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Theorem cnplimclemle 14008
Description: Lemma for cnplimccntop 14010. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
Hypotheses
Ref Expression
cnplimccntop.k  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
cnplimc.j  |-  J  =  ( Kt  A )
cnplimclemr.a  |-  ( ph  ->  A  C_  CC )
cnplimclemr.f  |-  ( ph  ->  F : A --> CC )
cnplimclemr.b  |-  ( ph  ->  B  e.  A )
cnplimclemr.l  |-  ( ph  ->  ( F `  B
)  e.  ( F lim
CC  B ) )
cnplimclemle.e  |-  ( ph  ->  E  e.  RR+ )
cnplimclemle.d  |-  ( ph  ->  D  e.  RR+ )
cnplimclemle.z  |-  ( ph  ->  Z  e.  A )
cnplimclemle.im  |-  ( (
ph  /\  Z #  B  /\  ( abs `  ( Z  -  B )
)  <  D )  ->  ( abs `  (
( F `  Z
)  -  ( F `
 B ) ) )  <  ( E  /  2 ) )
cnplimclemle.zd  |-  ( ph  ->  ( abs `  ( Z  -  B )
)  <  D )
Assertion
Ref Expression
cnplimclemle  |-  ( ph  ->  ( abs `  (
( F `  Z
)  -  ( F `
 B ) ) )  <  E )

Proof of Theorem cnplimclemle
StepHypRef Expression
1 simpr 110 . . 3  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )
2 cnplimclemr.f . . . . . . . 8  |-  ( ph  ->  F : A --> CC )
3 cnplimclemle.z . . . . . . . 8  |-  ( ph  ->  Z  e.  A )
42, 3ffvelcdmd 5651 . . . . . . 7  |-  ( ph  ->  ( F `  Z
)  e.  CC )
5 cnplimclemr.b . . . . . . . 8  |-  ( ph  ->  B  e.  A )
62, 5ffvelcdmd 5651 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  CC )
74, 6subcld 8264 . . . . . 6  |-  ( ph  ->  ( ( F `  Z )  -  ( F `  B )
)  e.  CC )
87abscld 11183 . . . . 5  |-  ( ph  ->  ( abs `  (
( F `  Z
)  -  ( F `
 B ) ) )  e.  RR )
98adantr 276 . . . 4  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  e.  RR )
10 cnplimclemle.e . . . . . . 7  |-  ( ph  ->  E  e.  RR+ )
1110rphalfcld 9705 . . . . . 6  |-  ( ph  ->  ( E  /  2
)  e.  RR+ )
1211rpred 9692 . . . . 5  |-  ( ph  ->  ( E  /  2
)  e.  RR )
1312adantr 276 . . . 4  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( E  /  2 )  e.  RR )
144adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( F `  Z )  e.  CC )
151adantr 276 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )
16 simpll 527 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  ph )
1716, 8syl 14 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  e.  RR )
1816, 12syl 14 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  ( E  /  2 )  e.  RR )
19 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  Z #  B )
20 cnplimclemle.zd . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( Z  -  B )
)  <  D )
2116, 20syl 14 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  ( abs `  ( Z  -  B ) )  < 
D )
22 cnplimclemle.im . . . . . . . . . . . 12  |-  ( (
ph  /\  Z #  B  /\  ( abs `  ( Z  -  B )
)  <  D )  ->  ( abs `  (
( F `  Z
)  -  ( F `
 B ) ) )  <  ( E  /  2 ) )
2316, 19, 21, 22syl3anc 1238 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  <  ( E  /  2 ) )
2417, 18, 23ltnsymd 8073 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  /\  Z #  B )  ->  -.  ( E  /  2
)  <  ( abs `  ( ( F `  Z )  -  ( F `  B )
) ) )
2515, 24pm2.65da 661 . . . . . . . . 9  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  -.  Z #  B )
26 cnplimclemr.a . . . . . . . . . . 11  |-  ( ph  ->  A  C_  CC )
2726, 3sseldd 3156 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  CC )
2826, 5sseldd 3156 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
2928adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  B  e.  CC )
30 apti 8575 . . . . . . . . . 10  |-  ( ( Z  e.  CC  /\  B  e.  CC )  ->  ( Z  =  B  <->  -.  Z #  B )
)
3127, 29, 30syl2an2r 595 . . . . . . . . 9  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( Z  =  B  <->  -.  Z #  B ) )
3225, 31mpbird 167 . . . . . . . 8  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  Z  =  B )
3332fveq2d 5518 . . . . . . 7  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( F `  Z )  =  ( F `  B ) )
3414, 33subeq0bd 8332 . . . . . 6  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  (
( F `  Z
)  -  ( F `
 B ) )  =  0 )
3534abs00bd 11068 . . . . 5  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  =  0 )
3611adantr 276 . . . . . 6  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( E  /  2 )  e.  RR+ )
3736rpgt0d 9695 . . . . 5  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  0  <  ( E  /  2
) )
3835, 37eqbrtrd 4024 . . . 4  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  <  ( E  /  2 ) )
399, 13, 38ltnsymd 8073 . . 3  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  -.  ( E  /  2
)  <  ( abs `  ( ( F `  Z )  -  ( F `  B )
) ) )
401, 39pm2.21dd 620 . 2  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  <  E
)
41 simpr 110 . 2  |-  ( (
ph  /\  ( abs `  ( ( F `  Z )  -  ( F `  B )
) )  <  E
)  ->  ( abs `  ( ( F `  Z )  -  ( F `  B )
) )  <  E
)
42 rphalflt 9679 . . . 4  |-  ( E  e.  RR+  ->  ( E  /  2 )  < 
E )
4310, 42syl 14 . . 3  |-  ( ph  ->  ( E  /  2
)  <  E )
4410rpred 9692 . . . 4  |-  ( ph  ->  E  e.  RR )
45 axltwlin 8021 . . . 4  |-  ( ( ( E  /  2
)  e.  RR  /\  E  e.  RR  /\  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  e.  RR )  ->  ( ( E  /  2 )  < 
E  ->  ( ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) )  \/  ( abs `  ( ( F `  Z )  -  ( F `  B )
) )  <  E
) ) )
4612, 44, 8, 45syl3anc 1238 . . 3  |-  ( ph  ->  ( ( E  / 
2 )  <  E  ->  ( ( E  / 
2 )  <  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  \/  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  <  E
) ) )
4743, 46mpd 13 . 2  |-  ( ph  ->  ( ( E  / 
2 )  <  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  \/  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  <  E
) )
4840, 41, 47mpjaodan 798 1  |-  ( ph  ->  ( abs `  (
( F `  Z
)  -  ( F `
 B ) ) )  <  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3129   class class class wbr 4002    o. ccom 4629   -->wf 5211   ` cfv 5215  (class class class)co 5872   CCcc 7806   RRcr 7807   0cc0 7808    < clt 7988    - cmin 8124   # cap 8534    / cdiv 8625   2c2 8966   RR+crp 9649   abscabs 10999   ↾t crest 12676   MetOpencmopn 13314   lim CC climc 13994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924  ax-pre-mulgt0 7925  ax-pre-mulext 7926  ax-arch 7927  ax-caucvg 7928
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-frec 6389  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-reap 8528  df-ap 8535  df-div 8626  df-inn 8916  df-2 8974  df-3 8975  df-4 8976  df-n0 9173  df-z 9250  df-uz 9525  df-rp 9650  df-seqfrec 10441  df-exp 10515  df-cj 10844  df-re 10845  df-im 10846  df-rsqrt 11000  df-abs 11001
This theorem is referenced by:  cnplimclemr  14009
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