Theorem climconst 13341

Description: An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
climconst.1	|- Z = ( ZZ>= ` M )
climconst.2	|- ( ph -> M e. ZZ )
climconst.3	|- ( ph -> F e. V )
climconst.4	|- ( ph -> A e. CC )
climconst.5	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )

Assertion

Ref	Expression
climconst	|- ( ph -> F ~~> A )

Distinct variable groups:   A, k   k, F   ph, k   k, Z

Allowed substitution hints:   M( k)   V( k)

Proof of Theorem climconst

Dummy variables j x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climconst.2	. . . . . . 7 |- ( ph -> M e. ZZ )
2		uzid 11106	. . . . . . 7 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
3	1, 2	syl 16	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` M ) )
4		climconst.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	syl6eleqr 2566	. . . . 5 |- ( ph -> M e. Z )
6	5	adantr 465	. . . 4 |- ( ( ph /\ x e. RR+ ) -> M e. Z )
7		climconst.4	. . . . . . . . . 10 |- ( ph -> A e. CC )
8	7	subidd 9928	. . . . . . . . 9 |- ( ph -> ( A - A ) = 0 )
9	8	fveq2d 5875	. . . . . . . 8 |- ( ph -> ( abs ` ( A - A ) ) = ( abs ` 0 ) )
10		abs0 13093	. . . . . . . 8 |- ( abs ` 0 ) = 0
11	9, 10	syl6eq 2524	. . . . . . 7 |- ( ph -> ( abs ` ( A - A ) ) = 0 )
12	11	adantr 465	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( abs ` ( A - A ) ) = 0 )
13		rpgt0 11241	. . . . . . 7 |- ( x e. RR+ -> 0 < x )
14	13	adantl 466	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> 0 < x )
15	12, 14	eqbrtrd 4472	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( abs ` ( A - A ) ) < x )
16	15	ralrimivw 2882	. . . 4 |- ( ( ph /\ x e. RR+ ) -> A. k e. Z ( abs ` ( A - A ) ) < x )
17		fveq2 5871	. . . . . . 7 |- ( j = M -> ( ZZ>= ` j ) = ( ZZ>= ` M ) )
18	17, 4	syl6eqr 2526	. . . . . 6 |- ( j = M -> ( ZZ>= ` j ) = Z )
19	18	raleqdv 3069	. . . . 5 |- ( j = M -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x <-> A. k e. Z ( abs ` ( A - A ) ) < x ) )
20	19	rspcev 3219	. . . 4 |- ( ( M e. Z /\ A. k e. Z ( abs ` ( A - A ) ) < x ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x )
21	6, 16, 20	syl2anc 661	. . 3 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x )
22	21	ralrimiva 2881	. 2 |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x )
23		climconst.3	. . 3 |- ( ph -> F e. V )
24		climconst.5	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
25	7	adantr 465	. . 3 |- ( ( ph /\ k e. Z ) -> A e. CC )
26	4, 1, 23, 24, 7, 25	clim2c 13303	. 2 |- ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x ) )
27	22, 26	mpbird 232	1 |- ( ph -> F ~~> A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2817   E.wrex 2818   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   CCcc 9500   0cc0 9502   < clt 9638   - cmin 9815   ZZcz 10874   ZZ>=cuz 11092   RR+crp 11230   abscabs 13042   ~~> cli 13282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-seq 12086  df-exp 12145  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286

This theorem is referenced by:  climconst2  13346  fsumcvg  13509  expcnv  13650  ntrivcvgfvn0  28928  fprodcvg  28957  fprodntriv  28969  faclim2  29068  clim1fr1  31434  climneg  31443  ioodvbdlimc1lem2  31553  ioodvbdlimc2lem  31555  fourierdlem103  31801  fourierdlem104  31802