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Theorem ellimcabssub0 37004
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.
StepHypRef Expression
1 ellimcabssub0.c . . . 4  |-  ( ph  ->  C  e.  CC )
2 0cnd 9621 . . . 4  |-  ( ph  ->  0  e.  CC )
31, 22thd 242 . . 3  |-  ( ph  ->  ( C  e.  CC  <->  0  e.  CC ) )
4 ellimcabssub0.b . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
51adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
64, 5subcld 9969 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  ( B  -  C )  e.  CC )
7 ellimcabssub0.g . . . . . . . . . . . . 13  |-  G  =  ( x  e.  A  |->  ( B  -  C
) )
86, 7fmptd 6035 . . . . . . . . . . . 12  |-  ( ph  ->  G : A --> CC )
98ffvelrnda 6011 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  A )  ->  ( G `  z )  e.  CC )
109subid1d 9958 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  A )  ->  (
( G `  z
)  -  0 )  =  ( G `  z ) )
11 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  A )  ->  z  e.  A )
12 vex 3064 . . . . . . . . . . . . . 14  |-  z  e. 
_V
13 csbov1g 6317 . . . . . . . . . . . . . 14  |-  ( z  e.  _V  ->  [_ z  /  x ]_ ( B  -  C )  =  ( [_ z  /  x ]_ B  -  C
) )
1412, 13ax-mp 5 . . . . . . . . . . . . 13  |-  [_ z  /  x ]_ ( B  -  C )  =  ( [_ z  /  x ]_ B  -  C
)
1514a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ ( B  -  C )  =  ( [_ z  /  x ]_ B  -  C
) )
16 sban 2166 . . . . . . . . . . . . . . . . 17  |-  ( [ z  /  x ]
( ph  /\  x  e.  A )  <->  ( [
z  /  x ] ph  /\  [ z  /  x ] x  e.  A
) )
17 nfv 1730 . . . . . . . . . . . . . . . . . . 19  |-  F/ x ph
1817sbf 2147 . . . . . . . . . . . . . . . . . 18  |-  ( [ z  /  x ] ph 
<-> 
ph )
19 clelsb3 2525 . . . . . . . . . . . . . . . . . 18  |-  ( [ z  /  x ]
x  e.  A  <->  z  e.  A )
2018, 19anbi12i 697 . . . . . . . . . . . . . . . . 17  |-  ( ( [ z  /  x ] ph  /\  [ z  /  x ] x  e.  A )  <->  ( ph  /\  z  e.  A ) )
2116, 20bitri 251 . . . . . . . . . . . . . . . 16  |-  ( [ z  /  x ]
( ph  /\  x  e.  A )  <->  ( ph  /\  z  e.  A ) )
224nfth 1648 . . . . . . . . . . . . . . . . . . 19  |-  F/ x
( ( ph  /\  x  e.  A )  ->  B  e.  CC )
2322sbf 2147 . . . . . . . . . . . . . . . . . 18  |-  ( [ z  /  x ]
( ( ph  /\  x  e.  A )  ->  B  e.  CC )  <-> 
( ( ph  /\  x  e.  A )  ->  B  e.  CC ) )
24 sbim 2162 . . . . . . . . . . . . . . . . . 18  |-  ( [ z  /  x ]
( ( ph  /\  x  e.  A )  ->  B  e.  CC )  <-> 
( [ z  /  x ] ( ph  /\  x  e.  A )  ->  [ z  /  x ] B  e.  CC ) )
2523, 24bitr3i 253 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  A )  ->  B  e.  CC )  <->  ( [
z  /  x ]
( ph  /\  x  e.  A )  ->  [ z  /  x ] B  e.  CC ) )
2625biimpi 196 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  ->  B  e.  CC )  ->  ( [ z  /  x ] ( ph  /\  x  e.  A )  ->  [ z  /  x ] B  e.  CC ) )
2721, 26syl5bir 220 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  ->  B  e.  CC )  ->  (
( ph  /\  z  e.  A )  ->  [ z  /  x ] B  e.  CC ) )
284, 27ax-mp 5 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  A )  ->  [ z  /  x ] B  e.  CC )
29 sbsbc 3283 . . . . . . . . . . . . . . 15  |-  ( [ z  /  x ] B  e.  CC  <->  [. z  /  x ]. B  e.  CC )
30 sbcel1g 3783 . . . . . . . . . . . . . . . 16  |-  ( z  e.  _V  ->  ( [. z  /  x ]. B  e.  CC  <->  [_ z  /  x ]_ B  e.  CC )
)
3112, 30ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( [. z  /  x ]. B  e.  CC  <->  [_ z  /  x ]_ B  e.  CC )
3229, 31bitri 251 . . . . . . . . . . . . . 14  |-  ( [ z  /  x ] B  e.  CC  <->  [_ z  /  x ]_ B  e.  CC )
