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Theorem limcimolemlt 14404
Description: Lemma for limcimo 14405. (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 14302 . . . 4  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2 ax-resscn 7916 . . . . . . 7  |-  RR  C_  CC
3 sseq1 3190 . . . . . . 7  |-  ( S  =  RR  ->  ( S  C_  CC  <->  RR  C_  CC ) )
42, 3mpbiri 168 . . . . . 6  |-  ( S  =  RR  ->  S  C_  CC )
54adantl 277 . . . . 5  |-  ( (
ph  /\  S  =  RR )  ->  S  C_  CC )
6 eqimss 3221 . . . . . 6  |-  ( S  =  CC  ->  S  C_  CC )
76adantl 277 . . . . 5  |-  ( (
ph  /\  S  =  CC )  ->  S  C_  CC )
8 limcimo.s . . . . . 6  |-  ( ph  ->  S  e.  { RR ,  CC } )
9 elpri 3627 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
108, 9syl 14 . . . . 5  |-  ( ph  ->  ( S  =  RR  \/  S  =  CC ) )
115, 7, 10mpjaodan 799 . . . 4  |-  ( ph  ->  S  C_  CC )
12 xmetres2 14150 . . . 4  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  S  C_  CC )  -> 
( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( *Met `  S ) )
131, 11, 12sylancr 414 . . 3  |-  ( ph  ->  ( ( abs  o.  -  )  |`  ( S  X.  S ) )  e.  ( *Met `  S ) )
14 limcimo.c . . . 4  |-  ( ph  ->  C  e.  ( Kt  S ) )
15 eqid 2187 . . . . . 6  |-  ( ( abs  o.  -  )  |`  ( S  X.  S
) )  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )
16 limcflfcntop.k . . . . . 6  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
17 eqid 2187 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( S  X.  S ) ) )
1815, 16, 17metrest 14277 . . . . 5  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  S  C_  CC )  -> 
( Kt  S )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( S  X.  S ) ) ) )
191, 11, 18sylancr 414 . . . 4  |-  ( ph  ->  ( Kt  S )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( S  X.  S ) ) ) )
2014, 19eleqtrd 2266 . . 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 11259 . . . 4  |-  ( ( D  e.  RR+  /\  G  e.  RR+ )  -> inf ( { D ,  G } ,  RR ,  <  )  e.  RR+ )
2522, 23, 24syl2anc 411 . . 3  |-  ( ph  -> inf ( { D ,  G } ,  RR ,  <  )  e.  RR+ )
2617mopni3 14255 . . 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 1251 . 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 14398 . . . . . . . . 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 1010 . . . . . . 7  |-  ( ph  ->  F : dom  F --> CC )
3230simp2d 1011 . . . . . . 7  |-  ( ph  ->  dom  F  C_  CC )
33 limcimo.b . . . . . . 7  |-  ( ph  ->  B  e.  CC )
3431, 32, 33ellimc3ap 14401 . . . . . 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 147 . . . . 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 112 . . . 4  |-  ( ph  ->  X  e.  CC )
3736adantr 276 . . 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 14401 . . . . . 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 147 . . . . 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 112 . . . 4  |-  ( ph  ->  Y  e.  CC )
4241adantr 276 . . 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 276 . . . 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 4018 . . . . . 6  |-  ( q  =  ( B  +  ( r  /  2
) )  ->  (
q #  B  <->  ( B  +  ( r  / 
2 ) ) #  B
) )
46 simprrr 540 . . . . . . 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 276 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  B  e.  S )
4947ad2antrr 488 . . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . . 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 2266 . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  e.  RR+ )
5352rphalfcld 9722 . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 9709 . . . . . . . . . . . . . 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 8000 . . . . . . . . . . . . 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 2267 . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 9711 . . . . . . . . . . . . . 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 7990 . . . . . . . . . . . . 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 110 . . . . . . . . . . . . 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 2267 . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 799 . . . . . . . . . . 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 6028 . . . . . . . . . 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 276 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  B  e.  CC )
6853rpcnd 9711 . . . . . . . . . . . 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 7990 . . . . . . . . . . 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 2187 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
7170cnmetdval 14300 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  ( B  +  (
r  /  2 ) )  e.  CC )  ->  ( B ( abs  o.  -  )
( B  +  ( r  /  2 ) ) )  =  ( abs `  ( B  -  ( B  +  ( r  /  2
) ) ) ) )
7267, 69, 71syl2anc 411 . . . . . . . . . 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 8312 . . . . . . . . . . . . . 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 8269 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  ( B  -  B )  =  0 )
7574oveq1d 5903 . . . . . . . . . . . . . 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 2222 . . . . . . . . . . . . 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 5531 . . . . . . . . . . . 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 7963 . . . . . . . . . . . . 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 11215 . . . . . . . . . . . 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 2220 . . . . . . . . . . 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 8270 . . . . . . . . . . . 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 5531 . . . . . . . . . . 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 9709 . . . . . . . . . . . 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 9713 . . . . . . . . . . . 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 11189 . . . . . . . . . . 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 2224 . . . . . . . . . 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 2224 . . . . . . . . 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 9696 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
8988ad2antrl 490 . . . . . . . . 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 4037 . . . . . . . 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 276 . . . . . . . . 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 9674 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9392ad2antrl 490 . . . . . . . . 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 14164 . . . . . . . . 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 1249 . . . . . . . 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 167 . . . . . . 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 3168 . . . . . 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 9715 . . . . . . . 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 8296 . . . . . . . . . . 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 8165 . . . . . . . . . . . 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 8216 . . . . . . . . . . . 12  |-  -u 0  =  0
