Theorem exp3vallem 10717

Description: Lemma for exp3val 10718. 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 5594	. . . . 5 |- ( w = 1 -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) )
3	2	breq1d 4064	. . . 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 5594	. . . . 5 |- ( w = k -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` k ) )
6	5	breq1d 4064	. . . 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 5594	. . . . 5 |- ( w = ( k + 1 ) -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( k + 1 ) ) )
9	8	breq1d 4064	. . . 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 5594	. . . . 5 |- ( w = N -> ( seq 1 ( x. , ( NN X. { A } ) ) ` w ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
12	11	breq1d 4064	. . . 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 9429	. . . . . 6 |- ( ph -> 1 e. ZZ )
15		exp3vallem.a	. . . . . . . 8 |- ( ph -> A e. CC )
16		elnnuz 9715	. . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
17	16	biimpri 133	. . . . . . . 8 |- ( x e. ( ZZ>= ` 1 ) -> x e. NN )
18		fvconst2g 5816	. . . . . . . 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 2283	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
22		mulcl 8082	. . . . . . 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 10639	. . . . 5 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = ( ( NN X. { A } ) ` 1 ) )
25		1nn 9077	. . . . . 6 |- 1 e. NN
26		fvconst2g 5816	. . . . . 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 2239	. . . 4 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) = A )
29		exp3vallem.ap	. . . 4 |- ( ph -> A # 0 )
30	28, 29	eqbrtrd 4076	. . 3 |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` 1 ) # 0 )
31		nnuz 9714	. . . . . . . . . . 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 10641	. . . . . . . . . 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 5734	. . . . . . . 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 8761	. . . . . 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 9715	. . . . . . . . . . . 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 10642	. . . . . . . . 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 9081	. . . . . . . . . . 11 |- ( ( k e. NN /\ ph ) -> ( k + 1 ) e. NN )
49		fvconst2g 5816	. . . . . . . . . . 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 5978	. . . . . . . . 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 2239	. . . . . . . 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 4064	. . . . . . 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 9082	. 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 1373   e. wcel 2177   {csn 3638   class class class wbr 4054   X. cxp 4686   -->wf 5281   ` cfv 5285  (class class class)co 5962   CCcc 7953   0cc0 7955   1c1 7956   + caddc 7958   x. cmul 7960   # cap 8684   NNcn 9066   ZZ>=cuz 9678   seqcseq 10624

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-seqfrec 10625

This theorem is referenced by:  exp3val  10718