Theorem climeq 11226

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 11210	. 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 2165	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
8	1, 2, 6, 7	clim2 11210	. 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 1342   e. wcel 2135   A.wral 2442   E.wrex 2443   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   CCcc 7742   < clt 7924   - cmin 8060   ZZcz 9182   ZZ>=cuz 9457   RR+crp 9580   abscabs 10925   ~~> cli 11205

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183  df-uz 9458  df-clim 11206

This theorem is referenced by:  climmpt  11227  climres  11230  climshft  11231  climshft2  11233  isumclim3  11350