Theorem ntrivcvgfvn0 14631

Description: Any value of a product sequence that converges to a nonzero value is itself nonzero. (Contributed by Scott Fenton, 20-Dec-2017.)

Hypotheses

Ref	Expression
ntrivcvgfvn0.1	|- Z = ( ZZ>= ` M )
ntrivcvgfvn0.2	|- ( ph -> N e. Z )
ntrivcvgfvn0.3	|- ( ph -> seq M ( x. , F ) ~~> X )
ntrivcvgfvn0.4	|- ( ph -> X =/= 0 )
ntrivcvgfvn0.5	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )

Assertion

Ref	Expression
ntrivcvgfvn0	|- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )

Distinct variable groups:   k, F   ph, k   k, M   k, N   k, Z

Allowed substitution hint:   X( k)

Proof of Theorem ntrivcvgfvn0

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ntrivcvgfvn0.4	. 2 |- ( ph -> X =/= 0 )
2		fclim 14284	. . . . . . . 8 |- ~~> : dom ~~> --> CC
3		ffun 6048	. . . . . . . 8 |- ( ~~> : dom ~~> --> CC -> Fun ~~> )
4	2, 3	ax-mp 5	. . . . . . 7 |- Fun ~~>
5		ntrivcvgfvn0.3	. . . . . . 7 |- ( ph -> seq M ( x. , F ) ~~> X )
6		funbrfv 6234	. . . . . . 7 |- ( Fun ~~> -> ( seq M ( x. , F ) ~~> X -> ( ~~> ` seq M ( x. , F ) ) = X ) )
7	4, 5, 6	mpsyl 68	. . . . . 6 |- ( ph -> ( ~~> ` seq M ( x. , F ) ) = X )
8	7	adantr 481	. . . . 5 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ~~> ` seq M ( x. , F ) ) = X )
9		eqid 2622	. . . . . . 7 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
10		ntrivcvgfvn0.1	. . . . . . . . . 10 |- Z = ( ZZ>= ` M )
11		uzssz 11707	. . . . . . . . . 10 |- ( ZZ>= ` M ) C_ ZZ
12	10, 11	eqsstri 3635	. . . . . . . . 9 |- Z C_ ZZ
13		ntrivcvgfvn0.2	. . . . . . . . 9 |- ( ph -> N e. Z )
14	12, 13	sseldi 3601	. . . . . . . 8 |- ( ph -> N e. ZZ )
15	14	adantr 481	. . . . . . 7 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> N e. ZZ )
16		seqex 12803	. . . . . . . 8 |- seq M ( x. , F ) e. _V
17	16	a1i 11	. . . . . . 7 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> seq M ( x. , F ) e. _V )
18		0cnd 10033	. . . . . . 7 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> 0 e. CC )
19		fveq2 6191	. . . . . . . . . . 11 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
20	19	eqeq1d 2624	. . . . . . . . . 10 |- ( m = N -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` N ) = 0 ) )
21	20	imbi2d 330	. . . . . . . . 9 |- ( m = N -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` N ) = 0 ) ) )
22		fveq2 6191	. . . . . . . . . . 11 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
23	22	eqeq1d 2624	. . . . . . . . . 10 |- ( m = n -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` n ) = 0 ) )
24	23	imbi2d 330	. . . . . . . . 9 |- ( m = n -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` n ) = 0 ) ) )
25		fveq2 6191	. . . . . . . . . . 11 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
26	25	eqeq1d 2624	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) )
27	26	imbi2d 330	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
28		fveq2 6191	. . . . . . . . . . 11 |- ( m = k -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` k ) )
29	28	eqeq1d 2624	. . . . . . . . . 10 |- ( m = k -> ( ( seq M ( x. , F ) ` m ) = 0 <-> ( seq M ( x. , F ) ` k ) = 0 ) )
30	29	imbi2d 330	. . . . . . . . 9 |- ( m = k -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` m ) = 0 ) <-> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` k ) = 0 ) ) )
31		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` N ) = 0 )
32	31	a1i 11	. . . . . . . . 9 |- ( N e. ZZ -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` N ) = 0 ) )
33	13, 10	syl6eleq 2711	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. ( ZZ>= ` M ) )
34		uztrn 11704	. . . . . . . . . . . . . . . 16 |- ( ( n e. ( ZZ>= ` N ) /\ N e. ( ZZ>= ` M ) ) -> n e. ( ZZ>= ` M ) )
