Theorem climconst 10129

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 8633	. . . . . . 7 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` M ) )
4		climconst.1	. . . . . 6 |- Z = ( ZZ>= ` M )
5	3, 4	syl6eleqr 2172	. . . . 5 |- ( ph -> M e. Z )
6	5	adantr 270	. . . 4 |- ( ( ph /\ x e. RR+ ) -> M e. Z )
7		climconst.4	. . . . . . . . . 10 |- ( ph -> A e. CC )
8	7	subidd 7407	. . . . . . . . 9 |- ( ph -> ( A - A ) = 0 )
9	8	fveq2d 5202	. . . . . . . 8 |- ( ph -> ( abs ` ( A - A ) ) = ( abs ` 0 ) )
10		abs0 9944	. . . . . . . 8 |- ( abs ` 0 ) = 0
11	9, 10	syl6eq 2129	. . . . . . 7 |- ( ph -> ( abs ` ( A - A ) ) = 0 )
12	11	adantr 270	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( abs ` ( A - A ) ) = 0 )
13		rpgt0 8745	. . . . . . 7 |- ( x e. RR+ -> 0 < x )
14	13	adantl 271	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> 0 < x )
15	12, 14	eqbrtrd 3805	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( abs ` ( A - A ) ) < x )
16	15	ralrimivw 2435	. . . 4 |- ( ( ph /\ x e. RR+ ) -> A. k e. Z ( abs ` ( A - A ) ) < x )
17		fveq2 5198	. . . . . . 7 |- ( j = M -> ( ZZ>= ` j ) = ( ZZ>= ` M ) )
18	17, 4	syl6eqr 2131	. . . . . 6 |- ( j = M -> ( ZZ>= ` j ) = Z )
19	18	raleqdv 2555	. . . . 5 |- ( j = M -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x <-> A. k e. Z ( abs ` ( A - A ) ) < x ) )
20	19	rspcev 2701	. . . 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 403	. . 3 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( A - A ) ) < x )
22	21	ralrimiva 2434	. 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 270	. . 3 |- ( ( ph /\ k e. Z ) -> A e. CC )
26	4, 1, 23, 24, 7, 25	clim2c 10123	. 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 165	1 |- ( ph -> F ~~> A )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   < clt 7153   - cmin 7279   ZZcz 8351   ZZ>=cuz 8619   RR+crp 8734   abscabs 9883   ~~> cli 10117

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-iseq 9432  df-iexp 9476  df-cj 9729  df-rsqrt 9884  df-abs 9885  df-clim 10118

This theorem is referenced by:  climconst2  10130