Theorem climeq 11939

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 11923	. 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 2232	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
8	1, 2, 6, 7	clim2 11923	. 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 191	1 |- ( ph -> ( F ~~> A <-> G ~~> A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   < clt 8273   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   RR+crp 9949   abscabs 11637   ~~> cli 11918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-clim 11919

This theorem is referenced by:  climmpt  11940  climres  11943  climshft  11944  climshft2  11946  isumclim3  12064