Theorem exp3vallem 10019

Description: Lemma for exp3val 10020. 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.)

Hypotheses

Ref	Expression
exp3vallem.a	|- ( ph -> A e. CC )
exp3vallem.ap	|- ( ph -> A # 0 )
exp3vallem.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
exp3vallem	|- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 )

Proof of Theorem exp3vallem

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

Step	Hyp	Ref	Expression
1		exp3vallem.n	. 2 |- ( ph -> N e. NN )
2		fveq2 5320	. . . . 5 |- ( w = 1 -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) )
3	2	breq1d 3863	. . . 4 |- ( w = 1 -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 ) )
4	3	imbi2d 229	. . 3 |- ( w = 1 -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 ) <-> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 ) ) )
5		fveq2 5320	. . . . 5 |- ( w = k -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) )
6	5	breq1d 3863	. . . 4 |- ( w = k -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) )
7	6	imbi2d 229	. . 3 |- ( w = k -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 ) <-> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) ) )
8		fveq2 5320	. . . . 5 |- ( w = ( k + 1 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) )
9	8	breq1d 3863	. . . 4 |- ( w = ( k + 1 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 ) )
10	9	imbi2d 229	. . 3 |- ( w = ( k + 1 ) -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 ) <-> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 ) ) )
11		fveq2 5320	. . . . 5 |- ( w = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
12	11	breq1d 3863	. . . 4 |- ( w = N -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 ) )
13	12	imbi2d 229	. . 3 |- ( w = N -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 ) <-> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 ) ) )
14		1zzd 8840	. . . . . 6 |- ( ph -> 1 e. ZZ )
15		exp3vallem.a	. . . . . . . 8 |- ( ph -> A e. CC )
16		elnnuz 9118	. . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
17	16	biimpri 132	. . . . . . . 8 |- ( x e. ( ZZ>= ` 1 ) -> x e. NN )
18		fvconst2g 5527	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
19	15, 17, 18	syl2an 284	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) = A )
20	15	adantr 271	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
21	19, 20	eqeltrd 2165	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
22		mulcl 7532	. . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
23	22	adantl 272	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
24	14, 21, 23	seq3-1 9940	. . . . 5 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
25		1nn 8496	. . . . . 6 |- 1 e. NN
26		fvconst2g 5527	. . . . . 6 |- ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
27	15, 25, 26	sylancl 405	. . . . 5 |- ( ph -> ( ( NN X. { A } ) ` 1 ) = A )
28	24, 27	eqtrd 2121	. . . 4 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = A )
29		exp3vallem.ap	. . . 4 |- ( ph -> A # 0 )
30	28, 29	eqbrtrd 3873	. . 3 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 )
31		nnuz 9117	. . . . . . . . . . 11 |- NN = ( ZZ>= ` 1 )
32	16, 21	sylan2b 282	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
33	31, 14, 32, 23	seqf 9943	. . . . . . . . . 10 |- ( ph -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
34	33	adantl 272	. . . . . . . . 9 |- ( ( k e. NN /\ ph ) -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
35		simpl 108	. . . . . . . . 9 |- ( ( k e. NN /\ ph ) -> k e. NN )
36	34, 35	ffvelrnd 5451	. . . . . . . 8 |- ( ( k e. NN /\ ph ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) e. CC )
37	36	adantr 271	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) e. CC )
38	15	ad2antlr 474	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> A e. CC )
39		simpr 109	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 )
40	29	ad2antlr 474	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> A # 0 )
41	37, 38, 39, 40	mulap0d 8190	. . . . . 6 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) x. A ) # 0 )
42		elnnuz 9118	. . . . . . . . . . . 12 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
43	42	biimpi 119	. . . . . . . . . . 11 |- ( k e. NN -> k e. ( ZZ>= ` 1 ) )
44	43	adantr 271	. . . . . . . . . 10 |- ( ( k e. NN /\ ph ) -> k e. ( ZZ>= ` 1 ) )
45	21	adantll 461	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ph ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
46	22	adantl 272	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ph ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
47	44, 45, 46	seq3p1 9947	. . . . . . . . 9 |- ( ( k e. NN /\ ph ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) = ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) x. ( ( NN X. { A } ) ` ( k + 1 ) ) ) )
48	35	peano2nnd 8500	. . . . . . . . . . 11 |- ( ( k e. NN /\ ph ) -> ( k + 1 ) e. NN )
49		fvconst2g 5527	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( k + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
50	15, 48, 49	syl2an2 562	. . . . . . . . . 10 |- ( ( k e. NN /\ ph ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
51	50	oveq2d 5684	. . . . . . . . 9 |- ( ( k e. NN /\ ph ) -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) x. ( ( NN X. { A } ) ` ( k + 1 ) ) ) = ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) x. A ) )
52	47, 51	eqtrd 2121	. . . . . . . 8 |- ( ( k e. NN /\ ph ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) = ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) x. A ) )
53	52	breq1d 3863	. . . . . . 7 |- ( ( k e. NN /\ ph ) -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 <-> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) x. A ) # 0 ) )
54	53	adantr 271	. . . . . 6 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 <-> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) x. A ) # 0 ) )
55	41, 54	mpbird 166	. . . . 5 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 )
56	55	exp31 357	. . . 4 |- ( k e. NN -> ( ph -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 -> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 ) ) )
57	56	a2d 26	. . 3 |- ( k e. NN -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 ) ) )
58	4, 7, 10, 13, 30, 57	nnind 8501	. 2 |- ( N e. NN -> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 ) )
59	1, 58	mpcom 36	1 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   e. wcel 1439   {csn 3452   class class class wbr 3853   X. cxp 4452   -->wf 5026   ` cfv 5030  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414   + caddc 7416   x. cmul 7418   # cap 8121   NNcn 8485   ZZ>=cuz 9082   seqcseq 9915

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-frec 6172  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-inn 8486  df-n0 8737  df-z 8814  df-uz 9083  df-iseq 9916  df-seq3 9917

This theorem is referenced by:  exp3val  10020