Theorem climeq 11309

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 11293	. 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 2178	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
8	1, 2, 6, 7	clim2 11293	. 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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   CCcc 7811   < clt 7994   - cmin 8130   ZZcz 9255   ZZ>=cuz 9530   RR+crp 9655   abscabs 11008   ~~> cli 11288

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-clim 11289

This theorem is referenced by:  climmpt  11310  climres  11313  climshft  11314  climshft2  11316  isumclim3  11433