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Theorem limcimolemlt 12589
Description: Lemma for limcimo 12590. (Contributed by Jim Kingdon, 3-Jul-2023.)
Hypotheses
Ref Expression
limcflf.f  |-  ( ph  ->  F : A --> CC )
limcflf.a  |-  ( ph  ->  A  C_  CC )
limcimo.b  |-  ( ph  ->  B  e.  CC )
limcimo.bc  |-  ( ph  ->  B  e.  C )
limcimo.bs  |-  ( ph  ->  B  e.  S )
limcimo.c  |-  ( ph  ->  C  e.  ( Kt  S ) )
limcimo.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
limcimo.ca  |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )
limcflfcntop.k  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
limcimo.d  |-  ( ph  ->  D  e.  RR+ )
limcimo.x  |-  ( ph  ->  X  e.  ( F lim
CC  B ) )
limcimo.y  |-  ( ph  ->  Y  e.  ( F lim
CC  B ) )
limcimo.z  |-  ( ph  ->  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  D )  ->  ( abs `  (
( F `  z
)  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) )
limcimo.g  |-  ( ph  ->  G  e.  RR+ )
limcimo.w  |-  ( ph  ->  A. w  e.  A  ( ( w #  B  /\  ( abs `  (
w  -  B ) )  <  G )  ->  ( abs `  (
( F `  w
)  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) )
Assertion
Ref Expression
limcimolemlt  |-  ( ph  ->  ( abs `  ( X  -  Y )
)  <  ( abs `  ( X  -  Y
) ) )
Distinct variable groups:    w, A    z, A    B, q    w, B   
z, B    C, q    z, D    w, F    z, F    w, G    w, X    z, X    w, Y    z, Y
Allowed substitution hints:    ph( z, w, q)    A( q)    C( z, w)    D( w, q)    S( z, w, q)    F( q)    G( z, q)    K( z, w, q)    X( q)    Y( q)

Proof of Theorem limcimolemlt
Dummy variables  a  b  c  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnxmet 12520 . . . 4  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2 ax-resscn 7637 . . . . . . 7  |-  RR  C_  CC
3 sseq1 3086 . . . . . . 7  |-  ( S  =  RR  ->  ( S  C_  CC  <->  RR  C_  CC ) )
42, 3mpbiri 167 . . . . . 6  |-  ( S  =  RR  ->  S  C_  CC )
54adantl 273 . . . . 5  |-  ( (
ph  /\  S  =  RR )  ->  S  C_  CC )
6 eqimss 3117 . . . . . 6  |-  ( S  =  CC  ->  S  C_  CC )
76adantl 273 . . . . 5  |-  ( (
ph  /\  S  =  CC )  ->  S  C_  CC )
8 limcimo.s . . . . . 6  |-  ( ph  ->  S  e.  { RR ,  CC } )
9 elpri 3516 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
108, 9syl 14 . . . . 5  |-  ( ph  ->  ( S  =  RR  \/  S  =  CC ) )
115, 7, 10mpjaodan 770 . . . 4  |-  ( ph  ->  S  C_  CC )
12 xmetres2 12368 . . . 4  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  S  C_  CC )  -> 
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( *Met `  S ) )
131, 11, 12sylancr 408 . . 3  |-  ( ph  ->  ( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( *Met `  S ) )
14 limcimo.c . . . 4  |-  ( ph  ->  C  e.  ( Kt  S ) )
15 eqid 2115 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
16 limcflfcntop.k . . . . . 6  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
17 eqid 2115 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) )
1815, 16, 17metrest 12495 . . . . 5  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  S  C_  CC )  -> 
( Kt  S )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( S  X.  S ) ) ) )
191, 11, 18sylancr 408 . . . 4  |-  ( ph  ->  ( Kt  S )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( S  X.  S ) ) ) )
2014, 19eleqtrd 2193 . . 3  |-  ( ph  ->  C  e.  ( MetOpen `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) )
21 limcimo.bc . . 3  |-  ( ph  ->  B  e.  C )
22 limcimo.d . . . 4  |-  ( ph  ->  D  e.  RR+ )
23 limcimo.g . . . 4  |-  ( ph  ->  G  e.  RR+ )
24 rpmincl 10901 . . . 4  |-  ( ( D  e.  RR+  /\  G  e.  RR+ )  -> inf ( { D ,  G } ,  RR ,  <  )  e.  RR+ )
2522, 23, 24syl2anc 406 . . 3  |-  ( ph  -> inf ( { D ,  G } ,  RR ,  <  )  e.  RR+ )
2617mopni3 12473 . . 3  |-  ( ( ( ( ( abs 
o.  -  )  |`  ( S  X.  S ) )  e.  ( *Met `  S )  /\  C  e.  ( MetOpen `  ( ( abs  o.  -  )  |`  ( S  X.  S
