Theorem neglimc 39879

Description: Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
neglimc.f	|- F = ( x e. A |-> B )
neglimc.g	|- G = ( x e. A |-> -u B )
neglimc.b	|- ( ( ph /\ x e. A ) -> B e. CC )
neglimc.c	|- ( ph -> C e. ( F lim CC D ) )

Assertion

Ref	Expression
neglimc	|- ( ph -> -u C e. ( G lim CC D ) )

Distinct variable groups:   x, A   ph, x

Allowed substitution hints:   B( x)   C( x)   D( x)   F( x)   G( x)

Proof of Theorem neglimc

Dummy variables v w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl 23639	. . . 4 |- ( F lim CC D ) C_ CC
2		neglimc.c	. . . 4 |- ( ph -> C e. ( F lim CC D ) )
3	1, 2	sseldi 3601	. . 3 |- ( ph -> C e. CC )
4	3	negcld 10379	. 2 |- ( ph -> -u C e. CC )
5		neglimc.b	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> B e. CC )
6		neglimc.f	. . . . . . . . 9 |- F = ( x e. A |-> B )
7	5, 6	fmptd 6385	. . . . . . . 8 |- ( ph -> F : A --> CC )
8	6, 5, 2	limcmptdm 39867	. . . . . . . 8 |- ( ph -> A C_ CC )
9		limcrcl 23638	. . . . . . . . . 10 |- ( C e. ( F lim CC D ) -> ( F : dom F --> CC /\ dom F C_ CC /\ D e. CC ) )
10	2, 9	syl 17	. . . . . . . . 9 |- ( ph -> ( F : dom F --> CC /\ dom F C_ CC /\ D e. CC ) )
11	10	simp3d 1075	. . . . . . . 8 |- ( ph -> D e. CC )
12	7, 8, 11	ellimc3 23643	. . . . . . 7 |- ( ph -> ( C e. ( F lim CC D ) <-> ( C e. CC /\ A. y e. RR+ E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) ) ) )
13	2, 12	mpbid 222	. . . . . 6 |- ( ph -> ( C e. CC /\ A. y e. RR+ E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) ) )
14	13	simprd 479	. . . . 5 |- ( ph -> A. y e. RR+ E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) )
15	14	r19.21bi 2932	. . . 4 |- ( ( ph /\ y e. RR+ ) -> E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) )
16		simplll 798	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) -> ph )
17	16	3ad2ant1 1082	. . . . . . . 8 |- ( ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) /\ ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) /\ ( v =/= D /\ ( abs ` ( v - D ) ) < w ) ) -> ph )
18		simp1r 1086	. . . . . . . 8 |- ( ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) /\ ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) /\ ( v =/= D /\ ( abs ` ( v - D ) ) < w ) ) -> v e. A )
19		simp3 1063	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) /\ ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) /\ ( v =/= D /\ ( abs ` ( v - D ) ) < w ) ) -> ( v =/= D /\ ( abs ` ( v - D ) ) < w ) )
20		simp2 1062	. . . . . . . . 9 |- ( ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) /\ ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) /\ ( v =/= D /\ ( abs ` ( v - D ) ) < w ) ) -> ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) )
21	19, 20	mpd 15	. . . . . . . 8 |- ( ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) /\ ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) /\ ( v =/= D /\ ( abs ` ( v - D ) ) < w ) ) -> ( abs ` ( ( F ` v ) - C ) ) < y )
22		nfv 1843	. . . . . . . . . . . . . . . 16 |- F/ x ( ph /\ v e. A )
23		neglimc.g	. . . . . . . . . . . . . . . . . . 19 |- G = ( x e. A |-> -u B )
24		nfmpt1 4747	. . . . . . . . . . . . . . . . . . 19 |- F/_ x ( x e. A |-> -u B )
25	23, 24	nfcxfr 2762	. . . . . . . . . . . . . . . . . 18 |- F/_ x G
26		nfcv 2764	. . . . . . . . . . . . . . . . . 18 |- F/_ x v
27	25, 26	nffv 6198	. . . . . . . . . . . . . . . . 17 |- F/_ x ( G ` v )
28		nfmpt1 4747	. . . . . . . . . . . . . . . . . . . 20 |- F/_ x ( x e. A |-> B )
29	6, 28	nfcxfr 2762	. . . . . . . . . . . . . . . . . . 19 |- F/_ x F
30	29, 26	nffv 6198	. . . . . . . . . . . . . . . . . 18 |- F/_ x ( F ` v )
31	30	nfneg 10277	. . . . . . . . . . . . . . . . 17 |- F/_ x -u ( F ` v )
32	27, 31	nfeq 2776	. . . . . . . . . . . . . . . 16 |- F/ x ( G ` v ) = -u ( F ` v )
33	22, 32	nfim 1825	. . . . . . . . . . . . . . 15 |- F/ x ( ( ph /\ v e. A ) -> ( G ` v ) = -u ( F ` v ) )
34		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( x = v -> ( x e. A <-> v e. A ) )
35	34	anbi2d 740	. . . . . . . . . . . . . . . 16 |- ( x = v -> ( ( ph /\ x e. A ) <-> ( ph /\ v e. A ) ) )
36		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( x = v -> ( G ` x ) = ( G ` v ) )
37		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( x = v -> ( F ` x ) = ( F ` v ) )
38	37	negeqd 10275	. . . . . . . . . . . . . . . . 17 |- ( x = v -> -u ( F ` x ) = -u ( F ` v ) )
39	36, 38	eqeq12d 2637	. . . . . . . . . . . . . . . 16 |- ( x = v -> ( ( G ` x ) = -u ( F ` x ) <-> ( G ` v ) = -u ( F ` v ) ) )
