Theorem climsubmpt 39892

Description: Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
climsubmpt.k	|- F/ k ph
climsubmpt.z	|- Z = ( ZZ>= ` M )
climsubmpt.m	|- ( ph -> M e. ZZ )
climsubmpt.a	|- ( ( ph /\ k e. Z ) -> A e. CC )
climsubmpt.b	|- ( ( ph /\ k e. Z ) -> B e. CC )
climsubmpt.c	|- ( ph -> ( k e. Z |-> A ) ~~> C )
climsubmpt.d	|- ( ph -> ( k e. Z |-> B ) ~~> D )

Assertion

Ref	Expression
climsubmpt	|- ( ph -> ( k e. Z |-> ( A - B ) ) ~~> ( C - D ) )

Distinct variable group:   k, Z

Allowed substitution hints:   ph( k)   A( k)   B( k)   C( k)   D( k)   M( k)

Proof of Theorem climsubmpt

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climsubmpt.z	. 2 |- Z = ( ZZ>= ` M )
2		climsubmpt.m	. 2 |- ( ph -> M e. ZZ )
3		climsubmpt.c	. 2 |- ( ph -> ( k e. Z |-> A ) ~~> C )
4		fvex 6201	. . . . 5 |- ( ZZ>= ` M ) e. _V
5	1, 4	eqeltri 2697	. . . 4 |- Z e. _V
6	5	mptex 6486	. . 3 |- ( k e. Z |-> ( A - B ) ) e. _V
7	6	a1i 11	. 2 |- ( ph -> ( k e. Z |-> ( A - B ) ) e. _V )
8		climsubmpt.d	. 2 |- ( ph -> ( k e. Z |-> B ) ~~> D )
9		simpr 477	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. Z )
10		climsubmpt.k	. . . . . . 7 |- F/ k ph
11		nfv 1843	. . . . . . 7 |- F/ k j e. Z
12	10, 11	nfan 1828	. . . . . 6 |- F/ k ( ph /\ j e. Z )
13		nfcv 2764	. . . . . . . 8 |- F/_ k j
14	13	nfcsb1 3548	. . . . . . 7 |- F/_ k [_ j / k ]_ A
15	14	nfel1 2779	. . . . . 6 |- F/ k [_ j / k ]_ A e. CC
16	12, 15	nfim 1825	. . . . 5 |- F/ k ( ( ph /\ j e. Z ) -> [_ j / k ]_ A e. CC )
17		eleq1 2689	. . . . . . 7 |- ( k = j -> ( k e. Z <-> j e. Z ) )
18	17	anbi2d 740	. . . . . 6 |- ( k = j -> ( ( ph /\ k e. Z ) <-> ( ph /\ j e. Z ) ) )
19		csbeq1a 3542	. . . . . . 7 |- ( k = j -> A = [_ j / k ]_ A )
20	19	eleq1d 2686	. . . . . 6 |- ( k = j -> ( A e. CC <-> [_ j / k ]_ A e. CC ) )
21	18, 20	imbi12d 334	. . . . 5 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> A e. CC ) <-> ( ( ph /\ j e. Z ) -> [_ j / k ]_ A e. CC ) ) )
22		climsubmpt.a	. . . . 5 |- ( ( ph /\ k e. Z ) -> A e. CC )
23	16, 21, 22	chvar 2262	. . . 4 |- ( ( ph /\ j e. Z ) -> [_ j / k ]_ A e. CC )
24		eqid 2622	. . . . 5 |- ( k e. Z |-> A ) = ( k e. Z |-> A )
25	13, 14, 19, 24	fvmptf 6301	. . . 4 |- ( ( j e. Z /\ [_ j / k ]_ A e. CC ) -> ( ( k e. Z |-> A ) ` j ) = [_ j / k ]_ A )
26	9, 23, 25	syl2anc 693	. . 3 |- ( ( ph /\ j e. Z ) -> ( ( k e. Z |-> A ) ` j ) = [_ j / k ]_ A )