) ) )  /\  B  e.  C )  /\ inf ( { D ,  G } ,  RR ,  <  )  e.  RR+ )  ->  E. r  e.  RR+  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) )
2713, 20, 21, 25, 26syl31anc 1202 . 2  |-  ( ph  ->  E. r  e.  RR+  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) )
28 limcimo.x . . . . . 6  |-  ( ph  ->  X  e.  ( F lim
CC  B ) )
29 limcrcl 12583 . . . . . . . . 9  |-  ( X  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
3028, 29syl 14 . . . . . . . 8  |-  ( ph  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
3130simp1d 976 . . . . . . 7  |-  ( ph  ->  F : dom  F --> CC )
3230simp2d 977 . . . . . . 7  |-  ( ph  ->  dom  F  C_  CC )
33 limcimo.b . . . . . . 7  |-  ( ph  ->  B  e.  CC )
3431, 32, 33ellimc3ap 12586 . . . . . 6  |-  ( ph  ->  ( X  e.  ( F lim CC  B )  <-> 
( X  e.  CC  /\ 
A. a  e.  RR+  E. b  e.  RR+  A. c  e.  dom  F ( ( c #  B  /\  ( abs `  ( c  -  B ) )  < 
b )  ->  ( abs `  ( ( F `
 c )  -  X ) )  < 
a ) ) ) )
3528, 34mpbid 146 . . . . 5  |-  ( ph  ->  ( X  e.  CC  /\ 
A. a  e.  RR+  E. b  e.  RR+  A. c  e.  dom  F ( ( c #  B  /\  ( abs `  ( c  -  B ) )  < 
b )  ->  ( abs `  ( ( F `
 c )  -  X ) )  < 
a ) ) )
3635simpld 111 . . . 4  |-  ( ph  ->  X  e.  CC )
3736adantr 272 . . 3  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  X  e.  CC )
38 limcimo.y . . . . . 6  |-  ( ph  ->  Y  e.  ( F lim
CC  B ) )
3931, 32, 33ellimc3ap 12586 . . . . . 6  |-  ( ph  ->  ( Y  e.  ( F lim CC  B )  <-> 
( Y  e.  CC  /\ 
A. a  e.  RR+  E. b  e.  RR+  A. c  e.  dom  F ( ( c #  B  /\  ( abs `  ( c  -  B ) )  < 
b )  ->  ( abs `  ( ( F `
 c )  -  Y ) )  < 
a ) ) ) )
4038, 39mpbid 146 . . . . 5  |-  ( ph  ->  ( Y  e.  CC  /\ 
A. a  e.  RR+  E. b  e.  RR+  A. c  e.  dom  F ( ( c #  B  /\  ( abs `  ( c  -  B ) )  < 
b )  ->  ( abs `  ( ( F `
 c )  -  Y ) )  < 
a ) ) )
4140simpld 111 . . . 4  |-  ( ph  ->  Y  e.  CC )
4241adantr 272 . . 3  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  Y  e.  CC )
43 limcflf.f . . . . 5  |-  ( ph  ->  F : A --> CC )
4443adantr 272 . . . 4  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  F : A --> CC )
45 breq1 3898 . . . . . 6  |-  ( q  =  ( B  +  ( r  /  2
) )  ->  (
q #  B  <->  ( B  +  ( r  / 
2 ) ) #  B
) )
46 simprrr 512 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
47 limcimo.bs . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  S )
4847adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  B  e.  S )
4947ad2antrr 477 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  RR )  ->  B  e.  S )
50 simpr 109 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  RR )  ->  S  =  RR )
5149, 50eleqtrd 2193 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  RR )  ->  B  e.  RR )
52 simprl 503 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  e.  RR+ )
5352rphalfcld 9395 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
r  /  2 )  e.  RR+ )
5453adantr 272 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  RR )  ->  ( r  /  2 )  e.  RR+ )
5554rpred 9382 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  RR )  ->  ( r  /  2 )  e.  RR )
5651, 55readdcld 7719 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  RR )  ->  ( B  +  ( r  / 
2 ) )  e.  RR )
5756, 50eleqtrrd 2194 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  RR )  ->  ( B  +  ( r  / 
2 ) )  e.  S )
5833ad2antrr 477 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  CC )  ->  B  e.  CC )
5953adantr 272 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  CC )  ->  ( r  /  2 )  e.  RR+ )
6059rpcnd 9384 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  CC )  ->  ( r  /  2 )  e.  CC )
6158, 60addcld 7709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  CC )  ->  ( B  +  ( r  / 
2 ) )  e.  CC )
62 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  CC )  ->  S  =  CC )
6361, 62eleqtrrd 2194 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  RR+  /\  (
