Theorem climneg 39842

Description: Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
climneg.1	|- F/ k ph
climneg.2	|- F/_ k F
climneg.3	|- Z = ( ZZ>= ` M )
climneg.4	|- ( ph -> M e. ZZ )
climneg.5	|- ( ph -> F ~~> A )
climneg.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )

Assertion

Ref	Expression
climneg	|- ( ph -> ( k e. Z |-> -u ( F ` k ) ) ~~> -u A )

Distinct variable group:   k, Z

Allowed substitution hints:   ph( k)   A( k)   F( k)   M( k)

Proof of Theorem climneg

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climneg.1	. . 3 |- F/ k ph
2		nfmpt1 4747	. . 3 |- F/_ k ( k e. Z |-> -u 1 )
3		climneg.2	. . 3 |- F/_ k F
4		nfmpt1 4747	. . 3 |- F/_ k ( k e. Z |-> -u ( F ` k ) )
5		climneg.3	. . 3 |- Z = ( ZZ>= ` M )
6		climneg.4	. . 3 |- ( ph -> M e. ZZ )
7		fvex 6201	. . . . . . 7 |- ( ZZ>= ` M ) e. _V
8	5, 7	eqeltri 2697	. . . . . 6 |- Z e. _V
9	8	mptex 6486	. . . . 5 |- ( k e. Z |-> -u 1 ) e. _V
10	9	a1i 11	. . . 4 |- ( ph -> ( k e. Z |-> -u 1 ) e. _V )
11		1cnd 10056	. . . . 5 |- ( ph -> 1 e. CC )
12	11	negcld 10379	. . . 4 |- ( ph -> -u 1 e. CC )
13		eqidd 2623	. . . . . 6 |- ( j e. Z -> ( k e. Z |-> -u 1 ) = ( k e. Z |-> -u 1 ) )
14		eqidd 2623	. . . . . 6 |- ( ( j e. Z /\ k = j ) -> -u 1 = -u 1 )
15		id 22	. . . . . 6 |- ( j e. Z -> j e. Z )
16		1cnd 10056	. . . . . . 7 |- ( j e. Z -> 1 e. CC )
17	16	negcld 10379	. . . . . 6 |- ( j e. Z -> -u 1 e. CC )
18	13, 14, 15, 17	fvmptd 6288	. . . . 5 |- ( j e. Z -> ( ( k e. Z |-> -u 1 ) ` j ) = -u 1 )
19	18	adantl 482	. . . 4 |- ( ( ph /\ j e. Z ) -> ( ( k e. Z |-> -u 1 ) ` j ) = -u 1 )
20	5, 6, 10, 12, 19	climconst 14274	. . 3 |- ( ph -> ( k e. Z |-> -u 1 ) ~~> -u 1 )
21	8	mptex 6486	. . . 4 |- ( k e. Z |-> -u ( F ` k ) ) e. _V
22	21	a1i 11	. . 3 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) e. _V )
23		climneg.5	. . 3 |- ( ph -> F ~~> A )
24		neg1cn 11124	. . . . . 6 |- -u 1 e. CC
25		eqid 2622	. . . . . . 7 |- ( k e. Z |-> -u 1 ) = ( k e. Z |-> -u 1 )
26	25	fvmpt2 6291	. . . . . 6 |- ( ( k e. Z /\ -u 1 e. CC ) -> ( ( k e. Z |-> -u 1 ) ` k ) = -u 1 )
27	24, 26	mpan2 707	. . . . 5 |- ( k e. Z -> ( ( k e. Z |-> -u 1 ) ` k ) = -u 1 )
28	27, 24	syl6eqel 2709	. . . 4 |- ( k e. Z -> ( ( k e. Z |-> -u 1 ) ` k ) e. CC )
29	28	adantl 482	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> -u 1 ) ` k ) e. CC )
30		climneg.6	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
31		simpr 477	. . . . 5 |- ( ( ph /\ k e. Z ) -> k e. Z )
32	30	negcld 10379	. . . . 5 |- ( ( ph /\ k e. Z ) -> -u ( F ` k ) e. CC )
33		eqid 2622	. . . . . 6 |- ( k e. Z |-> -u ( F ` k ) ) = ( k e. Z |-> -u ( F ` k ) )
34	33	fvmpt2 6291	. . . . 5 |- ( ( k e. Z /\ -u ( F ` k ) e. CC ) -> ( ( k e. Z |-> -u ( F ` k ) ) ` k ) = -u ( F ` k ) )
35	31, 32, 34	syl2anc 693	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> -u ( F ` k ) ) ` k ) = -u ( F ` k ) )
36	30	mulm1d 10482	. . . 4 |- ( ( ph /\ k e. Z ) -> ( -u 1 x. ( F ` k ) ) = -u ( F ` k ) )
37	27	eqcomd 2628	. . . . . 6 |- ( k e. Z -> -u 1 = ( ( k e. Z |-> -u 1 ) ` k ) )
38	37	adantl 482	. . . . 5 |- ( ( ph /\ k e. Z ) -> -u 1 = ( ( k e. Z |-> -u 1 ) ` k ) )
39	38	oveq1d 6665	. . . 4 |- ( ( ph /\ k e. Z ) -> ( -u 1 x. ( F ` k ) ) = ( ( ( k e. Z |-> -u 1 ) ` k ) x. ( F ` k ) ) )
40	35, 36, 39	3eqtr2d 2662	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> -u ( F ` k ) ) ` k ) = ( ( ( k e. Z |-> -u 1 ) ` k ) x. ( F ` k ) ) )
41	1, 2, 3, 4, 5, 6, 20, 22, 23, 29, 30, 40	climmulf 39836	. 2 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) ~~> ( -u 1 x. A ) )
42		climcl 14230	. . . 4 |- ( F ~~> A -> A e. CC )
43	23, 42	syl 17	. . 3 |- ( ph -> A e. CC )
44	43	mulm1d 10482	. 2 |- ( ph -> ( -u 1 x. A ) = -u A )
45	41, 44	breqtrd 4679	1 |- ( ph -> ( k e. Z |-> -u ( F ` k ) ) ~~> -u A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   x. cmul 9941   -ucneg 10267   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:  climliminflimsupd  40033