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Theorem reccn2ap 11094
Description: The reciprocal function is continuous. The class  T is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2139. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
Hypothesis
Ref Expression
reccn2ap.t  |-  T  =  (inf ( { 1 ,  ( ( abs `  A )  x.  B
) } ,  RR ,  <  )  x.  (
( abs `  A
)  /  2 ) )
Assertion
Ref Expression
reccn2ap  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  E. y  e.  RR+  A. z  e. 
{ w  e.  CC  |  w #  0 } 
( ( abs `  (
z  -  A ) )  <  y  -> 
( abs `  (
( 1  /  z
)  -  ( 1  /  A ) ) )  <  B ) )
Distinct variable groups:    y, w, z, A    w, B, y, z    y, T, z
Allowed substitution hint:    T( w)

Proof of Theorem reccn2ap
StepHypRef Expression
1 reccn2ap.t . . 3  |-  T  =  (inf ( { 1 ,  ( ( abs `  A )  x.  B
) } ,  RR ,  <  )  x.  (
( abs `  A
)  /  2 ) )
2 1red 7793 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  1  e.  RR )
3 simp1 981 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  A  e.  CC )
4 simp2 982 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  A #  0 )
53, 4absrpclapd 10972 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( abs `  A )  e.  RR+ )
6 simp3 983 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  B  e.  RR+ )
75, 6rpmulcld 9512 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  x.  B )  e.  RR+ )
87rpred 9495 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  x.  B )  e.  RR )
9 mincl 11014 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  e.  RR )
102, 8, 9syl2anc 408 . . . . 5  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  e.  RR )
117rpgt0d 9498 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  < 
( ( abs `  A
)  x.  B ) )
12 0lt1 7901 . . . . . . 7  |-  0  <  1
1311, 12jctil 310 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( 0  <  1  /\  0  <  ( ( abs `  A
)  x.  B ) ) )
14 0red 7779 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  e.  RR )
15 ltmininf 11018 . . . . . . 7  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  (
( abs `  A
)  x.  B )  e.  RR )  -> 
( 0  < inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <->  ( 0  <  1  /\  0  < 
( ( abs `  A
)  x.  B ) ) ) )
1614, 2, 8, 15syl3anc 1216 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( 0  < inf ( { 1 ,  ( ( abs `  A )  x.  B
) } ,  RR ,  <  )  <->  ( 0  <  1  /\  0  <  ( ( abs `  A
)  x.  B ) ) ) )
1713, 16mpbird 166 . . . . 5  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  < inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  ) )
1810, 17elrpd 9493 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  e.  RR+ )
195rphalfcld 9508 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  /  2 )  e.  RR+ )
2018, 19rpmulcld 9512 . . 3  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )  e.  RR+ )
211, 20eqeltrid 2226 . 2  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  T  e.  RR+ )
223adantr 274 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  A  e.  CC )
23 simprl 520 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  z  e.  { w  e.  CC  |  w #  0 }
)
24 breq1 3932 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
w #  0  <->  z #  0
) )
2524elrab 2840 . . . . . . . . . . 11  |-  ( z  e.  { w  e.  CC  |  w #  0 }  <->  ( z  e.  CC  /\  z #  0 ) )
2623, 25sylib 121 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
z  e.  CC  /\  z #  0 ) )
2726simpld 111 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  z  e.  CC )
2822, 27mulcld 7798 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( A  x.  z )  e.  CC )
294adantr 274 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  A #  0 )
3026simprd 113 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  z #  0 )
3122, 27, 29, 30mulap0d 8431 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( A  x.  z ) #  0 )
3222, 27, 28, 31divsubdirapd 8602 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( A  -  z
)  /  ( A  x.  z ) )  =  ( ( A  /  ( A  x.  z ) )  -  ( z  /  ( A  x.  z )
) ) )
3322mulid1d 7795 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( A  x.  1 )  =  A )
3433oveq1d 5789 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( A  x.  1 )  /  ( A  x.  z ) )  =  ( A  / 
( A  x.  z
) ) )
35 1cnd 7794 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  1  e.  CC )
3635, 27, 22, 30, 29divcanap5d 8589 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( A  x.  1 )  /  ( A  x.  z ) )  =  ( 1  / 
z ) )
3734, 36eqtr3d 2174 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( A  /  ( A  x.  z ) )  =  ( 1  /  z
) )
3827mulid1d 7795 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
z  x.  1 )  =  z )
3927, 22mulcomd 7799 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
z  x.  A )  =  ( A  x.  z ) )
4038, 39oveq12d 5792 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( z  x.  1 )  /  ( z  x.  A ) )  =  ( z  / 
( A  x.  z
) ) )