r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ) r )  C_  C ) ) )  /\  S  =  CC )  ->  ( B  +  ( r  / 
2 ) )  e.  S )
6410adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( S  =  RR  \/  S  =  CC )
)
6557, 63, 64mpjaodan 770 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  +  ( r  /  2 ) )  e.  S )
6648, 65ovresd 5865 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B ( ( abs 
o.  -  )  |`  ( S  X.  S ) ) ( B  +  ( r  /  2 ) ) )  =  ( B ( abs  o.  -  ) ( B  +  ( r  / 
2 ) ) ) )
6733adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  B  e.  CC )
6853rpcnd 9384 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
r  /  2 )  e.  CC )
6967, 68addcld 7709 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  +  ( r  /  2 ) )  e.  CC )
70 eqid 2115 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
7170cnmetdval 12518 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  ( B  +  (
r  /  2 ) )  e.  CC )  ->  ( B ( abs  o.  -  )
( B  +  ( r  /  2 ) ) )  =  ( abs `  ( B  -  ( B  +  ( r  /  2
) ) ) ) )
7267, 69, 71syl2anc 406 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B ( abs  o.  -  ) ( B  +  ( r  / 
2 ) ) )  =  ( abs `  ( B  -  ( B  +  ( r  / 
2 ) ) ) ) )
7367, 67, 68subsub4d 8027 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( B  -  B
)  -  ( r  /  2 ) )  =  ( B  -  ( B  +  (
r  /  2 ) ) ) )
7467subidd 7984 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  -  B )  =  0 )
7574oveq1d 5743 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( B  -  B
)  -  ( r  /  2 ) )  =  ( 0  -  ( r  /  2
) ) )
7673, 75eqtr3d 2149 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  -  ( B  +  ( r  / 
2 ) ) )  =  ( 0  -  ( r  /  2
) ) )
7776fveq2d 5379 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( B  -  ( B  +  (
r  /  2 ) ) ) )  =  ( abs `  (
0  -  ( r  /  2 ) ) ) )
78 0cnd 7683 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  0  e.  CC )
7978, 68abssubd 10857 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( 0  -  ( r  /  2
) ) )  =  ( abs `  (
( r  /  2
)  -  0 ) ) )
8077, 79eqtrd 2147 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( B  -  ( B  +  (
r  /  2 ) ) ) )  =  ( abs `  (
( r  /  2
)  -  0 ) ) )
8168subid1d 7985 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( r  /  2
)  -  0 )  =  ( r  / 
2 ) )
8281fveq2d 5379 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( r  /  2 )  - 
0 ) )  =  ( abs `  (
r  /  2 ) ) )
8353rpred 9382 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
r  /  2 )  e.  RR )
8453rpge0d 9386 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  0  <_  ( r  /  2
) )
8583, 84absidd 10831 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( r  / 
2 ) )  =  ( r  /  2
) )
8680, 82, 853eqtrd 2151 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( B  -  ( B  +  (
r  /  2 ) ) ) )  =  ( r  /  2
) )
8766, 72, 863eqtrd 2151 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B ( ( abs 
o.  -  )  |`  ( S  X.  S ) ) ( B  +  ( r  /  2 ) ) )  =  ( r  /  2 ) )
88 rphalflt 9372 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
8988ad2antrl 479 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
r  /  2 )  <  r )
9087, 89eqbrtrd 3915 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B ( ( abs 
o.  -  )  |`  ( S  X.  S ) ) ( B  +  ( r  /  2 ) ) )  <  r
)
9113adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( abs  o.  -  )  |`  ( S  X.  S
) )  e.  ( *Met `  S
) )
92 rpxr 9350 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9392ad2antrl 479 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  e.  RR* )
94 elbl2 12382 . . . . . . . . 9  |-  ( ( ( ( ( abs 
o.  -  )  |`  ( S  X.  S ) )  e.  ( *Met `  S )  /\  r  e.  RR* )  /\  ( B  e.  S  /\  ( B  +  (
r  /  2 ) )  e.  S ) )  ->  ( ( B  +  ( r  /  2 ) )  e.  ( B (
