Theorem climfveqmpt 39903

Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
climfveqmpt.k	|- F/ k ph
climfveqmpt.m	|- ( ph -> M e. ZZ )
climfveqmpt.z	|- Z = ( ZZ>= ` M )
climfveqmpt.A	|- ( ph -> A e. R )
climfveqmpt.i	|- ( ph -> Z C_ A )
climfveqmpt.b	|- ( ( ph /\ k e. A ) -> B e. V )
climfveqmpt.t	|- ( ph -> C e. S )
climfveqmpt.l	|- ( ph -> Z C_ C )
climfveqmpt.c	|- ( ( ph /\ k e. C ) -> D e. W )
climfveqmpt.e	|- ( ( ph /\ k e. Z ) -> B = D )

Assertion

Ref	Expression
climfveqmpt	|- ( ph -> ( ~~> ` ( k e. A |-> B ) ) = ( ~~> ` ( k e. C |-> D ) ) )

Distinct variable groups:   A, k   C, k   k, V   k, W   k, Z

Allowed substitution hints:   ph( k)   B( k)   D( k)   R( k)   S( k)   M( k)

Proof of Theorem climfveqmpt

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climfveqmpt.z	. 2 |- Z = ( ZZ>= ` M )
2		climfveqmpt.A	. . 3 |- ( ph -> A e. R )
3	2	mptexd 6487	. 2 |- ( ph -> ( k e. A |-> B ) e. _V )
4		climfveqmpt.t	. . 3 |- ( ph -> C e. S )
5	4	mptexd 6487	. 2 |- ( ph -> ( k e. C |-> D ) e. _V )
6		climfveqmpt.m	. 2 |- ( ph -> M e. ZZ )
7		climfveqmpt.k	. . . . . 6 |- F/ k ph
8		nfv 1843	. . . . . 6 |- F/ k j e. Z
9	7, 8	nfan 1828	. . . . 5 |- F/ k ( ph /\ j e. Z )
10		nfcv 2764	. . . . . . 7 |- F/_ k j
11	10	nfcsb1 3548	. . . . . 6 |- F/_ k [_ j / k ]_ B
12	10	nfcsb1 3548	. . . . . 6 |- F/_ k [_ j / k ]_ D
13	11, 12	nfeq 2776	. . . . 5 |- F/ k [_ j / k ]_ B = [_ j / k ]_ D
14	9, 13	nfim 1825	. . . 4 |- F/ k ( ( ph /\ j e. Z ) -> [_ j / k ]_ B = [_ j / k ]_ D )
15		eleq1 2689	. . . . . 6 |- ( k = j -> ( k e. Z <-> j e. Z ) )
16	15	anbi2d 740	. . . . 5 |- ( k = j -> ( ( ph /\ k e. Z ) <-> ( ph /\ j e. Z ) ) )
17		csbeq1a 3542	. . . . . 6 |- ( k = j -> B = [_ j / k ]_ B )
18		csbeq1a 3542	. . . . . 6 |- ( k = j -> D = [_ j / k ]_ D )
19	17, 18	eqeq12d 2637	. . . . 5 |- ( k = j -> ( B = D <-> [_ j / k ]_ B = [_ j / k ]_ D ) )
20	16, 19	imbi12d 334	. . . 4 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> B = D ) <-> ( ( ph /\ j e. Z ) -> [_ j / k ]_ B = [_ j / k ]_ D ) ) )
21		climfveqmpt.e	. . . 4 |- ( ( ph /\ k e. Z ) -> B = D )
22	14, 20, 21	chvar 2262	. . 3 |- ( ( ph /\ j e. Z ) -> [_ j / k ]_ B = [_ j / k ]_ D )
23		climfveqmpt.i	. . . . . 6 |- ( ph -> Z C_ A )
24	23	adantr 481	. . . . 5 |- ( ( ph /\ j e. Z ) -> Z C_ A )
25		simpr 477	. . . . 5 |- ( ( ph /\ j e. Z ) -> j e. Z )
26	24, 25	sseldd 3604	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. A )
27		simpr 477	. . . . 5 |- ( ( ph /\ j e. A ) -> j e. A )
28		nfv 1843	. . . . . . . 8 |- F/ k j e. A
29	7, 28	nfan 1828	. . . . . . 7 |- F/ k ( ph /\ j e. A )
30		nfcv 2764	. . . . . . . 8 |- F/_ k V
31	11, 30	nfel 2777	. . . . . . 7 |- F/ k [_ j / k ]_ B e. V
32	29, 31	nfim 1825	. . . . . 6 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. V )
