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Theorem rlim0lt 12295
Description: Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
Hypotheses
Ref Expression
rlim0.1  |-  ( ph  ->  A. z  e.  A  B  e.  CC )
rlim0.2  |-  ( ph  ->  A  C_  RR )
Assertion
Ref Expression
rlim0lt  |-  ( ph  ->  ( ( z  e.  A  |->  B )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  < 
z  ->  ( abs `  B )  <  x
) ) )
Distinct variable groups:    x, y,
z, A    x, B, y    ph, x, y
Allowed substitution hints:    ph( z)    B( z)

Proof of Theorem rlim0lt
StepHypRef Expression
1 rlim0.1 . . 3  |-  ( ph  ->  A. z  e.  A  B  e.  CC )
2 rlim0.2 . . 3  |-  ( ph  ->  A  C_  RR )
3 0cn 9076 . . . 4  |-  0  e.  CC
43a1i 11 . . 3  |-  ( ph  ->  0  e.  CC )
51, 2, 4rlim2lt 12283 . 2  |-  ( ph  ->  ( ( z  e.  A  |->  B )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  < 
z  ->  ( abs `  ( B  -  0 ) )  <  x
) ) )
6 subid1 9314 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B  -  0 )  =  B )
76fveq2d 5724 . . . . . . . 8  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
87breq1d 4214 . . . . . . 7  |-  ( B  e.  CC  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
98imbi2d 308 . . . . . 6  |-  ( B  e.  CC  ->  (
( y  <  z  ->  ( abs `  ( B  -  0 ) )  <  x )  <-> 
( y  <  z  ->  ( abs `  B
)  <  x )
) )
109ralimi 2773 . . . . 5  |-  ( A. z  e.  A  B  e.  CC  ->  A. z  e.  A  ( (
y  <  z  ->  ( abs `  ( B  -  0 ) )  <  x )  <->  ( y  <  z  ->  ( abs `  B )  <  x
) ) )
11 ralbi 2834 . . . . 5  |-  ( A. z  e.  A  (
( y  <  z  ->  ( abs `  ( B  -  0 ) )  <  x )  <-> 
( y  <  z  ->  ( abs `  B
)  <  x )
)  ->  ( A. z  e.  A  (
y  <  z  ->  ( abs `  ( B  -  0 ) )  <  x )  <->  A. z  e.  A  ( y  <  z  ->  ( abs `  B )  <  x
) ) )
121, 10, 113syl 19 . . . 4  |-  ( ph  ->  ( A. z  e.  A  ( y  < 
z  ->  ( abs `  ( B  -  0 ) )  <  x
)  <->  A. z  e.  A  ( y  <  z  ->  ( abs `  B
)  <  x )
) )
1312rexbidv 2718 . . 3  |-  ( ph  ->  ( E. y  e.  RR  A. z  e.  A  ( y  < 
z  ->  ( abs `  ( B  -  0 ) )  <  x
)  <->  E. y  e.  RR  A. z  e.  A  ( y  <  z  -> 
( abs `  B
)  <  x )
) )
1413ralbidv 2717 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  <  z  -> 
( abs `  ( B  -  0 ) )  <  x )  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  < 
z  ->  ( abs `  B )  <  x
) ) )
155, 14bitrd 245 1  |-  ( ph  ->  ( ( z  e.  A  |->  B )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  < 
z  ->  ( abs `  B )  <  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    < clt 9112    - cmin 9283   RR+crp 10604   abscabs 12031    ~~> r crli 12271
This theorem is referenced by:  divrcnv  12624  divlogrlim  20518  cxplim  20802  cxploglim  20808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-rlim 12275
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