ball `  ( ( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  <->  ( B ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ( B  +  ( r  /  2 ) ) )  <  r ) )
9591, 93, 48, 65, 94syl22anc 1200 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( B  +  ( r  /  2 ) )  e.  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  <->  ( B ( ( abs  o.  -  )  |`  ( S  X.  S ) ) ( B  +  ( r  /  2 ) ) )  <  r ) )
9690, 95mpbird 166 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  +  ( r  /  2 ) )  e.  ( B (
ball `  ( ( abs  o.  -  )  |`  ( S  X.  S
) ) ) r ) )
9746, 96sseldd 3064 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  +  ( r  /  2 ) )  e.  C )
9853rpap0d 9388 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
r  /  2 ) #  0 )
9967, 67negsubdid 8011 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  -u ( B  -  B )  =  ( -u B  +  B ) )
10074negeqd 7880 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  -u ( B  -  B )  =  -u 0 )
101 neg0 7931 . . . . . . . . . . . 12  |-  -u 0  =  0
102100, 101syl6eq 2163 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  -u ( B  -  B )  =  0 )
10399, 102eqtr3d 2149 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( -u B  +  B )  =  0 )
104103oveq1d 5743 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( -u B  +  B
)  +  ( r  /  2 ) )  =  ( 0  +  ( r  /  2
) ) )
10567negcld 7983 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  -u B  e.  CC )
106105, 67, 68addassd 7712 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( -u B  +  B
)  +  ( r  /  2 ) )  =  ( -u B  +  ( B  +  ( r  /  2
) ) ) )
10768addid2d 7835 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
0  +  ( r  /  2 ) )  =  ( r  / 
2 ) )
108104, 106, 1073eqtr3d 2155 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( -u B  +  ( B  +  ( r  / 
2 ) ) )  =  ( r  / 
2 ) )
10998, 108, 1033brtr4d 3925 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( -u B  +  ( B  +  ( r  / 
2 ) ) ) #  ( -u B  +  B ) )
110 apadd2 8289 . . . . . . . 8  |-  ( ( ( B  +  ( r  /  2 ) )  e.  CC  /\  B  e.  CC  /\  -u B  e.  CC )  ->  (
( B  +  ( r  /  2 ) ) #  B  <->  ( -u B  +  ( B  +  ( r  /  2
) ) ) #  (
-u B  +  B
) ) )
11169, 67, 105, 110syl3anc 1199 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( B  +  ( r  /  2 ) ) #  B  <->  ( -u B  +  ( B  +  ( r  /  2
) ) ) #  (
-u B  +  B
) ) )
112109, 111mpbird 166 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  +  ( r  /  2 ) ) #  B )
11345, 97, 112elrabd 2811 . . . . 5  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  +  ( r  /  2 ) )  e.  { q  e.  C  |  q #  B } )
114 limcimo.ca . . . . . . 7  |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )
115114sseld 3062 . . . . . 6  |-  ( ph  ->  ( ( B  +  ( r  /  2
) )  e.  {
q  e.  C  | 
q #  B }  ->  ( B  +  ( r  /  2 ) )  e.  A ) )
116115adantr 272 . . . . 5  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( B  +  ( r  /  2 ) )  e.  { q  e.  C  |  q #  B }  ->  ( B  +  ( r  /  2 ) )  e.  A ) )
117113, 116mpd 13 . . . 4  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  +  ( r  /  2 ) )  e.  A )
11844, 117ffvelrnd 5510 . . 3  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( F `  ( B  +  ( r  / 
2 ) ) )  e.  CC )
11937, 42subcld 7996 . . . 4  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( X  -  Y )  e.  CC )
120119abscld 10845 . . 3  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( X  -  Y ) )  e.  RR )
12137, 118abssubd 10857 . . . 4  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( X  -  ( F `  ( B  +  ( r  / 
2 ) ) ) ) )  =  ( abs `  ( ( F `  ( B  +  ( r  / 
2 ) ) )  -  X ) ) )
12269, 67subcld 7996 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( B  +  ( r  /  2 ) )  -  B )  e.  CC )
123122abscld 10845 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( B  +  ( r  / 
2 ) )  -  B ) )  e.  RR )
12452rpred 9382 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  e.  RR )
