Theorem expivallem 9477

Description: Lemma for expival 9478. 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 5198	. . . . . 6 |- ( n = 1 -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) )
2	1	breq1d 3795	. . . . 5 |- ( n = 1 -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) # 0 ) )
3	2	imbi2d 228	. . . 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 5198	. . . . . 6 |- ( n = k -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) )
5	4	breq1d 3795	. . . . 5 |- ( n = k -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) )
6	5	imbi2d 228	. . . 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 5198	. . . . . 6 |- ( n = ( k + 1 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( k + 1 ) ) )
8	7	breq1d 3795	. . . . 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 228	. . . 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 5198	. . . . . 6 |- ( n = N -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) )
11	10	breq1d 3795	. . . . 5 |- ( n = N -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` n ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 ) )
12	11	imbi2d 228	. . . 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 108	. . . . 5 |- ( ( A e. CC /\ A # 0 ) -> A # 0 )
14		1zzd 8378	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 ) -> 1 e. ZZ )
15		cnex 7097	. . . . . . . . 9 |- CC e. _V
16	15	a1i 9	. . . . . . . 8 |- ( ( A e. CC /\ A # 0 ) -> CC e. _V )
17		elnnuz 8655	. . . . . . . . . . 11 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
18		fvconst2g 5396	. . . . . . . . . . 11 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
19	17, 18	sylan2br 282	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) = A )
20	19	adantlr 460	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) = A )
21		simpll 495	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
22	20, 21	eqeltrd 2155	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
23		mulcl 7100	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
24	23	adantl 271	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
25	14, 16, 22, 24	iseq1 9442	. . . . . . 7 |- ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
26		1nn 8050	. . . . . . . . 9 |- 1 e. NN
27		fvconst2g 5396	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
28	26, 27	mpan2 415	. . . . . . . 8 |- ( A e. CC -> ( ( NN X. { A } ) ` 1 ) = A )
29	28	adantr 270	. . . . . . 7 |- ( ( A e. CC /\ A # 0 ) -> ( ( NN X. { A } ) ` 1 ) = A )
30	25, 29	eqtrd 2113	. . . . . 6 |- ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) = A )
31	30	breq1d 3795	. . . . 5 |- ( ( A e. CC /\ A # 0 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) # 0 <-> A # 0 ) )
32	13, 31	mpbird 165	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` 1 ) # 0 )
33		simpl 107	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> k e. NN )
34		elnnuz 8655	. . . . . . . . . . 11 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
35	33, 34	sylib 120	. . . . . . . . . 10 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> k e. ( ZZ>= ` 1 ) )
36	35	adantr 270	. . . . . . . . 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 459	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
39	38	adantlr 460	. . . . . . . . 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 271	. . . . . . . . 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 9443	. . . . . . . 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 501	. . . . . . . 8 |- ( ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) /\ ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` k ) # 0 ) -> A e. CC )
43		simpr 108	. . . . . . . 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 502	. . . . . . . 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 7748	. . . . . . 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 271	. . . . . . . . . . 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 9445	. . . . . . . . . 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 497	. . . . . . . . . . . 12 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> A e. CC )
50	33	peano2nnd 8054	. . . . . . . . . . . 12 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( k + 1 ) e. NN )
51		fvconst2g 5396	. . . . . . . . . . . 12 |- ( ( A e. CC /\ ( k + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
52	49, 50, 51	syl2anc 403	. . . . . . . . . . 11 |- ( ( k e. NN /\ ( A e. CC /\ A # 0 ) ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
53	52	oveq2d 5548	. . . . . . . . . 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 2113	. . . . . . . . 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 3795	. . . . . . . 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 270	. . . . . . 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 165	. . . . . 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 356	. . . . 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 26	. . . 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 8055	. . 3 |- ( N e. NN -> ( ( A e. CC /\ A # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 ) )
61	60	impcom 123	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 )
62	61	3impa 1133	1 |- ( ( A e. CC /\ A # 0 /\ N e. NN ) -> ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) # 0 )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   class class class wbr 3785   X. cxp 4361   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   # cap 7681   NNcn 8039   ZZ>=cuz 8619   seqcseq 9431

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  expival  9478