Theorem ellimcabssub0 39849

Description: An equivalent condition for being a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
ellimcabssub0.f	|- F = ( x e. A |-> B )
ellimcabssub0.g	|- G = ( x e. A |-> ( B - C ) )
ellimcabssub0.a	|- ( ph -> A C_ CC )
ellimcabssub0.b	|- ( ( ph /\ x e. A ) -> B e. CC )
ellimcabssub0.p	|- ( ph -> D e. CC )
ellimcabssub0.c	|- ( ph -> C e. CC )

Assertion

Ref	Expression
ellimcabssub0	|- ( ph -> ( C e. ( F lim CC D ) <-> 0 e. ( G lim CC D ) ) )

Distinct variable groups:   x, A   x, C   ph, x

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

Proof of Theorem ellimcabssub0

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

Step	Hyp	Ref	Expression
1		ellimcabssub0.c	. . . 4 |- ( ph -> C e. CC )
2		0cnd 10033	. . . 4 |- ( ph -> 0 e. CC )
3	1, 2	2thd 255	. . 3 |- ( ph -> ( C e. CC <-> 0 e. CC ) )
4		ellimcabssub0.b	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> B e. CC )
5	1	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. A ) -> C e. CC )
6	4, 5	subcld 10392	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. A ) -> ( B - C ) e. CC )
7		ellimcabssub0.g	. . . . . . . . . . . . 13 |- G = ( x e. A |-> ( B - C ) )
8	6, 7	fmptd 6385	. . . . . . . . . . . 12 |- ( ph -> G : A --> CC )
9	8	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ z e. A ) -> ( G ` z ) e. CC )
10	9	subid1d 10381	. . . . . . . . . 10 |- ( ( ph /\ z e. A ) -> ( ( G ` z ) - 0 ) = ( G ` z ) )
11		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ z e. A ) -> z e. A )
12		vex 3203	. . . . . . . . . . . . . 14 |- z e. _V
13		csbov1g 6690	. . . . . . . . . . . . . 14 |- ( z e. _V -> [_ z / x ]_ ( B - C ) = ( [_ z / x ]_ B - C ) )
14	12, 13	ax-mp 5	. . . . . . . . . . . . 13 |- [_ z / x ]_ ( B - C ) = ( [_ z / x ]_ B - C )
15	14	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ z e. A ) -> [_ z / x ]_ ( B - C ) = ( [_ z / x ]_ B - C ) )
16		sban 2399	. . . . . . . . . . . . . . . . 17 |- ( [ z / x ] ( ph /\ x e. A ) <-> ( [ z / x ] ph /\ [ z / x ] x e. A ) )
17		nfv 1843	. . . . . . . . . . . . . . . . . . 19 |- F/ x ph
18	17	sbf 2380	. . . . . . . . . . . . . . . . . 18 |- ( [ z / x ] ph <-> ph )
19		clelsb3 2729	. . . . . . . . . . . . . . . . . 18 |- ( [ z / x ] x e. A <-> z e. A )
20	18, 19	anbi12i 733	. . . . . . . . . . . . . . . . 17 |- ( ( [ z / x ] ph /\ [ z / x ] x e. A ) <-> ( ph /\ z e. A ) )
21	16, 20	bitri 264	. . . . . . . . . . . . . . . 16 |- ( [ z / x ] ( ph /\ x e. A ) <-> ( ph /\ z e. A ) )
22	4	nfth 1727	. . . . . . . . . . . . . . . . . 18 |- F/ x ( ( ph /\ x e. A ) -> B e. CC )
23	22	sbf 2380	. . . . . . . . . . . . . . . . 17 |- ( [ z / x ] ( ( ph /\ x e. A ) -> B e. CC ) <-> ( ( ph /\ x e. A ) -> B e. CC ) )
24		sbim 2395	. . . . . . . . . . . . . . . . 17 |- ( [ z / x ] ( ( ph /\ x e. A ) -> B e. CC ) <-> ( [ z / x ] ( ph /\ x e. A ) -> [ z / x ] B e. CC ) )
25	23, 24	sylbb1 227	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. A ) -> B e. CC ) -> ( [ z / x ] ( ph /\ x e. A ) -> [ z / x ] B e. CC ) )
26	21, 25	syl5bir 233	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. A ) -> B e. CC ) -> ( ( ph /\ z e. A ) -> [ z / x ] B e. CC ) )
27	4, 26	ax-mp 5	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. A ) -> [ z / x ] B e. CC )
28		sbsbc 3439	. . . . . . . . . . . . . . 15 |- ( [ z / x ] B e. CC <-> [. z / x ]. B e. CC )
29		sbcel1g 3987	. . . . . . . . . . . . . . . 16 |- ( z e. _V -> ( [. z / x ]. B e. CC <-> [_ z / x ]_ B e. CC ) )
30	12, 29	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. z / x ]. B e. CC <-> [_ z / x ]_ B e. CC )
31	28, 30	bitri 264	. . . . . . . . . . . . . 14 |- ( [ z / x ] B e. CC <-> [_ z / x ]_ B e. CC )
