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Metamath Proof Explorer

Theorem climfveqf

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

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

Proof

Step Hyp Ref Expression
1 climfveqf.p 
 |-  F/ k ph
2 climfveqf.n 
 |-  F/_ k F
3 climfveqf.o 
 |-  F/_ k G
4 climfveqf.z 
 |-  Z = ( ZZ>= ` M )
5 climfveqf.f 
 |-  ( ph -> F e. V )
6 climfveqf.g 
 |-  ( ph -> G e. W )
7 climfveqf.m 
 |-  ( ph -> M e. ZZ )
8 climfveqf.e 
 |-  ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )
9 climdm 
 |-  ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
10 9 bilani 
 |-  ( ( ph /\ F e. dom ~~> ) -> F ~~> ( ~~> ` F ) )
11 10 9 sylibr 
 |-  ( ( ph /\ F e. dom ~~> ) -> F e. dom ~~> )
12 nfcv 
 |-  F/_ k j
13 12 nfel1 
 |-  F/ k j e. Z
14 1 13 nfan 
 |-  F/ k ( ph /\ j e. Z )
15 2 12 nffv 
 |-  F/_ k ( F ` j )
16 3 12 nffv 
 |-  F/_ k ( G ` j )
17 15 16 nfeq 
 |-  F/ k ( F ` j ) = ( G ` j )
18 14 17 nfim 
 |-  F/ k ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) )
19 eleq1w 
 |-  ( k = j -> ( k e. Z <-> j e. Z ) )
20 19 anbi2d 
 |-  ( k = j -> ( ( ph /\ k e. Z ) <-> ( ph /\ j e. Z ) ) )
21 fveq2 
 |-  ( k = j -> ( F ` k ) = ( F ` j ) )
22 fveq2 
 |-  ( k = j -> ( G ` k ) = ( G ` j ) )
23 21 22 eqeq12d 
 |-  ( k = j -> ( ( F ` k ) = ( G ` k ) <-> ( F ` j ) = ( G ` j ) ) )
24 20 23 imbi12d 
 |-  ( k = j -> ( ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) ) <-> ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) ) ) )
25 18 24 8 chvarfv 
 |-  ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) )
26 4 5 6 7 25 climeldmeq 
 |-  ( ph -> ( F e. dom ~~> <-> G e. dom ~~> ) )
27 26 adantr 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( F e. dom ~~> <-> G e. dom ~~> ) )
28 11 27 mpbid 
 |-  ( ( ph /\ F e. dom ~~> ) -> G e. dom ~~> )
29 climdm 
 |-  ( G e. dom ~~> <-> G ~~> ( ~~> ` G ) )
30 28 29 sylib 
 |-  ( ( ph /\ F e. dom ~~> ) -> G ~~> ( ~~> ` G ) )
31 6 adantr 
 |-  ( ( ph /\ F e. dom ~~> ) -> G e. W )
32 5 adantr 
 |-  ( ( ph /\ F e. dom ~~> ) -> F e. V )
33 7 adantr 
 |-  ( ( ph /\ F e. dom ~~> ) -> M e. ZZ )
34 25 eqcomd 
 |-  ( ( ph /\ j e. Z ) -> ( G ` j ) = ( F ` j ) )
35 34 adantlr 
 |-  ( ( ( ph /\ F e. dom ~~> ) /\ j e. Z ) -> ( G ` j ) = ( F ` j ) )
36 4 31 32 33 35 climeq 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( G ~~> ( ~~> ` G ) <-> F ~~> ( ~~> ` G ) ) )
37 30 36 mpbid 
 |-  ( ( ph /\ F e. dom ~~> ) -> F ~~> ( ~~> ` G ) )
38 climuni 
 |-  ( ( F ~~> ( ~~> ` F ) /\ F ~~> ( ~~> ` G ) ) -> ( ~~> ` F ) = ( ~~> ` G ) )
39 10 37 38 syl2anc 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( ~~> ` F ) = ( ~~> ` G ) )
40 ndmfv 
 |-  ( -. F e. dom ~~> -> ( ~~> ` F ) = (/) )
41 40 adantl 
 |-  ( ( ph /\ -. F e. dom ~~> ) -> ( ~~> ` F ) = (/) )
42 simpr 
 |-  ( ( ph /\ -. F e. dom ~~> ) -> -. F e. dom ~~> )
43 26 adantr 
 |-  ( ( ph /\ -. F e. dom ~~> ) -> ( F e. dom ~~> <-> G e. dom ~~> ) )
44 42 43 mtbid 
 |-  ( ( ph /\ -. F e. dom ~~> ) -> -. G e. dom ~~> )
45 ndmfv 
 |-  ( -. G e. dom ~~> -> ( ~~> ` G ) = (/) )
46 44 45 syl 
 |-  ( ( ph /\ -. F e. dom ~~> ) -> ( ~~> ` G ) = (/) )
47 41 46 eqtr4d 
 |-  ( ( ph /\ -. F e. dom ~~> ) -> ( ~~> ` F ) = ( ~~> ` G ) )
48 39 47 pm2.61dan 
 |-  ( ph -> ( ~~> ` F ) = ( ~~> ` G ) )