40	35, 39	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( x = v -> ( ( ( ph /\ x e. A ) -> ( G ` x ) = -u ( F ` x ) ) <-> ( ( ph /\ v e. A ) -> ( G ` v ) = -u ( F ` v ) ) ) )
41		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. A ) -> x e. A )
42	5	negcld 10379	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. A ) -> -u B e. CC )
43	23	fvmpt2 6291	. . . . . . . . . . . . . . . . 17 |- ( ( x e. A /\ -u B e. CC ) -> ( G ` x ) = -u B )
44	41, 42, 43	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. A ) -> ( G ` x ) = -u B )
45	6	fvmpt2 6291	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. A /\ B e. CC ) -> ( F ` x ) = B )
46	41, 5, 45	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
47	46	negeqd 10275	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. A ) -> -u ( F ` x ) = -u B )
48	44, 47	eqtr4d 2659	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. A ) -> ( G ` x ) = -u ( F ` x ) )
49	33, 40, 48	chvar 2262	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. A ) -> ( G ` v ) = -u ( F ` v ) )
50	49	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. A ) -> ( ( G ` v ) - -u C ) = ( -u ( F ` v ) - -u C ) )
51	7	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. A ) -> ( F ` v ) e. CC )
52	3	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ v e. A ) -> C e. CC )
53	51, 52	negsubdi3d 39506	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. A ) -> -u ( ( F ` v ) - C ) = ( -u ( F ` v ) - -u C ) )
54	50, 53	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ( ph /\ v e. A ) -> ( ( G ` v ) - -u C ) = -u ( ( F ` v ) - C ) )
55	54	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ v e. A ) -> ( abs ` ( ( G ` v ) - -u C ) ) = ( abs ` -u ( ( F ` v ) - C ) ) )
56	51, 52	subcld 10392	. . . . . . . . . . . 12 |- ( ( ph /\ v e. A ) -> ( ( F ` v ) - C ) e. CC )
57	56	absnegd 14188	. . . . . . . . . . 11 |- ( ( ph /\ v e. A ) -> ( abs ` -u ( ( F ` v ) - C ) ) = ( abs ` ( ( F ` v ) - C ) ) )
58	55, 57	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ v e. A ) -> ( abs ` ( ( G ` v ) - -u C ) ) = ( abs ` ( ( F ` v ) - C ) ) )
59	58	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ v e. A ) /\ ( abs ` ( ( F ` v ) - C ) ) < y ) -> ( abs ` ( ( G ` v ) - -u C ) ) = ( abs ` ( ( F ` v ) - C ) ) )
60		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ v e. A ) /\ ( abs ` ( ( F ` v ) - C ) ) < y ) -> ( abs ` ( ( F ` v ) - C ) ) < y )
61	59, 60	eqbrtrd 4675	. . . . . . . 8 |- ( ( ( ph /\ v e. A ) /\ ( abs ` ( ( F ` v ) - C ) ) < y ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y )
62	17, 18, 21, 61	syl21anc 1325	. . . . . . 7 |- ( ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) /\ ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) /\ ( v =/= D /\ ( abs ` ( v - D ) ) < w ) ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y )
63	62	3exp 1264	. . . . . 6 |- ( ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) /\ v e. A ) -> ( ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) -> ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y ) ) )
64	63	ralimdva 2962	. . . . 5 |- ( ( ( ph /\ y e. RR+ ) /\ w e. RR+ ) -> ( A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) -> A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y ) ) )
65	64	reximdva 3017	. . . 4 |- ( ( ph /\ y e. RR+ ) -> ( E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( F ` v ) - C ) ) < y ) -> E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y ) ) )
66	15, 65	mpd 15	. . 3 |- ( ( ph /\ y e. RR+ ) -> E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y ) )
67	66	ralrimiva 2966	. 2 |- ( ph -> A. y e. RR+ E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y ) )
68	42, 23	fmptd 6385	. . 3 |- ( ph -> G : A --> CC )
69	68, 8, 11	ellimc3 23643	. 2 |- ( ph -> ( -u C e. ( G lim CC D ) <-> ( -u C e. CC /\ A. y e. RR+ E. w e. RR+ A. v e. A ( ( v =/= D /\ ( abs ` ( v - D ) ) < w ) -> ( abs ` ( ( G ` v ) - -u C ) ) < y ) ) ) )
70	4, 67, 69	mpbir2and 957	1 |- ( ph -> -u C e. ( G lim CC D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   < clt 10074   - cmin 10266   -ucneg 10267   RR+crp 11832   abscabs 13974   lim CC climc 23626

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-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cnp 21032  df-xms 22125  df-ms 22126  df-limc 23630

This theorem is referenced by:  sublimc  39884  reclimc  39885