Theorem climeq 10138

Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
climeq.1	|- Z = ( ZZ>= ` M )
climeq.2	|- ( ph -> F e. V )
climeq.3	|- ( ph -> G e. W )
climeq.5	|- ( ph -> M e. ZZ )
climeq.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )

Assertion

Ref	Expression
climeq	|- ( ph -> ( F ~~> A <-> G ~~> A ) )

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

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

Proof of Theorem climeq

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

Step	Hyp	Ref	Expression
1		climeq.1	. . 3 |- Z = ( ZZ>= ` M )
2		climeq.5	. . 3 |- ( ph -> M e. ZZ )
3		climeq.2	. . 3 |- ( ph -> F e. V )
4		climeq.6	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )
5	1, 2, 3, 4	clim2 10122	. 2 |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. y e. Z A. k e. ( ZZ>= ` y ) ( ( G ` k ) e. CC /\ ( abs ` ( ( G ` k ) - A ) ) < x ) ) ) )
6		climeq.3	. . 3 |- ( ph -> G e. W )
7		eqidd 2082	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
8	1, 2, 6, 7	clim2 10122	. 2 |- ( ph -> ( G ~~> A <-> ( A e. CC /\ A. x e. RR+ E. y e. Z A. k e. ( ZZ>= ` y ) ( ( G ` k ) e. CC /\ ( abs ` ( ( G ` k ) - A ) ) < x ) ) ) )
9	5, 8	bitr4d 189	1 |- ( ph -> ( F ~~> A <-> G ~~> A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = 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   < 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-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  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-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

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-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  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-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-clim 10118

This theorem is referenced by:  climmpt  10139  climres  10142  climshft  10143  climshft2  10145