Theorem limsupequzmptf 39963

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

Hypotheses

Ref	Expression
limsupequzmptf.j	|- F/ j ph
limsupequzmptf.o	|- F/_ j A
limsupequzmptf.p	|- F/_ j B
limsupequzmptf.m	|- ( ph -> M e. ZZ )
limsupequzmptf.n	|- ( ph -> N e. ZZ )
limsupequzmptf.a	|- A = ( ZZ>= ` M )
limsupequzmptf.b	|- B = ( ZZ>= ` N )
limsupequzmptf.c	|- ( ( ph /\ j e. A ) -> C e. V )
limsupequzmptf.d	|- ( ( ph /\ j e. B ) -> C e. W )

Assertion

Ref	Expression
limsupequzmptf	|- ( ph -> ( limsup ` ( j e. A |-> C ) ) = ( limsup ` ( j e. B |-> C ) ) )

Distinct variable groups:   j, V   j, W

Allowed substitution hints:   ph( j)   A( j)   B( j)   C( j)   M( j)   N( j)

Proof of Theorem limsupequzmptf

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ k ph
2		limsupequzmptf.m	. . 3 |- ( ph -> M e. ZZ )
3		limsupequzmptf.n	. . 3 |- ( ph -> N e. ZZ )
4		limsupequzmptf.a	. . 3 |- A = ( ZZ>= ` M )
5		limsupequzmptf.b	. . 3 |- B = ( ZZ>= ` N )
6		limsupequzmptf.j	. . . . . 6 |- F/ j ph
7		limsupequzmptf.o	. . . . . . 7 |- F/_ j A
8	7	nfcri 2758	. . . . . 6 |- F/ j k e. A
9	6, 8	nfan 1828	. . . . 5 |- F/ j ( ph /\ k e. A )
10		nfcsb1v 3549	. . . . . 6 |- F/_ j [_ k / j ]_ C
11		nfcv 2764	. . . . . 6 |- F/_ j V
12	10, 11	nfel 2777	. . . . 5 |- F/ j [_ k / j ]_ C e. V
13	9, 12	nfim 1825	. . . 4 |- F/ j ( ( ph /\ k e. A ) -> [_ k / j ]_ C e. V )
14		eleq1 2689	. . . . . 6 |- ( j = k -> ( j e. A <-> k e. A ) )
15	14	anbi2d 740	. . . . 5 |- ( j = k -> ( ( ph /\ j e. A ) <-> ( ph /\ k e. A ) ) )
16		csbeq1a 3542	. . . . . 6 |- ( j = k -> C = [_ k / j ]_ C )
17	16	eleq1d 2686	. . . . 5 |- ( j = k -> ( C e. V <-> [_ k / j ]_ C e. V ) )
18	15, 17	imbi12d 334	. . . 4 |- ( j = k -> ( ( ( ph /\ j e. A ) -> C e. V ) <-> ( ( ph /\ k e. A ) -> [_ k / j ]_ C e. V ) ) )
19		limsupequzmptf.c	. . . 4 |- ( ( ph /\ j e. A ) -> C e. V )
20	13, 18, 19	chvar 2262	. . 3 |- ( ( ph /\ k e. A ) -> [_ k / j ]_ C e. V )
21		limsupequzmptf.p	. . . . . . 7 |- F/_ j B
22	21	nfcri 2758	. . . . . 6 |- F/ j k e. B
23	6, 22	nfan 1828	. . . . 5 |- F/ j ( ph /\ k e. B )
24		nfcv 2764	. . . . . 6 |- F/_ j W
25	10, 24	nfel 2777	. . . . 5 |- F/ j [_ k / j ]_ C e. W
26	23, 25	nfim 1825	. . . 4 |- F/ j ( ( ph /\ k e. B ) -> [_ k / j ]_ C e. W )
27		eleq1 2689	. . . . . 6 |- ( j = k -> ( j e. B <-> k e. B ) )
28	27	anbi2d 740	. . . . 5 |- ( j = k -> ( ( ph /\ j e. B ) <-> ( ph /\ k e. B ) ) )
29	16	eleq1d 2686	. . . . 5 |- ( j = k -> ( C e. W <-> [_ k / j ]_ C e. W ) )
30	28, 29	imbi12d 334	. . . 4 |- ( j = k -> ( ( ( ph /\ j e. B ) -> C e. W ) <-> ( ( ph /\ k e. B ) -> [_ k / j ]_ C e. W ) ) )
31		limsupequzmptf.d	. . . 4 |- ( ( ph /\ j e. B ) -> C e. W )
32	26, 30, 31	chvar 2262	. . 3 |- ( ( ph /\ k e. B ) -> [_ k / j ]_ C e. W )
33	1, 2, 3, 4, 5, 20, 32	limsupequzmpt 39961	. 2 |- ( ph -> ( limsup ` ( k e. A |-> [_ k / j ]_ C ) ) = ( limsup ` ( k e. B |-> [_ k / j ]_ C ) ) )
34		nfcv 2764	. . . . 5 |- F/_ k A
35		nfcv 2764	. . . . 5 |- F/_ k C
36	7, 34, 35, 10, 16	cbvmptf 4748	. . . 4 |- ( j e. A |-> C ) = ( k e. A |-> [_ k / j ]_ C )
37	36	fveq2i 6194	. . 3 |- ( limsup ` ( j e. A |-> C ) ) = ( limsup ` ( k e. A |-> [_ k / j ]_ C ) )
38	37	a1i 11	. 2 |- ( ph -> ( limsup ` ( j e. A |-> C ) ) = ( limsup ` ( k e. A |-> [_ k / j ]_ C ) ) )
39		nfcv 2764	. . . . 5 |- F/_ k B
40	21, 39, 35, 10, 16	cbvmptf 4748	. . . 4 |- ( j e. B |-> C ) = ( k e. B |-> [_ k / j ]_ C )
41	40	fveq2i 6194	. . 3 |- ( limsup ` ( j e. B |-> C ) ) = ( limsup ` ( k e. B |-> [_ k / j ]_ C ) )
42	41	a1i 11	. 2 |- ( ph -> ( limsup ` ( j e. B |-> C ) ) = ( limsup ` ( k e. B |-> [_ k / j ]_ C ) ) )
43	33, 38, 42	3eqtr4d 2666	1 |- ( ph -> ( limsup ` ( j e. A |-> C ) ) = ( limsup ` ( j e. B |-> C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   [_csb 3533   |-> cmpt 4729   ` cfv 5888   ZZcz 11377   ZZ>=cuz 11687   limsupclsp 14201

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-int 4476  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ico 12181  df-limsup 14202

This theorem is referenced by: (None)