Theorem climmpt 12320

Description: Exhibit a function G with the same convergence properties as the not-quite-function F. (Contributed by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
2clim.1	|- Z = ( ZZ>= ` M )
climmpt.2	|- G = ( k e. Z |-> ( F ` k ) )

Assertion

Ref	Expression
climmpt	|- ( ( M e. ZZ /\ F e. V ) -> ( F ~~> A <-> G ~~> A ) )

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

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

Proof of Theorem climmpt

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2clim.1	. 2 |- Z = ( ZZ>= ` M )
2		simpr 448	. 2 |- ( ( M e. ZZ /\ F e. V ) -> F e. V )
3		climmpt.2	. . . 4 |- G = ( k e. Z |-> ( F ` k ) )
4		fvex 5701	. . . . . 6 |- ( ZZ>= ` M ) e. _V
5	1, 4	eqeltri 2474	. . . . 5 |- Z e. _V
6	5	mptex 5925	. . . 4 |- ( k e. Z |-> ( F ` k ) ) e. _V
7	3, 6	eqeltri 2474	. . 3 |- G e. _V
8	7	a1i 11	. 2 |- ( ( M e. ZZ /\ F e. V ) -> G e. _V )
9		simpl 444	. 2 |- ( ( M e. ZZ /\ F e. V ) -> M e. ZZ )
10		fveq2 5687	. . . . 5 |- ( k = m -> ( F ` k ) = ( F ` m ) )
11		fvex 5701	. . . . 5 |- ( F ` m ) e. _V
12	10, 3, 11	fvmpt 5765	. . . 4 |- ( m e. Z -> ( G ` m ) = ( F ` m ) )
13	12	eqcomd 2409	. . 3 |- ( m e. Z -> ( F ` m ) = ( G ` m ) )
14	13	adantl 453	. 2 |- ( ( ( M e. ZZ /\ F e. V ) /\ m e. Z ) -> ( F ` m ) = ( G ` m ) )
15	1, 2, 8, 9, 14	climeq 12316	1 |- ( ( M e. ZZ /\ F e. V ) -> ( F ~~> A <-> G ~~> A ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   class class class wbr 4172   e. cmpt 4226   ` cfv 5413   ZZcz 10238   ZZ>=cuz 10444   ~~> cli 12233

This theorem is referenced by:  climmpt2  12322  climrecl  12332  climge0  12333  caurcvg2  12426  caucvg  12427  climfsum  12554  dstfrvclim1  24688

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-neg 9250  df-z 10239  df-uz 10445  df-clim 12237