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Theorem cnplimclemle 14222
Description: Lemma for cnplimccntop 14224. 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 5654 . . . . . . 7  |-  ( ph  ->  ( F `  Z
)  e.  CC )
5 cnplimclemr.b . . . . . . . 8  |-  ( ph  ->  B  e.  A )
62, 5ffvelcdmd 5654 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  CC )
74, 6subcld 8270 . . . . . 6  |-  ( ph  ->  ( ( F `  Z )  -  ( F `  B )
)  e.  CC )
87abscld 11192 . . . . 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 9711 . . . . . 6  |-  ( ph  ->  ( E  /  2
)  e.  RR+ )
1211rpred 9698 . . . . 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 8079 . . . . . . . . . 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 3158 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  CC )
2826, 5sseldd 3158 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
2928adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  B  e.  CC )
30 apti 8581 . . . . . . . . . 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 5521 . . . . . . 7  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( F `  Z )  =  ( F `  B ) )
3414, 33subeq0bd 8338 . . . . . 6  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  (
( F `  Z
)  -  ( F `
 B ) )  =  0 )
3534abs00bd 11077 . . . . 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 9701 . . . . 5  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  0  <  ( E  /  2
) )
3835, 37eqbrtrd 4027 . . . 4  |-  ( (
ph  /\  ( E  /  2 )  < 
( abs `  (
( F `  Z
)  -  ( F `
 B ) ) ) )  ->  ( abs `  ( ( F `
 Z )  -  ( F `  B ) ) )  <  ( E  /  2 ) )
399, 13, 38ltnsymd 8079 . . 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 9685 . . . 4  |-  ( E  e.  RR+  ->  ( E  /  2 )  < 
E )
4310, 42syl 14 . . 3  |-  ( ph  ->  ( E  /  2
)  <  E )
4410rpred 9698 . . . 4  |-  ( ph  ->  E  e.  RR )
45 axltwlin 8027 . . . 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 3131   class class class wbr 4005    o. ccom 4632   -->wf 5214   ` cfv 5218  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813    < clt 7994    - cmin 8130   # cap 8540    / cdiv 8631   2c2 8972   RR+crp 9655   abscabs 11008   ↾t crest 12693   MetOpencmopn 13530   lim CC climc 14208
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-rp 9656  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010
This theorem is referenced by:  cnplimclemr  14223
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