3328, 32sylib 198 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ B  e.  CC )
341adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  A )  ->  C  e.  CC )
3533, 34subcld 9969 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  A )  ->  ( [_ z  /  x ]_ B  -  C
)  e.  CC )
3615, 35eqeltrd 2492 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ ( B  -  C )  e.  CC )
377fvmpts 5937 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  [_ z  /  x ]_ ( B  -  C
)  e.  CC )  ->  ( G `  z )  =  [_ z  /  x ]_ ( B  -  C )
)
3811, 36, 37syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  A )  ->  ( G `  z )  =  [_ z  /  x ]_ ( B  -  C
) )
39 ellimcabssub0.f . . . . . . . . . . . . . 14  |-  F  =  ( x  e.  A  |->  B )
4039fvmpts 5937 . . . . . . . . . . . . 13  |-  ( ( z  e.  A  /\  [_ z  /  x ]_ B  e.  CC )  ->  ( F `  z
)  =  [_ z  /  x ]_ B )
4111, 33, 40syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  =  [_ z  /  x ]_ B )
4241oveq1d 6295 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  A )  ->  (
( F `  z
)  -  C )  =  ( [_ z  /  x ]_ B  -  C ) )
4342, 14syl6reqr 2464 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ ( B  -  C )  =  ( ( F `  z )  -  C
) )
4410, 38, 433eqtrrd 2450 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  A )  ->  (
( F `  z
)  -  C )  =  ( ( G `
 z )  - 
0 ) )
4544fveq2d 5855 . . . . . . . 8  |-  ( (
ph  /\  z  e.  A )  ->  ( abs `  ( ( F `
 z )  -  C ) )  =  ( abs `  (
( G `  z
)  -  0 ) ) )
4645breq1d 4407 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  (
( abs `  (
( F `  z
)  -  C ) )  <  y  <->  ( abs `  ( ( G `  z )  -  0 ) )  <  y
) )
4746imbi2d 316 . . . . . 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 ) ) )
4847ralbidva 2842 . . . . 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 ) ) )
4948rexbidv 2920 . . . 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 ) ) )
5049ralbidv 2845 . . 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 ) ) )
513, 50anbi12d 711 . 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 ) ) ) )
524, 39fmptd 6035 . . 3  |-  ( ph  ->  F : A --> CC )
53 ellimcabssub0.a . . 3  |-  ( ph  ->  A  C_  CC )
54 ellimcabssub0.p . . 3  |-  ( ph  ->  D  e.  CC )
5552, 53, 54ellimc3 22577 . 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 ) ) ) )
568, 53, 54ellimc3 22577 . 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 ) ) ) )
5751, 55, 563bitr4d 287 1  |-  ( ph  ->  ( C  e.  ( F lim CC  D )  <->  0  e.  ( G lim
CC  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407   [wsb 1765    e. wcel 1844    =/= wne 2600   A.wral 2756   E.wrex 2757   _Vcvv 3061   [.wsbc 3279   [_csb 3375    C_ wss 3416   class class class wbr 4397    |-> cmpt 4455   ` cfv 5571  (class class class)co 6280   CCcc 9522   0cc0 9524    < clt 9660    - cmin 9843   RR+crp 11267   abscabs 13218   lim CC climc 22560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fi 7907  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-fz 11729  df-seq 12154  df-exp 12213  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-plusg 14924  df-mulr 14925  df-starv 14926  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-rest 15039  df-topn 15040  df-topgen 15060  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-cnfld 18743  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cnp 20024  df-xms 21117  df-ms 21118  df-limc 22564
This theorem is referenced by:  reclimc  37040
  Copyright terms: Public domain W3C validator