4135, 22, 27, 29, 30divcanap5d 8589 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( z  x.  1 )  /  ( z  x.  A ) )  =  ( 1  /  A ) )
4240, 41eqtr3d 2174 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
z  /  ( A  x.  z ) )  =  ( 1  /  A ) )
4337, 42oveq12d 5792 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( A  /  ( A  x.  z )
)  -  ( z  /  ( A  x.  z ) ) )  =  ( ( 1  /  z )  -  ( 1  /  A
) ) )
4432, 43eqtrd 2172 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( A  -  z
)  /  ( A  x.  z ) )  =  ( ( 1  /  z )  -  ( 1  /  A
) ) )
4544fveq2d 5425 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( ( A  -  z )  / 
( A  x.  z
) ) )  =  ( abs `  (
( 1  /  z
)  -  ( 1  /  A ) ) ) )
4622, 27subcld 8085 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( A  -  z )  e.  CC )
4746, 28, 31absdivapd 10979 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( ( A  -  z )  / 
( A  x.  z
) ) )  =  ( ( abs `  ( A  -  z )
)  /  ( abs `  ( A  x.  z
) ) ) )
4845, 47eqtr3d 2174 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( ( 1  /  z )  -  ( 1  /  A
) ) )  =  ( ( abs `  ( A  -  z )
)  /  ( abs `  ( A  x.  z
) ) ) )
4946abscld 10965 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  -  z ) )  e.  RR )
5021adantr 274 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  e.  RR+ )
5150rpred 9495 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  e.  RR )
5228abscld 10965 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  x.  z ) )  e.  RR )
536rpred 9495 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  B  e.  RR )
5453adantr 274 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  B  e.  RR )
5552, 54remulcld 7808 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  ( A  x.  z )
)  x.  B )  e.  RR )
5622, 27abssubd 10977 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  -  z ) )  =  ( abs `  (
z  -  A ) ) )
57 simprr 521 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( z  -  A ) )  < 
T )
5856, 57eqbrtrd 3950 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  -  z ) )  < 
T )
597adantr 274 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  x.  B )  e.  RR+ )
6059rpred 9495 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  x.  B )  e.  RR )
6119adantr 274 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  /  2 )  e.  RR+ )
6261rpred 9495 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  /  2 )  e.  RR )
6360, 62remulcld 7808 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  x.  B )  x.  ( ( abs `  A )  /  2
) )  e.  RR )
64 1re 7777 . . . . . . . . . . 11  |-  1  e.  RR
65 min2inf 11016 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  ( ( abs `  A )  x.  B ) )
6664, 60, 65sylancr 410 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  <_  ( ( abs `  A )  x.  B
) )
6710adantr 274 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  e.  RR )
6867, 60, 61lemul1d 9539 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  ( ( abs `  A )  x.  B )  <->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )  <_  ( ( ( abs `  A )  x.  B )  x.  ( ( abs `  A
)  /  2 ) ) ) )
6966, 68mpbid 146 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )  <_  ( ( ( abs `  A )  x.  B )  x.  ( ( abs `  A
)  /  2 ) ) )
701, 69eqbrtrid 3963 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  <_  ( ( ( abs `  A )  x.  B
)  x.  ( ( abs `  A )  /  2 ) ) )
7127abscld 10965 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  z )  e.  RR )
7222abscld 10965 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  A )  e.  RR )
7372recnd 7806 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  A )  e.  CC )
74732halvesd 8977 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  /  2 )  +  ( ( abs `  A )  /  2
) )  =  ( abs `  A ) )
7572, 71resubcld 8155 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  -  ( abs `  z ) )  e.  RR )
7627, 22subcld 8085 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
z  -  A )  e.  CC )
7776abscld 10965 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( z  -  A ) )  e.  RR )
7856, 77eqeltrd 2216 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  -  z ) )  e.  RR )
7922, 27abs2difd 10981 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  -  ( abs `  z ) )  <_ 
( abs `  ( A  -  z )
) )
80 min1inf 11015 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  1 )
8164, 60, 80sylancr 410 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  <_  1 )
82 1red 7793 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  1  e.  RR )
8367, 82, 61lemul1d 9539 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  1  <->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )  <_  ( 1  x.  ( ( abs `  A
)  /  2 ) ) ) )
8481, 83mpbid 146 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )  <_  ( 1  x.  ( ( abs `  A
)  /  2 ) ) )
851, 84eqbrtrid 3963 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  <_  ( 1  x.  (
( abs `  A
)  /  2 ) ) )
