Theorem climeq 11240

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 11224	. 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 2166	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
8	1, 2, 6, 7	clim2 11224	. 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 190	1 |- ( ph -> ( F ~~> A <-> G ~~> A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   < clt 7933   - cmin 8069   ZZcz 9191   ZZ>=cuz 9466   RR+crp 9589   abscabs 10939   ~~> cli 11219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-clim 11220

This theorem is referenced by:  climmpt  11241  climres  11244  climshft  11245  climshft2  11247  isumclim3  11364