Theorem ntrivcvgtail 14632

Description: A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.)

Hypotheses

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

Assertion

Ref	Expression
ntrivcvgtail	|- ( ph -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) )

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

Allowed substitution hint:   X( k)

Proof of Theorem ntrivcvgtail

Step	Hyp	Ref	Expression
1		fclim 14284	. . . . . . . 8 |- ~~> : dom ~~> --> CC
2		ffun 6048	. . . . . . . 8 |- ( ~~> : dom ~~> --> CC -> Fun ~~> )
3	1, 2	ax-mp 5	. . . . . . 7 |- Fun ~~>
4		ntrivcvgtail.3	. . . . . . 7 |- ( ph -> seq M ( x. , F ) ~~> X )
5		funbrfv 6234	. . . . . . 7 |- ( Fun ~~> -> ( seq M ( x. , F ) ~~> X -> ( ~~> ` seq M ( x. , F ) ) = X ) )
6	3, 4, 5	mpsyl 68	. . . . . 6 |- ( ph -> ( ~~> ` seq M ( x. , F ) ) = X )
7		ntrivcvgtail.4	. . . . . 6 |- ( ph -> X =/= 0 )
8	6, 7	eqnetrd 2861	. . . . 5 |- ( ph -> ( ~~> ` seq M ( x. , F ) ) =/= 0 )
9	4, 6	breqtrrd 4681	. . . . 5 |- ( ph -> seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) )
10	8, 9	jca 554	. . . 4 |- ( ph -> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) )
11	10	adantr 481	. . 3 |- ( ( ph /\ N = M ) -> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) )
12		seqeq1 12804	. . . . . . 7 |- ( N = M -> seq N ( x. , F ) = seq M ( x. , F ) )
13	12	fveq2d 6195	. . . . . 6 |- ( N = M -> ( ~~> ` seq N ( x. , F ) ) = ( ~~> ` seq M ( x. , F ) ) )
14	13	neeq1d 2853	. . . . 5 |- ( N = M -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 <-> ( ~~> ` seq M ( x. , F ) ) =/= 0 ) )
15	12, 13	breq12d 4666	. . . . 5 |- ( N = M -> ( seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) <-> seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) )
16	14, 15	anbi12d 747	. . . 4 |- ( N = M -> ( ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) <-> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) ) )
17	16	adantl 482	. . 3 |- ( ( ph /\ N = M ) -> ( ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) <-> ( ( ~~> ` seq M ( x. , F ) ) =/= 0 /\ seq M ( x. , F ) ~~> ( ~~> ` seq M ( x. , F ) ) ) ) )
18	11, 17	mpbird 247	. 2 |- ( ( ph /\ N = M ) -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) )
19		ntrivcvgtail.1	. . . . . 6 |- Z = ( ZZ>= ` M )
20		simpr 477	. . . . . . 7 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
21	20, 19	syl6eleqr 2712	. . . . . 6 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. Z )
22		ntrivcvgtail.5	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
23	22	adantlr 751	. . . . . 6 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ k e. Z ) -> ( F ` k ) e. CC )
24	4	adantr 481	. . . . . 6 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq M ( x. , F ) ~~> X )
25	7	adantr 481	. . . . . . 7 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> X =/= 0 )
26	19, 21, 24, 25, 23	ntrivcvgfvn0 14631	. . . . . 6 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ` ( N - 1 ) ) =/= 0 )
27	19, 21, 23, 24, 26	clim2div 14621	. . . . 5 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) )
28		funbrfv 6234	. . . . 5 |- ( Fun ~~> -> ( seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) = ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) ) )
29	3, 27, 28	mpsyl 68	. . . 4 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) = ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) )
30		climcl 14230	. . . . . . 7 |- ( seq M ( x. , F ) ~~> X -> X e. CC )
31	4, 30	syl 17	. . . . . 6 |- ( ph -> X e. CC )
32	31	adantr 481	. . . . 5 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> X e. CC )
33		ntrivcvgtail.2	. . . . . . . . 9 |- ( ph -> N e. Z )
34		eluzel2 11692	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
35	34, 19	eleq2s 2719	. . . . . . . . 9 |- ( N e. Z -> M e. ZZ )
36	33, 35	syl 17	. . . . . . . 8 |- ( ph -> M e. ZZ )
37	19, 36, 22	prodf 14619	. . . . . . 7 |- ( ph -> seq M ( x. , F ) : Z --> CC )
38	19	feq2i 6037	. . . . . . 7 |- ( seq M ( x. , F ) : Z --> CC <-> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
39	37, 38	sylib 208	. . . . . 6 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
40	39	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ` ( N - 1 ) ) e. CC )
41	32, 40, 25, 26	divne0d 10817	. . . 4 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( X / ( seq M ( x. , F ) ` ( N - 1 ) ) ) =/= 0 )
42	29, 41	eqnetrd 2861	. . 3 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) =/= 0 )
43	27, 29	breqtrrd 4681	. . 3 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) )
44		uzssz 11707	. . . . . . . . . . . 12 |- ( ZZ>= ` M ) C_ ZZ
45	19, 44	eqsstri 3635	. . . . . . . . . . 11 |- Z C_ ZZ
46	45, 33	sseldi 3601	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
47	46	zcnd 11483	. . . . . . . . 9 |- ( ph -> N e. CC )
48	47	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. CC )
49		1cnd 10056	. . . . . . . 8 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> 1 e. CC )
50	48, 49	npcand 10396	. . . . . . 7 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( N - 1 ) + 1 ) = N )
51	50	seqeq1d 12807	. . . . . 6 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> seq ( ( N - 1 ) + 1 ) ( x. , F ) = seq N ( x. , F ) )
52	51	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) = ( ~~> ` seq N ( x. , F ) ) )
53	52	neeq1d 2853	. . . 4 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) =/= 0 <-> ( ~~> ` seq N ( x. , F ) ) =/= 0 ) )
54	51, 52	breq12d 4666	. . . 4 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) <-> seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) )
55	53, 54	anbi12d 747	. . 3 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) =/= 0 /\ seq ( ( N - 1 ) + 1 ) ( x. , F ) ~~> ( ~~> ` seq ( ( N - 1 ) + 1 ) ( x. , F ) ) ) <-> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) ) )
56	42, 43, 55	mpbi2and 956	. 2 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) )
57	33, 19	syl6eleq 2711	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
58		uzm1 11718	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
59	57, 58	syl 17	. 2 |- ( ph -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
60	18, 56, 59	mpjaodan 827	1 |- ( ph -> ( ( ~~> ` seq N ( x. , F ) ) =/= 0 /\ seq N ( x. , F ) ~~> ( ~~> ` seq N ( x. , F ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   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   - cmin 10266   / cdiv 10684   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-1st 7168  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-fz 12327  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:  ntrivcvgmullem  14633