Theorem climfveqf 39912

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

Hypotheses

Ref	Expression
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

Ref	Expression
climfveqf	|- ( ph -> ( ~~> ` F ) = ( ~~> ` G ) )

Distinct variable group:   k, Z

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

Proof of Theorem climfveqf

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climdm 14285	. . . . 5 |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
2	1	biimpi 206	. . . 4 |- ( F e. dom ~~> -> F ~~> ( ~~> ` F ) )
3	2	adantl 482	. . 3 |- ( ( ph /\ F e. dom ~~> ) -> F ~~> ( ~~> ` F ) )
4	3, 1	sylibr 224	. . . . . 6 |- ( ( ph /\ F e. dom ~~> ) -> F e. dom ~~> )
5		climfveqf.z	. . . . . . . 8 |- Z = ( ZZ>= ` M )
6		climfveqf.f	. . . . . . . 8 |- ( ph -> F e. V )
7		climfveqf.g	. . . . . . . 8 |- ( ph -> G e. W )
8		climfveqf.m	. . . . . . . 8 |- ( ph -> M e. ZZ )
9		climfveqf.p	. . . . . . . . . . 11 |- F/ k ph
10		nfcv 2764	. . . . . . . . . . . 12 |- F/_ k j
11	10	nfel1 2779	. . . . . . . . . . 11 |- F/ k j e. Z
12	9, 11	nfan 1828	. . . . . . . . . 10 |- F/ k ( ph /\ j e. Z )
13		climfveqf.n	. . . . . . . . . . . 12 |- F/_ k F
14	13, 10	nffv 6198	. . . . . . . . . . 11 |- F/_ k ( F ` j )
15		climfveqf.o	. . . . . . . . . . . 12 |- F/_ k G
16	15, 10	nffv 6198	. . . . . . . . . . 11 |- F/_ k ( G ` j )
17	14, 16	nfeq 2776	. . . . . . . . . 10 |- F/ k ( F ` j ) = ( G ` j )
18	12, 17	nfim 1825	. . . . . . . . 9 |- F/ k ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) )
19		eleq1 2689	. . . . . . . . . . 11 |- ( k = j -> ( k e. Z <-> j e. Z ) )
20	19	anbi2d 740	. . . . . . . . . 10 |- ( k = j -> ( ( ph /\ k e. Z ) <-> ( ph /\ j e. Z ) ) )
21		fveq2 6191	. . . . . . . . . . 11 |- ( k = j -> ( F ` k ) = ( F ` j ) )
22		fveq2 6191	. . . . . . . . . . 11 |- ( k = j -> ( G ` k ) = ( G ` j ) )
23	21, 22	eqeq12d 2637	. . . . . . . . . 10 |- ( k = j -> ( ( F ` k ) = ( G ` k ) <-> ( F ` j ) = ( G ` j ) ) )
24	20, 23	imbi12d 334	. . . . . . . . 9 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) ) <-> ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) ) ) )
25		climfveqf.e	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )
26	18, 24, 25	chvar 2262	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> ( F ` j ) = ( G ` j ) )
27	5, 6, 7, 8, 26	climeldmeq 39897	. . . . . . 7 |- ( ph -> ( F e. dom ~~> <-> G e. dom ~~> ) )
28	27	adantr 481	. . . . . 6 |- ( ( ph /\ F e. dom ~~> ) -> ( F e. dom ~~> <-> G e. dom ~~> ) )
29	4, 28	mpbid 222	. . . . 5 |- ( ( ph /\ F e. dom ~~> ) -> G e. dom ~~> )
30		climdm 14285	. . . . 5 |- ( G e. dom ~~> <-> G ~~> ( ~~> ` G ) )
31	29, 30	sylib 208	. . . 4 |- ( ( ph /\ F e. dom ~~> ) -> G ~~> ( ~~> ` G ) )
32	7	adantr 481	. . . . 5 |- ( ( ph /\ F e. dom ~~> ) -> G e. W )
33	6	adantr 481	. . . . 5 |- ( ( ph /\ F e. dom ~~> ) -> F e. V )
34	8	adantr 481	. . . . 5 |- ( ( ph /\ F e. dom ~~> ) -> M e. ZZ )
35	26	eqcomd 2628	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( G ` j ) = ( F ` j ) )
36	35	adantlr 751	. . . . 5 |- ( ( ( ph /\ F e. dom ~~> ) /\ j e. Z ) -> ( G ` j ) = ( F ` j ) )
37	5, 32, 33, 34, 36	climeq 14298	. . . 4 |- ( ( ph /\ F e. dom ~~> ) -> ( G ~~> ( ~~> ` G ) <-> F ~~> ( ~~> ` G ) ) )
38	31, 37	mpbid 222	. . 3 |- ( ( ph /\ F e. dom ~~> ) -> F ~~> ( ~~> ` G ) )
39		climuni 14283	. . 3 |- ( ( F ~~> ( ~~> ` F ) /\ F ~~> ( ~~> ` G ) ) -> ( ~~> ` F ) = ( ~~> ` G ) )
40	3, 38, 39	syl2anc 693	. 2 |- ( ( ph /\ F e. dom ~~> ) -> ( ~~> ` F ) = ( ~~> ` G ) )
41		ndmfv 6218	. . . 4 |- ( -. F e. dom ~~> -> ( ~~> ` F ) = (/) )
42	41	adantl 482	. . 3 |- ( ( ph /\ -. F e. dom ~~> ) -> ( ~~> ` F ) = (/) )
43		simpr 477	. . . . 5 |- ( ( ph /\ -. F e. dom ~~> ) -> -. F e. dom ~~> )
44	27	adantr 481	. . . . 5 |- ( ( ph /\ -. F e. dom ~~> ) -> ( F e. dom ~~> <-> G e. dom ~~> ) )
45	43, 44	mtbid 314	. . . 4 |- ( ( ph /\ -. F e. dom ~~> ) -> -. G e. dom ~~> )
46		ndmfv 6218	. . . 4 |- ( -. G e. dom ~~> -> ( ~~> ` G ) = (/) )
47	45, 46	syl 17	. . 3 |- ( ( ph /\ -. F e. dom ~~> ) -> ( ~~> ` G ) = (/) )
48	42, 47	eqtr4d 2659	. 2 |- ( ( ph /\ -. F e. dom ~~> ) -> ( ~~> ` F ) = ( ~~> ` G ) )
49	40, 48	pm2.61dan 832	1 |- ( ph -> ( ~~> ` F ) = ( ~~> ` G ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ` 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-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  climfveqmpt2  39925