12522rpred 9382 . . . . . . 7  |-  ( ph  ->  D  e.  RR )
126125adantr 272 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  D  e.  RR )
12767, 68pncan2d 7998 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( B  +  ( r  /  2 ) )  -  B )  =  ( r  / 
2 ) )
128127fveq2d 5379 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( B  +  ( r  / 
2 ) )  -  B ) )  =  ( abs `  (
r  /  2 ) ) )
129128, 85eqtrd 2147 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( B  +  ( r  / 
2 ) )  -  B ) )  =  ( r  /  2
) )
130129, 89eqbrtrd 3915 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( B  +  ( r  / 
2 ) )  -  B ) )  < 
r )
13123rpred 9382 . . . . . . . . 9  |-  ( ph  ->  G  e.  RR )
132131adantr 272 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  G  e.  RR )
133 mincl 10894 . . . . . . . 8  |-  ( ( D  e.  RR  /\  G  e.  RR )  -> inf ( { D ,  G } ,  RR ,  <  )  e.  RR )
134126, 132, 133syl2anc 406 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  -> inf ( { D ,  G } ,  RR ,  <  )  e.  RR )
135 simprrl 511 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  < inf ( { D ,  G } ,  RR ,  <  ) )
136 min1inf 10895 . . . . . . . 8  |-  ( ( D  e.  RR  /\  G  e.  RR )  -> inf ( { D ,  G } ,  RR ,  <  )  <_  D )
137126, 132, 136syl2anc 406 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  -> inf ( { D ,  G } ,  RR ,  <  )  <_  D )
138124, 134, 126, 135, 137ltletrd 8104 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  <  D )
139123, 124, 126, 130, 138lttrd 7811 . . . . 5  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( B  +  ( r  / 
2 ) )  -  B ) )  < 
D )
140 breq1 3898 . . . . . . . 8  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  (
z #  B  <->  ( B  +  ( r  / 
2 ) ) #  B
) )
141 fvoveq1 5751 . . . . . . . . 9  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  ( abs `  ( z  -  B ) )  =  ( abs `  (
( B  +  ( r  /  2 ) )  -  B ) ) )
142141breq1d 3905 . . . . . . . 8  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  (
( abs `  (
z  -  B ) )  <  D  <->  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  D
) )
143140, 142anbi12d 462 . . . . . . 7  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  (
( z #  B  /\  ( abs `  ( z  -  B ) )  <  D )  <->  ( ( B  +  ( r  /  2 ) ) #  B  /\  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  D
) ) )
144143imbrov2fvoveq 5753 . . . . . 6  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  (
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  D )  ->  ( abs `  (
( F `  z
)  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) )  <-> 
( ( ( B  +  ( r  / 
2 ) ) #  B  /\  ( abs `  (
( B  +  ( r  /  2 ) )  -  B ) )  <  D )  ->  ( abs `  (
( F `  ( B  +  ( r  /  2 ) ) )  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) ) )
145 limcimo.z . . . . . . 7  |-  ( ph  ->  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  D )  ->  ( abs `  (
( F `  z
)  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) )
146145adantr 272 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  A. z  e.  A  ( (
z #  B  /\  ( abs `  ( z  -  B ) )  < 
D )  ->  ( abs `  ( ( F `
 z )  -  X ) )  < 
( ( abs `  ( X  -  Y )
)  /  2 ) ) )
147144, 146, 117rspcdva 2765 . . . . 5  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( ( B  +  ( r  /  2
) ) #  B  /\  ( abs `  ( ( B  +  ( r  /  2 ) )  -  B ) )  <  D )  -> 
( abs `  (
( F `  ( B  +  ( r  /  2 ) ) )  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) )
148112, 139, 147mp2and 427 . . . 4  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( F `
 ( B  +  ( r  /  2
) ) )  -  X ) )  < 
( ( abs `  ( X  -  Y )
)  /  2 ) )
149121, 148eqbrtrd 3915 . . 3  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( X  -  ( F `  ( B  +  ( r  / 
2 ) ) ) ) )  <  (
( abs `  ( X  -  Y )
)  /  2 ) )