27	26, 23	eqeltrd 2701	. 2 |- ( ( ph /\ j e. Z ) -> ( ( k e. Z |-> A ) ` j ) e. CC )
28	13	nfcsb1 3548	. . . . . . 7 |- F/_ k [_ j / k ]_ B
29		nfcv 2764	. . . . . . 7 |- F/_ k CC
30	28, 29	nfel 2777	. . . . . 6 |- F/ k [_ j / k ]_ B e. CC
31	12, 30	nfim 1825	. . . . 5 |- F/ k ( ( ph /\ j e. Z ) -> [_ j / k ]_ B e. CC )
32		csbeq1a 3542	. . . . . . 7 |- ( k = j -> B = [_ j / k ]_ B )
33	32	eleq1d 2686	. . . . . 6 |- ( k = j -> ( B e. CC <-> [_ j / k ]_ B e. CC ) )
34	18, 33	imbi12d 334	. . . . 5 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> B e. CC ) <-> ( ( ph /\ j e. Z ) -> [_ j / k ]_ B e. CC ) ) )
35		climsubmpt.b	. . . . 5 |- ( ( ph /\ k e. Z ) -> B e. CC )
36	31, 34, 35	chvar 2262	. . . 4 |- ( ( ph /\ j e. Z ) -> [_ j / k ]_ B e. CC )
37		eqid 2622	. . . . 5 |- ( k e. Z |-> B ) = ( k e. Z |-> B )
38	13, 28, 32, 37	fvmptf 6301	. . . 4 |- ( ( j e. Z /\ [_ j / k ]_ B e. CC ) -> ( ( k e. Z |-> B ) ` j ) = [_ j / k ]_ B )
39	9, 36, 38	syl2anc 693	. . 3 |- ( ( ph /\ j e. Z ) -> ( ( k e. Z |-> B ) ` j ) = [_ j / k ]_ B )
40	39, 36	eqeltrd 2701	. 2 |- ( ( ph /\ j e. Z ) -> ( ( k e. Z |-> B ) ` j ) e. CC )
41		ovexd 6680	. . . 4 |- ( ( ph /\ j e. Z ) -> ( [_ j / k ]_ A - [_ j / k ]_ B ) e. _V )
42		nfcv 2764	. . . . . 6 |- F/_ k -
43	14, 42, 28	nfov 6676	. . . . 5 |- F/_ k ( [_ j / k ]_ A - [_ j / k ]_ B )
44	19, 32	oveq12d 6668	. . . . 5 |- ( k = j -> ( A - B ) = ( [_ j / k ]_ A - [_ j / k ]_ B ) )
45		eqid 2622	. . . . 5 |- ( k e. Z |-> ( A - B ) ) = ( k e. Z |-> ( A - B ) )
46	13, 43, 44, 45	fvmptf 6301	. . . 4 |- ( ( j e. Z /\ ( [_ j / k ]_ A - [_ j / k ]_ B ) e. _V ) -> ( ( k e. Z |-> ( A - B ) ) ` j ) = ( [_ j / k ]_ A - [_ j / k ]_ B ) )
47	9, 41, 46	syl2anc 693	. . 3 |- ( ( ph /\ j e. Z ) -> ( ( k e. Z |-> ( A - B ) ) ` j ) = ( [_ j / k ]_ A - [_ j / k ]_ B ) )
48	26, 39	oveq12d 6668	. . 3 |- ( ( ph /\ j e. Z ) -> ( ( ( k e. Z |-> A ) ` j ) - ( ( k e. Z |-> B ) ` j ) ) = ( [_ j / k ]_ A - [_ j / k ]_ B ) )
49	47, 48	eqtr4d 2659	. 2 |- ( ( ph /\ j e. Z ) -> ( ( k e. Z |-> ( A - B ) ) ` j ) = ( ( ( k e. Z |-> A ) ` j ) - ( ( k e. Z |-> B ) ` j ) ) )
50	1, 2, 3, 7, 8, 27, 40, 49	climsub 14364	1 |- ( ph -> ( k e. Z |-> ( A - B ) ) ~~> ( C - D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   _Vcvv 3200   [_csb 3533   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   - cmin 10266   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:  climsubc2mpt  39893  climsubc1mpt  39894