Theorem exp3vallem 10955

Description: Lemma for exp3val 10956. 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 5690	. . . . 5 |- ( w = 1 -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) )
3	2	breq1d 4135	. . . 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 5690	. . . . 5 |- ( w = k -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) )
6	5	breq1d 4135	. . . 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 5690	. . . . 5 |- ( w = ( k + 1 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) )
9	8	breq1d 4135	. . . 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 5690	. . . . 5 |- ( w = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
12	11	breq1d 4135	. . . 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 9650	. . . . . 6 |- ( ph -> 1 e. ZZ )
15		exp3vallem.a	. . . . . . . 8 |- ( ph -> A e. CC )
16		elnnuz 9938	. . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
17	16	biimpri 133	. . . . . . . 8 |- ( x e. ( ZZ>= ` 1 ) -> x e. NN )
18		fvconst2g 5920	. . . . . . . 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 2315	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
22		mulcl 8296	. . . . . . 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 10877	. . . . 5 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
25		1nn 9294	. . . . . 6 |- 1 e. NN
26		fvconst2g 5920	. . . . . 6 |- ( ( A e. CC /\ 1 e. NN ) -> ( ( NN X. { A } ) ` 1 ) = A )
27	15, 25, 26	sylancl 417	. . . . 5 |- ( ph -> ( ( NN X. { A } ) ` 1 ) = A )
28	24, 27	eqtrd 2271	. . . 4 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = A )
29		exp3vallem.ap	. . . 4 |- ( ph -> A # 0 )
30	28, 29	eqbrtrd 4147	. . 3 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 )
31		nnuz 9937	. . . . . . . . . . 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 10879	. . . . . . . . . 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 5835	. . . . . . . 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 493	. . . . . . 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 493	. . . . . . 7 |- ( ( ( k e. NN /\ ph ) /\ ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) # 0 ) -> A # 0 )
41	37, 38, 39, 40	mulap0d 8976	. . . . . 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 9938	. . . . . . . . . . . 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 480	. . . . . . . . . 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 10880	. . . . . . . . 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 9298	. . . . . . . . . . 11 |- ( ( k e. NN /\ ph ) -> ( k + 1 ) e. NN )
49		fvconst2g 5920	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( k + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
50	15, 48, 49	syl2an2 602	. . . . . . . . . 10 |- ( ( k e. NN /\ ph ) -> ( ( NN X. { A } ) ` ( k + 1 ) ) = A )
51	50	oveq2d 6091	. . . . . . . . 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 2271	. . . . . . . 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 4135	. . . . . . 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 9299	. 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 1402   e. wcel 2209   {csn 3705   class class class wbr 4125   X. cxp 4767   -->wf 5368   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   # cap 8899   NNcn 9283   ZZ>=cuz 9900   seqcseq 10862

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863

This theorem is referenced by:  exp3val  10956