102100, 101eqtrdi 2236 . . . . . . . . . . 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 2222 . . . . . . . . . 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 5903 . . . . . . . . 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 8268 . . . . . . . . . 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 7993 . . . . . . . . 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 8120 . . . . . . . . 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 2228 . . . . . . . 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 4047 . . . . . . 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 8579 . . . . . . . 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 1248 . . . . . . 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 167 . . . . . 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 2907 . . . . 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 3166 . . . . . 6  |-  ( ph  ->  ( ( B  +  ( r  /  2
) )  e.  {
q  e.  C  | 
q #  B }  ->  ( B  +  ( r  /  2 ) )  e.  A ) )
116115adantr 276 . . . . 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, 117ffvelcdmd 5665 . . 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 8281 . . . 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 11203 . . 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 11215 . . . 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 8281 . . . . . . 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 11203 . . . . . 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 9709 . . . . . 6  |-  ( (
ph  /\  ( r  e.  RR+  /\  ( r  < inf ( { D ,  G } ,  RR ,  <  )  /\  ( B ( ball `  (
( abs  o.  -  )  |`  ( S  X.  S
) ) ) r )  C_  C )
) )  ->  r  e.  RR )
12522rpred 9709 . . . . . . 7  |-  ( ph  ->  D  e.  RR )
126125adantr 276 . . . . . 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 8283 . . . . . . . . 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 5531 . . . . . . . 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 2220 . . . . . . 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 4037 . . . . . 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 9709 . . . . . . . . 9  |-  ( ph  ->  G  e.  RR )
132131adantr 276 . . . . . . . 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 11252 . . . . . . . 8  |-  ( ( D  e.  RR  /\  G  e.  RR )  -> inf ( { D ,  G } ,  RR ,  <  )  e.  RR )
134126, 132, 133syl2anc 411 . . . . . . 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 539 . . . . . . 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 11253 . . . . . . . 8  |-  ( ( D  e.  RR  /\  G  e.  RR )  -> inf ( { D ,  G } ,  RR ,  <  )  <_  D )
137126, 132, 136syl2anc 411 . . . . . . 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 8393 . . . . . 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 8096 . . . . 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 4018 . . . . . . . 8  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  (
z #  B  <->  ( B  +  ( r  / 
2 ) ) #  B
) )
141 fvoveq1 5911 . . . . . . . . 9  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  ( abs `  ( z  -  B ) )  =  ( abs `  (
( B  +  ( r  /  2 ) )  -  B ) ) )
142141breq1d 4025 . . . . . . . 8  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  (
( abs `  (
z  -  B ) )  <  D  <->  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  D
) )
143140, 142anbi12d 473 . . . . . . 7  |-  ( z  =  ( B  +  ( r  /  2
) )  ->  (
( z #  B  /\  ( abs `  ( z  -  B ) )  <  D )  <->  ( ( B  +  ( r  /  2 ) ) #  B  /\  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  D
) ) )
144143imbrov2fvoveq 5913 . . . . . 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 276 . . . . . 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 2858 . . . . 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 433 . . . 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 4037 . . 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 11254 . . . . . . 7  |-  ( ( D  e.  RR  /\  G  e.  RR )  -> inf ( { D ,  G } ,  RR ,  <  )  <_  G )
151126, 132, 150syl2anc 411 . . . . . 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 8393 . . . . 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 8096 . . . 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 4018 . . . . . . 7  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  (
w #  B  <->  ( B  +  ( r  / 
2 ) ) #  B
) )
155 fvoveq1 5911 . . . . . . . 8  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  ( abs `  ( w  -  B ) )  =  ( abs `  (
( B  +  ( r  /  2 ) )  -  B ) ) )
156155breq1d 4025 . . . . . . 7  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  (
( abs `  (
w  -  B ) )  <  G  <->  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  G
) )
157154, 156anbi12d 473 . . . . . 6  |-  ( w  =  ( B  +  ( r  /  2
) )  ->  (
( w #  B  /\  ( abs `  ( w  -  B ) )  <  G )  <->  ( ( B  +  ( r  /  2 ) ) #  B  /\  ( abs `  ( ( B  +  ( r  /  2
) )  -  B
) )  <  G
) ) )
158157imbrov2fvoveq 5913 . . . . 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 276 . . . . 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 2858 . . . 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 433 . . 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 11223 . 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 2609 1  |-  ( ph  ->  ( abs `  ( X  -  Y )
)  <  ( abs `  ( X  -  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    /\ w3a 979    = wceq 1363    e. wcel 2158   A.wral 2465   E.wrex 2466   {crab 2469    C_ wss 3141   {cpr 3605   class class class wbr 4015    X. cxp 4636   dom cdm 4638    |` cres 4640    o. ccom 4642   -->wf 5224   ` cfv 5228  (class class class)co 5888  infcinf 6995   CCcc 7822   RRcr 7823   0cc0 7824    + caddc 7827   RR*cxr 8004    < clt 8005    <_ cle 8006    - cmin 8141   -ucneg 8142   # cap 8551    / cdiv 8642   2c2 8983   RR+crp 9666   abscabs 11019   ↾t crest 12705   *Metcxmet 13697   ballcbl 13699   MetOpencmopn 13702   lim CC climc 14394
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-map 6663  df-pm 6664  df-sup 6996  df-inf 6997  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-xneg 9785  df-xadd 9786  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-rest 12707  df-topgen 12726  df-psmet 13704  df-xmet 13705  df-met 13706  df-bl 13707  df-mopn 13708  df-top 13769  df-topon 13782  df-bases 13814  df-limced 14396
This theorem is referenced by:  limcimo  14405
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