32	27, 31	sylib 208	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. A ) -> [_ z / x ]_ B e. CC )
33	1	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. A ) -> C e. CC )
34	32, 33	subcld 10392	. . . . . . . . . . . 12 |- ( ( ph /\ z e. A ) -> ( [_ z / x ]_ B - C ) e. CC )
35	15, 34	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ph /\ z e. A ) -> [_ z / x ]_ ( B - C ) e. CC )
36	7	fvmpts 6285	. . . . . . . . . . 11 |- ( ( z e. A /\ [_ z / x ]_ ( B - C ) e. CC ) -> ( G ` z ) = [_ z / x ]_ ( B - C ) )
37	11, 35, 36	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ z e. A ) -> ( G ` z ) = [_ z / x ]_ ( B - C ) )
38		ellimcabssub0.f	. . . . . . . . . . . . . 14 |- F = ( x e. A |-> B )
39	38	fvmpts 6285	. . . . . . . . . . . . 13 |- ( ( z e. A /\ [_ z / x ]_ B e. CC ) -> ( F ` z ) = [_ z / x ]_ B )
40	11, 32, 39	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ z e. A ) -> ( F ` z ) = [_ z / x ]_ B )
41	40	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ph /\ z e. A ) -> ( ( F ` z ) - C ) = ( [_ z / x ]_ B - C ) )
42	41, 14	syl6reqr 2675	. . . . . . . . . 10 |- ( ( ph /\ z e. A ) -> [_ z / x ]_ ( B - C ) = ( ( F ` z ) - C ) )
43	10, 37, 42	3eqtrrd 2661	. . . . . . . . 9 |- ( ( ph /\ z e. A ) -> ( ( F ` z ) - C ) = ( ( G ` z ) - 0 ) )
44	43	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ z e. A ) -> ( abs ` ( ( F ` z ) - C ) ) = ( abs ` ( ( G ` z ) - 0 ) ) )
45	44	breq1d 4663	. . . . . . 7 |- ( ( ph /\ z e. A ) -> ( ( abs ` ( ( F ` z ) - C ) ) < y <-> ( abs ` ( ( G ` z ) - 0 ) ) < y ) )
46	45	imbi2d 330	. . . . . 6 |- ( ( ph /\ z e. A ) -> ( ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( F ` z ) - C ) ) < y ) <-> ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( G ` z ) - 0 ) ) < y ) ) )
47	46	ralbidva 2985	. . . . 5 |- ( ph -> ( A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( F ` z ) - C ) ) < y ) <-> A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( G ` z ) - 0 ) ) < y ) ) )
48	47	rexbidv 3052	. . . 4 |- ( ph -> ( E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( F ` z ) - C ) ) < y ) <-> E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( G ` z ) - 0 ) ) < y ) ) )
49	48	ralbidv 2986	. . 3 |- ( ph -> ( A. y e. RR+ E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( F ` z ) - C ) ) < y ) <-> A. y e. RR+ E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( G ` z ) - 0 ) ) < y ) ) )
50	3, 49	anbi12d 747	. 2 |- ( ph -> ( ( C e. CC /\ A. y e. RR+ E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( F ` z ) - C ) ) < y ) ) <-> ( 0 e. CC /\ A. y e. RR+ E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( G ` z ) - 0 ) ) < y ) ) ) )
51	4, 38	fmptd 6385	. . 3 |- ( ph -> F : A --> CC )
52		ellimcabssub0.a	. . 3 |- ( ph -> A C_ CC )
53		ellimcabssub0.p	. . 3 |- ( ph -> D e. CC )
54	51, 52, 53	ellimc3 23643	. 2 |- ( ph -> ( C e. ( F lim CC D ) <-> ( C e. CC /\ A. y e. RR+ E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( F ` z ) - C ) ) < y ) ) ) )
55	8, 52, 53	ellimc3 23643	. 2 |- ( ph -> ( 0 e. ( G lim CC D ) <-> ( 0 e. CC /\ A. y e. RR+ E. w e. RR+ A. z e. A ( ( z =/= D /\ ( abs ` ( z - D ) ) < w ) -> ( abs ` ( ( G ` z ) - 0 ) ) < y ) ) ) )
56	50, 54, 55	3bitr4d 300	1 |- ( ph -> ( C e. ( F lim CC D ) <-> 0 e. ( G lim CC D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   [wsb 1880   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   [_csb 3533   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   < clt 10074   - cmin 10266   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-fal 1489  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:  reclimc  39885