Theorem exp3vallem 10523

Description: Lemma for exp3val 10524. 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 5517	. . . . 5 |- ( w = 1 -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) )
3	2	breq1d 4015	. . . 4 |- ( w = 1 -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 ) )
4	3	imbi2d 230	. . 3 |- ( w = 1 -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 ) <-> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 ) ) )
5		fveq2 5517	. . . . 5 |- ( w = k -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) )
6	5	breq1d 4015	. . . 4 |- ( w = k -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) )
7	6	imbi2d 230	. . 3 |- ( w = k -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 ) <-> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) ) )
8		fveq2 5517	. . . . 5 |- ( w = ( k + 1 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) )
9	8	breq1d 4015	. . . 4 |- ( w = ( k + 1 ) -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) # 0 ) )
10	9	imbi2d 230	. . 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 5517	. . . . 5 |- ( w = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
12	11	breq1d 4015	. . . 4 |- ( w = N -> ( ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 <-> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 ) )
13	12	imbi2d 230	. . 3 |- ( w = N -> ( ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) # 0 ) <-> ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 ) ) )
14		1zzd 9282	. . . . . 6 |- ( ph -> 1 e. ZZ )
15		exp3vallem.a	. . . . . . . 8 |- ( ph -> A e. CC )
16		elnnuz 9566	. . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
17	16	biimpri 133	. . . . . . . 8 |- ( x e. ( ZZ>= ` 1 ) -> x e. NN )
18		fvconst2g 5732	. . . . . . . 8 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
19	15, 17, 18	syl2an 289	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) = A )
20	15	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
21	19, 20	eqeltrd 2254	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
22		mulcl 7940	. . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
23	22	adantl 277	. . . . . 6 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
24	14, 21, 23	seq3-1 10462	. . . . 5 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
25		1nn 8932	. . . . . 6 |- 1 e. NN
26		fvconst2g 5732	. . . . . 6 |- ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
27	15, 25, 26	sylancl 413	. . . . 5 |- ( ph -> ( ( NN X. { A } ) ` 1 ) = A )
28	24, 27	eqtrd 2210	. . . 4 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = A )
29		exp3vallem.ap	. . . 4 |- ( ph -> A # 0 )
30	28, 29	eqbrtrd 4027	. . 3 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 )
31		nnuz 9565	. . . . . . . . . . 11 |- NN = ( ZZ>= ` 1 )
32	16, 21	sylan2b 287	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) e. CC )
33	31, 14, 32, 23	seqf 10463	. . . . . . . . . 10 |- ( ph -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
34	33	adantl 277	. . . . . . . . 9 |- ( ( k e. NN /\ ph ) -> seq 1 ( x. , ( NN X. { A } ) ) : NN --> CC )
35		simpl 109	. . . . . . . . 9 |- ( ( k e. NN /\ ph ) -> k e. NN )
36	34, 35	ffvelcdmd 5654	. . . . . . . 8 |- ( ( k e. NN /\ ph ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) e. CC )
37	36	adantr 276	. . . . . . 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 489	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> A e. CC )
39		simpr 110	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 )
40	29	ad2antlr 489	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> A # 0 )
41	37, 38, 39, 40	mulap0d 8617	. . . . . 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 9566	. . . . . . . . . . . 12 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
43	42	biimpi 120	. . . . . . . . . . 11 |- ( k e. NN -> k e. ( ZZ>= ` 1 ) )
44	43	adantr 276	. . . . . . . . . 10 |- ( ( k e. NN /\ ph ) -> k e. ( ZZ>= ` 1 ) )
45	21	adantll 476	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ph ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
46	22	adantl 277	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ph ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
47	44, 45, 46	seq3p1 10464	. . . . . . . . 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 8936	. . . . . . . . . . 11 |- ( ( k e. NN /\ ph ) -> ( k + 1 ) e. NN )
49		fvconst2g 5732	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( k + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
50	15, 48, 49	syl2an2 594	. . . . . . . . . 10 |- ( ( k e. NN /\ ph ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
51	50	oveq2d 5893	. . . . . . . . 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 2210	. . . . . . . 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 4015	. . . . . . 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 276	. . . . . 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 167	. . . . 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 364	. . . 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 8937	. 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 104   <-> wb 105   = wceq 1353   e. wcel 2148   {csn 3594   class class class wbr 4005   X. cxp 4626   -->wf 5214   ` cfv 5218  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   # cap 8540   NNcn 8921   ZZ>=cuz 9530   seqcseq 10447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-seqfrec 10448

This theorem is referenced by:  exp3val  10524