8662recnd 7806 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  /  2 )  e.  CC )
8786mulid2d 7796 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
1  x.  ( ( abs `  A )  /  2 ) )  =  ( ( abs `  A )  /  2
) )
8885, 87breqtrd 3954 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  <_  ( ( abs `  A
)  /  2 ) )
8978, 51, 62, 58, 88ltletrd 8197 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  -  z ) )  < 
( ( abs `  A
)  /  2 ) )
9075, 78, 62, 79, 89lelttrd 7899 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  -  ( abs `  z ) )  < 
( ( abs `  A
)  /  2 ) )
9172, 71, 62ltsubadd2d 8317 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  -  ( abs `  z ) )  < 
( ( abs `  A
)  /  2 )  <-> 
( abs `  A
)  <  ( ( abs `  z )  +  ( ( abs `  A
)  /  2 ) ) ) )
9290, 91mpbid 146 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  A )  < 
( ( abs `  z
)  +  ( ( abs `  A )  /  2 ) ) )
9374, 92eqbrtrd 3950 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  /  2 )  +  ( ( abs `  A )  /  2
) )  <  (
( abs `  z
)  +  ( ( abs `  A )  /  2 ) ) )
9462, 71, 62ltadd1d 8312 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  /  2 )  <  ( abs `  z
)  <->  ( ( ( abs `  A )  /  2 )  +  ( ( abs `  A
)  /  2 ) )  <  ( ( abs `  z )  +  ( ( abs `  A )  /  2
) ) ) )
9593, 94mpbird 166 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  A
)  /  2 )  <  ( abs `  z
) )
9662, 71, 59, 95ltmul2dd 9552 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  x.  B )  x.  ( ( abs `  A )  /  2
) )  <  (
( ( abs `  A
)  x.  B )  x.  ( abs `  z
) ) )
9722, 27absmuld 10978 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  x.  z ) )  =  ( ( abs `  A
)  x.  ( abs `  z ) ) )
9897oveq1d 5789 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  ( A  x.  z )
)  x.  B )  =  ( ( ( abs `  A )  x.  ( abs `  z
) )  x.  B
) )
9971recnd 7806 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  z )  e.  CC )
10054recnd 7806 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  B  e.  CC )
10173, 99, 100mul32d 7927 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  x.  ( abs `  z ) )  x.  B )  =  ( ( ( abs `  A
)  x.  B )  x.  ( abs `  z
) ) )
10298, 101eqtrd 2172 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  ( A  x.  z )
)  x.  B )  =  ( ( ( abs `  A )  x.  B )  x.  ( abs `  z
) ) )
10396, 102breqtrrd 3956 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  A
)  x.  B )  x.  ( ( abs `  A )  /  2
) )  <  (
( abs `  ( A  x.  z )
)  x.  B ) )
10451, 63, 55, 70, 103lelttrd 7899 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  <  ( ( abs `  ( A  x.  z )
)  x.  B ) )
10549, 51, 55, 58, 104lttrd 7900 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  -  z ) )  < 
( ( abs `  ( A  x.  z )
)  x.  B ) )
10628, 31absrpclapd 10972 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( A  x.  z ) )  e.  RR+ )
10749, 54, 106ltdivmuld 9547 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( ( abs `  ( A  -  z )
)  /  ( abs `  ( A  x.  z
) ) )  < 
B  <->  ( abs `  ( A  -  z )
)  <  ( ( abs `  ( A  x.  z ) )  x.  B ) ) )
108105, 107mpbird 166 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  (
( abs `  ( A  -  z )
)  /  ( abs `  ( A  x.  z
) ) )  < 
B )
10948, 108eqbrtrd 3950 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  ( abs `  ( ( 1  /  z )  -  ( 1  /  A
) ) )  < 
B )
110109expr 372 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  z  e.  { w  e.  CC  |  w #  0 }
)  ->  ( ( abs `  ( z  -  A ) )  < 
T  ->  ( abs `  ( ( 1  / 
z )  -  (
1  /  A ) ) )  <  B
) )
111110ralrimiva 2505 . 2  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  A. z  e.  { w  e.  CC  |  w #  0 } 
( ( abs `  (
z  -  A ) )  <  T  -> 
( abs `  (
( 1  /  z
)  -  ( 1  /  A ) ) )  <  B ) )
112 breq2 3933 . . 3  |-  ( y  =  T  ->  (
( abs `  (
z  -  A ) )  <  y  <->  ( abs `  ( z  -  A
) )  <  T
) )
113112rspceaimv 2797 . 2  |-  ( ( T  e.  RR+  /\  A. z  e.  { w  e.  CC  |  w #  0 }  ( ( abs `  ( z  -  A
) )  <  T  ->  ( abs `  (
( 1  /  z
)  -  ( 1  /  A ) ) )  <  B ) )  ->  E. y  e.  RR+  A. z  e. 
{ w  e.  CC  |  w #  0 } 
( ( abs `  (
z  -  A ) )  <  y  -> 
( abs `  (
( 1  /  z
)  -  ( 1  /  A ) ) )  <  B ) )
11421, 111, 113syl2anc 408 1  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  E. y  e.  RR+  A. z  e. 
{ w  e.  CC  |  w #  0 } 
( ( abs `  (
z  -  A ) )  <  y  -> 
( abs `  (
( 1  /  z
)  -  ( 1  /  A ) ) )  <  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   {cpr 3528   class class class wbr 3929   ` cfv 5123  (class class class)co 5774  infcinf 6870   CCcc 7630   RRcr 7631   0cc0 7632   1c1 7633    + caddc 7635    x. cmul 7637    < clt 7812    <_ cle 7813    - cmin 7945   # cap 8355    / cdiv 8444   2c2 8783   RR+crp 9453   abscabs 10781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-rp 9454  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783
This theorem is referenced by:  divcnap  12738  cdivcncfap  12770
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