d 3774	. . . . 5 |- ( n = k -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) )
6	5	imbi2d 219	. . . 4 |- ( n = k -> ( ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 ) <-> ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) ) )
7		fveq2 5178	. . . . . 6 |- ( n = ( k + 1 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) )
8	7	breq1d 3774	. . . . 5 |- ( n = ( k + 1 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) # 0 ) )
9	8	imbi2d 219	. . . 4 |- ( n = ( k + 1 ) -> ( ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 ) <-> ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) # 0 ) ) )
10		fveq2 5178	. . . . . 6 |- ( n = N -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) )
11	10	breq1d 3774	. . . . 5 |- ( n = N -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 ) )
12	11	imbi2d 219	. . . 4 |- ( n = N -> ( ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 ) <-> ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 ) ) )
13		simpr 103	. . . . 5 |- ( ( A e. CC /\ A # 0 ) -> A # 0 )
14		1zzd 8272	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 ) -> 1 e. ZZ )
15		cnex 7005	. . . . . . . . 9 |- CC e. _V
16	15	a1i 9	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 ) -> CC e. _V )
17		elnnuz 8509	. . . . . . . . . . 11 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
18		fvconst2g 5375	. . . . . . . . . . 11 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
19	17, 18	sylan2br 272	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) = A )
20	19	adantlr 446	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) = A )
21		simpll 481	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
22	20, 21	eqeltrd 2114	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
23		mulcl 7008	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
24	23	adantl 262	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
25	14, 16, 22, 24	iseq1 9222	. . . . . . 7 |- ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
26		1nn 7925	. . . . . . . . 9 |- 1 e. NN
27		fvconst2g 5375	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
28	26, 27	mpan2 401	. . . . . . . 8 |- ( A e. CC -> ( ( NN X. { A } ) ` 1 ) = A )
29	28	adantr 261	. . . . . . 7 |- ( ( A e. CC /\ A # 0 ) -> ( ( NN X. { A } ) ` 1 ) = A )
30	25, 29	eqtrd 2072	. . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) = A )
31	30	breq1d 3774	. . . . 5 |- ( ( A e. CC /\ A # 0 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) # 0 <-> A # 0 ) )
32	13, 31	mpbird 156	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) # 0 )
33		simpl 102	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> k e. NN )
34		elnnuz 8509	. . . . . . . . . . 11 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
35	33, 34	sylib 127	. . . . . . . . . 10 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> k e. ( ZZ>= ` 1 ) )
36	35	adantr 261	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> k e. ( ZZ>= ` 1 ) )
37	15	a1i 9	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> CC e. _V )
38	22	adantll 445	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
39	38	adantlr 446	. . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
40	23	adantl 262	. . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
41	36, 37, 39, 40	iseqcl 9223	. . . . . . . 8 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) e. CC )
42		simplrl 487	. . . . . . . 8 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> A e. CC )
43		simpr 103	. . . . . . . 8 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 )
44		simplrr 488	. . . . . . . 8 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> A # 0 )
45	41, 42, 43, 44	mulap0d 7639	. . . . . . 7 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) x. A ) # 0 )
46	15	a1i 9	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> CC e. _V )
47	23	adantl 262	. . . . . . . . . . 11 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
48	35, 46, 38, 47	iseqp1 9225	. . . . . . . . . 10 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) = ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) x. ( ( NN X. { A } ) ` ( k + 1 ) ) ) )
49		simprl 483	. . . . . . . . . . . 12 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> A e. CC )
50	33	peano2nnd 7929	. . . . . . . . . . . 12 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( k + 1 ) e. NN )
51		fvconst2g 5375	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( k + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
52	49, 50, 51	syl2anc 391	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
53	52	oveq2d 5528	. . . . . . . . . 10 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) x. ( ( NN X. { A } ) ` ( k + 1 ) ) ) = ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) x. A ) )
54	48, 53	eqtrd 2072	. . . . . . . . 9 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) = ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) x. A ) )
55	54	breq1d 3774	. . . . . . . 8 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) # 0 <-> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) x. A ) # 0 ) )
56	55	adantr 261	. . . . . . 7 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) # 0 <-> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) x. A ) # 0 ) )
57	45, 56	mpbird 156	. . . . . 6 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) # 0 )
58	57	exp31 346	. . . . 5 |- ( k e. NN -> ( ( A e. CC /\ A # 0 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) # 0 ) ) )
59	58	a2d 23	. . . 4 |- ( k e. NN -> ( ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) # 0 ) ) )
60	3, 6, 9, 12, 32, 59	nnind 7930	. . 3 |- ( N e. NN -> ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 ) )
61	60	impcom 116	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 )
62	61	3impa 1099	1 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   _Vcvv 2557   {csn 3375   class class class wbr 3764   X. cxp 4343   ` cfv 4902  (class class class)co 5512   CCcc 6887   0cc0 6889   1c1 6890   + caddc 6892   x. cmul 6894   # cap 7572   NNcn 7914   ZZ>=cuz 8473   seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212

This theorem is referenced by:  expival  9257