Theorem exp3vallem 10325

Description: Lemma for exp3val 10326. 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 5429	. . . . 5 |- ( w = 1 -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) )
3	2	breq1d 3947	. . . 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 5429	. . . . 5 |- ( w = k -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) )
6	5	breq1d 3947	. . . 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 5429	. . . . 5 |- ( w = ( k + 1 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) )
9	8	breq1d 3947	. . . 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 5429	. . . . 5 |- ( w = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
12	11	breq1d 3947	. . . 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 9105	. . . . . 6 |- ( ph -> 1 e. ZZ )
15		exp3vallem.a	. . . . . . . 8 |- ( ph -> A e. CC )
16		elnnuz 9386	. . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
17	16	biimpri 132	. . . . . . . 8 |- ( x e. ( ZZ>= ` 1 ) -> x e. NN )
18		fvconst2g 5642	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
19	15, 17, 18	syl2an 287	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) = A )
20	15	adantr 274	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
21	19, 20	eqeltrd 2217	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
22		mulcl 7771	. . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
23	22	adantl 275	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
24	14, 21, 23	seq3-1 10264	. . . . 5 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
25		1nn 8755	. . . . . 6 |- 1 e. NN
26		fvconst2g 5642	. . . . . 6 |- ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
27	15, 25, 26	sylancl 410	. . . . 5 |- ( ph -> ( ( NN X. { A } ) ` 1 ) = A )
28	24, 27	eqtrd 2173	. . . 4 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = A )
29		exp3vallem.ap	. . . 4 |- ( ph -> A # 0 )
30	28, 29	eqbrtrd 3958	. . 3 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 )
31		nnuz 9385	. . . . . . . . . . 11 |- NN = ( ZZ>= ` 1 )
32	16, 21	sylan2b 285	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
33	31, 14, 32, 23	seqf 10265	. . . . . . . . . 10 |- ( ph -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
34	33	adantl 275	. . . . . . . . 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 5564	. . . . . . . 8 |- ( ( k e. NN /\ ph ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) e. CC )
37	36	adantr 274	. . . . . . 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 481	. . . . . . 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 481	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> A # 0 )
41	37, 38, 39, 40	mulap0d 8443	. . . . . 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 9386	. . . . . . . . . . . 12 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
43	42	biimpi 119	. . . . . . . . . . 11 |- ( k e. NN -> k e. ( ZZ>= ` 1 ) )
44	43	adantr 274	. . . . . . . . . 10 |- ( ( k e. NN /\ ph ) -> k e. ( ZZ>= ` 1 ) )
45	21	adantll 468	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ph ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
46	22	adantl 275	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ph ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
47	44, 45, 46	seq3p1 10266	. . . . . . . . 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 8759	. . . . . . . . . . 11 |- ( ( k e. NN /\ ph ) -> ( k + 1 ) e. NN )
49		fvconst2g 5642	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( k + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
50	15, 48, 49	syl2an2 584	. . . . . . . . . 10 |- ( ( k e. NN /\ ph ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
51	50	oveq2d 5798	. . . . . . . . 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 2173	. . . . . . . 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 3947	. . . . . . 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 274	. . . . . 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 362	. . . 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 8760	. 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 1332   e. wcel 1481   {csn 3532   class class class wbr 3937   X. cxp 4545   -->wf 5127   ` cfv 5131  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   # cap 8367   NNcn 8744   ZZ>=cuz 9350   seqcseq 10249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250

This theorem is referenced by:  exp3val  10326