33		eleq1 2689	. . . . . . . 8 |- ( k = j -> ( k e. A <-> j e. A ) )
34	33	anbi2d 740	. . . . . . 7 |- ( k = j -> ( ( ph /\ k e. A ) <-> ( ph /\ j e. A ) ) )
35	17	eleq1d 2686	. . . . . . 7 |- ( k = j -> ( B e. V <-> [_ j / k ]_ B e. V ) )
36	34, 35	imbi12d 334	. . . . . 6 |- ( k = j -> ( ( ( ph /\ k e. A ) -> B e. V ) <-> ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. V ) ) )
37		climfveqmpt.b	. . . . . 6 |- ( ( ph /\ k e. A ) -> B e. V )
38	32, 36, 37	chvar 2262	. . . . 5 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. V )
39		eqid 2622	. . . . . 6 |- ( k e. A |-> B ) = ( k e. A |-> B )
40	10, 11, 17, 39	fvmptf 6301	. . . . 5 |- ( ( j e. A /\ [_ j / k ]_ B e. V ) -> ( ( k e. A |-> B ) ` j ) = [_ j / k ]_ B )
41	27, 38, 40	syl2anc 693	. . . 4 |- ( ( ph /\ j e. A ) -> ( ( k e. A |-> B ) ` j ) = [_ j / k ]_ B )
42	26, 41	syldan 487	. . 3 |- ( ( ph /\ j e. Z ) -> ( ( k e. A |-> B ) ` j ) = [_ j / k ]_ B )
43		climfveqmpt.l	. . . . . 6 |- ( ph -> Z C_ C )
44	43	adantr 481	. . . . 5 |- ( ( ph /\ j e. Z ) -> Z C_ C )
45	44, 25	sseldd 3604	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. C )
46		simpr 477	. . . . 5 |- ( ( ph /\ j e. C ) -> j e. C )
47		nfv 1843	. . . . . . . 8 |- F/ k j e. C
48	7, 47	nfan 1828	. . . . . . 7 |- F/ k ( ph /\ j e. C )
49		nfcv 2764	. . . . . . . 8 |- F/_ k W
50	12, 49	nfel 2777	. . . . . . 7 |- F/ k [_ j / k ]_ D e. W
51	48, 50	nfim 1825	. . . . . 6 |- F/ k ( ( ph /\ j e. C ) -> [_ j / k ]_ D e. W )
52		eleq1 2689	. . . . . . . 8 |- ( k = j -> ( k e. C <-> j e. C ) )
53	52	anbi2d 740	. . . . . . 7 |- ( k = j -> ( ( ph /\ k e. C ) <-> ( ph /\ j e. C ) ) )
54	18	eleq1d 2686	. . . . . . 7 |- ( k = j -> ( D e. W <-> [_ j / k ]_ D e. W ) )
55	53, 54	imbi12d 334	. . . . . 6 |- ( k = j -> ( ( ( ph /\ k e. C ) -> D e. W ) <-> ( ( ph /\ j e. C ) -> [_ j / k ]_ D e. W ) ) )
56		climfveqmpt.c	. . . . . 6 |- ( ( ph /\ k e. C ) -> D e. W )
57	51, 55, 56	chvar 2262	. . . . 5 |- ( ( ph /\ j e. C ) -> [_ j / k ]_ D e. W )
58		eqid 2622	. . . . . 6 |- ( k e. C |-> D ) = ( k e. C |-> D )
59	10, 12, 18, 58	fvmptf 6301	. . . . 5 |- ( ( j e. C /\ [_ j / k ]_ D e. W ) -> ( ( k e. C |-> D ) ` j ) = [_ j / k ]_ D )
60	46, 57, 59	syl2anc 693	. . . 4 |- ( ( ph /\ j e. C ) -> ( ( k e. C |-> D ) ` j ) = [_ j / k ]_ D )
61	45, 60	syldan 487	. . 3 |- ( ( ph /\ j e. Z ) -> ( ( k e. C |-> D ) ` j ) = [_ j / k ]_ D )
62	22, 42, 61	3eqtr4d 2666	. 2 |- ( ( ph /\ j e. Z ) -> ( ( k e. A |-> B ) ` j ) = ( ( k e. C |-> D ) ` j ) )
63	1, 3, 5, 6, 62	climfveq 39901	1 |- ( ph -> ( ~~> ` ( k e. A |-> B ) ) = ( ~~> ` ( k e. C |-> D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   _Vcvv 3200   [_csb 3533   C_ wss 3574   |-> cmpt 4729   ` 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-rep 4771  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:  fnlimfvre  39906