150 min2inf 10896 . . . . . . 7  |-  ( ( D  e.  RR  /\  G  e.  RR )  -> inf ( { D ,  G } ,  RR ,  <  )  <_  G )
151126, 132, 150syl2anc 406 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  -> inf ( { D ,  G } ,  RR ,  <  )  <_  G )
152124, 134, 132, 135, 151ltletrd 8104 . . . . 5  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  <  G )
153123, 124, 132, 130, 152lttrd 7811 . . . 4  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( B  +  ( r  / 
2 ) )  -  B ) )  < 
G )
154 breq1 3898 . . . . . . 7  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  (
w #  B  <->  ( B  +  ( r  / 
2 ) ) #  B
) )
155 fvoveq1 5751 . . . . . . . 8  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  ( abs `  ( w  -  B ) )  =  ( abs `  (
( B  +  ( r  /  2 ) )  -  B ) ) )
156155breq1d 3905 . . . . . . 7  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  (
( abs `  (
w  -  B ) )  <  G  <->  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  G
) )
157154, 156anbi12d 462 . . . . . 6  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  (
( w #  B  /\  ( abs `  ( w  -  B ) )  <  G )  <->  ( ( B  +  ( r  /  2 ) ) #  B  /\  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  G
) ) )
158157imbrov2fvoveq 5753 . . . . 5  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  (
( ( w #  B  /\  ( abs `  (
w  -  B ) )  <  G )  ->  ( abs `  (
( F `  w
)  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) )  <-> 
( ( ( B  +  ( r  / 
2 ) ) #  B  /\  ( abs `  (
( B  +  ( r  /  2 ) )  -  B ) )  <  G )  ->  ( abs `  (
( F `  ( B  +  ( r  /  2 ) ) )  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) ) )
159 limcimo.w . . . . . 6  |-  ( ph  ->  A. w  e.  A  ( ( w #  B  /\  ( abs `  (
w  -  B ) )  <  G )  ->  ( abs `  (
( F `  w
)  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) )
160159adantr 272 . . . . 5  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  A. w  e.  A  ( (
w #  B  /\  ( abs `  ( w  -  B ) )  < 
G )  ->  ( abs `  ( ( F `
 w )  -  Y ) )  < 
( ( abs `  ( X  -  Y )
)  /  2 ) ) )
161158, 160, 117rspcdva 2765 . . . 4  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  (
( ( B  +  ( r  /  2
) ) #  B  /\  ( abs `  ( ( B  +  ( r  /  2 ) )  -  B ) )  <  G )  -> 
( abs `  (
( F `  ( B  +  ( r  /  2 ) ) )  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  /  2 ) ) )
162112, 153, 161mp2and 427 . . 3  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( ( F `
 ( B  +  ( r  /  2
) ) )  -  Y ) )  < 
( ( abs `  ( X  -  Y )
)  /  2 ) )
16337, 42, 118, 120, 149, 162abs3lemd 10865 . 2  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( abs `  ( X  -  Y ) )  < 
( abs `  ( X  -  Y )
) )
16427, 163rexlimddv 2528 1  |-  ( ph  ->  ( abs `  ( X  -  Y )
)  <  ( abs `  ( X  -  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   A.wral 2390   E.wrex 2391   {crab 2394    C_ wss 3037   {cpr 3494   class class class wbr 3895    X. cxp 4497   dom cdm 4499    |` cres 4501    o. ccom 4503   -->wf 5077   ` cfv 5081  (class class class)co 5728  infcinf 6822   CCcc 7545   RRcr 7546   0cc0 7547    + caddc 7550   RR*cxr 7723    < clt 7724    <_ cle 7725    - cmin 7856   -ucneg 7857   # cap 8261    / cdiv 8345   2c2 8681   RR+crp 9343   abscabs 10661   ↾t crest 11963   *Metcxmet 11992   ballcbl 11994   MetOpencmopn 11997   lim CC climc 12579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-isom 5090  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-map 6498  df-pm 6499  df-sup 6823  df-inf 6824  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-xneg 9452  df-xadd 9453  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-rest 11965  df-topgen 11984  df-psmet 11999  df-xmet 12000  df-met 12001  df-bl 12002  df-mopn 12003  df-top 12008  df-topon 12021  df-bases 12053  df-limced 12581
This theorem is referenced by:  limcimo  12590
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