Theorem climeqf 39920

Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
climeqf.p	|- F/ k ph
climeqf.k	|- F/_ k F
climeqf.n	|- F/_ k G
climeqf.m	|- ( ph -> M e. ZZ )
climeqf.z	|- Z = ( ZZ>= ` M )
climeqf.f	|- ( ph -> F e. V )
climeqf.g	|- ( ph -> G e. W )
climeqf.e	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )

Assertion

Ref	Expression
climeqf	|- ( ph -> ( F ~~> A <-> G ~~> A ) )

Distinct variable group:   k, Z

Allowed substitution hints:   ph( k)   A( k)   F( k)   G( k)   M( k)   V( k)   W( k)

Proof of Theorem climeqf

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climeqf.z	. 2 |- Z = ( ZZ>= ` M )
2		climeqf.f	. 2 |- ( ph -> F e. V )
3		climeqf.g	. 2 |- ( ph -> G e. W )
4		climeqf.m	. 2 |- ( ph -> M e. ZZ )
5		climeqf.p	. . . . 5 |- F/ k ph
6		nfv 1843	. . . . 5 |- F/ k j e. Z
7	5, 6	nfan 1828	. . . 4 |- F/ k ( ph /\ j e. Z )
8		climeqf.k	. . . . . 6 |- F/_ k F
9		nfcv 2764	. . . . . 6 |- F/_ k j
10	8, 9	nffv 6198	. . . . 5 |- F/_ k ( F ` j )
11		climeqf.n	. . . . . 6 |- F/_ k G
12	11, 9	nffv 6198	. . . . 5 |- F/_ k ( G ` j )
13	10, 12	nfeq 2776	. . . 4 |- F/ k ( F ` j ) = ( G ` j )
14	7, 13	nfim 1825	. . 3 |- F/ k ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) )
15		eleq1 2689	. . . . 5 |- ( k = j -> ( k e. Z <-> j e. Z ) )
16	15	anbi2d 740	. . . 4 |- ( k = j -> ( ( ph /\ k e. Z ) <-> ( ph /\ j e. Z ) ) )
17		fveq2 6191	. . . . 5 |- ( k = j -> ( F ` k ) = ( F ` j ) )
18		fveq2 6191	. . . . 5 |- ( k = j -> ( G ` k ) = ( G ` j ) )
19	17, 18	eqeq12d 2637	. . . 4 |- ( k = j -> ( ( F ` k ) = ( G ` k ) <-> ( F ` j ) = ( G ` j ) ) )
20	16, 19	imbi12d 334	. . 3 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) ) <-> ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) ) ) )
21		climeqf.e	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )
22	14, 20, 21	chvar 2262	. 2 |- ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) )
23	1, 2, 3, 4, 22	climeq 14298	1 |- ( ph -> ( F ~~> A <-> G ~~> A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   class class class wbr 4653   ` cfv 5888   ZZcz 11377   ZZ>=cuz 11687   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219

This theorem is referenced by:  climeqmpt  39929