35	33, 34	sylan2 491	. . . . . . . . . . . . . . 15 |- ( ( n e. ( ZZ>= ` N ) /\ ph ) -> n e. ( ZZ>= ` M ) )
36	35	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> n e. ( ZZ>= ` M ) )
37		seqp1 12816	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
38	36, 37	syl 17	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
39		oveq1 6657	. . . . . . . . . . . . . 14 |- ( ( seq M ( x. , F ) ` n ) = 0 -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( 0 x. ( F ` ( n + 1 ) ) ) )
40	39	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) = ( 0 x. ( F ` ( n + 1 ) ) ) )
41		peano2uz 11741	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( ZZ>= ` N ) -> ( n + 1 ) e. ( ZZ>= ` N ) )
42	10	uztrn2 11705	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. Z /\ ( n + 1 ) e. ( ZZ>= ` N ) ) -> ( n + 1 ) e. Z )
43	13, 41, 42	syl2an 494	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( n + 1 ) e. Z )
44		ntrivcvgfvn0.5	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
45	44	ralrimiva 2966	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
46		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
47	46	eleq1d 2686	. . . . . . . . . . . . . . . . . . 19 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
48	47	rspcv 3305	. . . . . . . . . . . . . . . . . 18 |- ( ( n + 1 ) e. Z -> ( A. k e. Z ( F ` k ) e. CC -> ( F ` ( n + 1 ) ) e. CC ) )
49	45, 48	mpan9 486	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( n + 1 ) e. Z ) -> ( F ` ( n + 1 ) ) e. CC )
50	43, 49	syldan 487	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( F ` ( n + 1 ) ) e. CC )
51	50	ancoms 469	. . . . . . . . . . . . . . 15 |- ( ( n e. ( ZZ>= ` N ) /\ ph ) -> ( F ` ( n + 1 ) ) e. CC )
52	51	mul02d 10234	. . . . . . . . . . . . . 14 |- ( ( n e. ( ZZ>= ` N ) /\ ph ) -> ( 0 x. ( F ` ( n + 1 ) ) ) = 0 )
53	52	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( 0 x. ( F ` ( n + 1 ) ) ) = 0 )
54	38, 40, 53	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( n e. ( ZZ>= ` N ) /\ ph /\ ( seq M ( x. , F ) ` n ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 )
55	54	3exp 1264	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` N ) -> ( ph -> ( ( seq M ( x. , F ) ` n ) = 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
56	55	adantrd 484	. . . . . . . . . 10 |- ( n e. ( ZZ>= ` N ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ( seq M ( x. , F ) ` n ) = 0 -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
57	56	a2d 29	. . . . . . . . 9 |- ( n e. ( ZZ>= ` N ) -> ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` n ) = 0 ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = 0 ) ) )
58	21, 24, 27, 30, 32, 57	uzind4 11746	. . . . . . . 8 |- ( k e. ( ZZ>= ` N ) -> ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( seq M ( x. , F ) ` k ) = 0 ) )
59	58	impcom 446	. . . . . . 7 |- ( ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) /\ k e. ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) ` k ) = 0 )
60	9, 15, 17, 18, 59	climconst 14274	. . . . . 6 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> seq M ( x. , F ) ~~> 0 )
61		funbrfv 6234	. . . . . 6 |- ( Fun ~~> -> ( seq M ( x. , F ) ~~> 0 -> ( ~~> ` seq M ( x. , F ) ) = 0 ) )
62	4, 60, 61	mpsyl 68	. . . . 5 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> ( ~~> ` seq M ( x. , F ) ) = 0 )
63	8, 62	eqtr3d 2658	. . . 4 |- ( ( ph /\ ( seq M ( x. , F ) ` N ) = 0 ) -> X = 0 )
64	63	ex 450	. . 3 |- ( ph -> ( ( seq M ( x. , F ) ` N ) = 0 -> X = 0 ) )
65	64	necon3d 2815	. 2 |- ( ph -> ( X =/= 0 -> ( seq M ( x. , F ) ` N ) =/= 0 ) )
66	1, 65	mpd 15	1 |- ( ph -> ( seq M ( x. , F ) ` N ) =/= 0 )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  ntrivcvgtail  14632