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Theorem reccn2ap 10921
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 2100. (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 7653 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  1  e.  RR )
3 simp1 949 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  A  e.  CC )
4 simp2 950 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  A #  0 )
53, 4absrpclapd 10800 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( abs `  A )  e.  RR+ )
6 simp3 951 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  B  e.  RR+ )
75, 6rpmulcld 9347 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  x.  B )  e.  RR+ )
87rpred 9330 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  x.  B )  e.  RR )
9 mincl 10841 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  e.  RR )
102, 8, 9syl2anc 406 . . . . 5  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  e.  RR )
117rpgt0d 9333 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  < 
( ( abs `  A
)  x.  B ) )
12 0lt1 7760 . . . . . . 7  |-  0  <  1
1311, 12jctil 308 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( 0  <  1  /\  0  <  ( ( abs `  A
)  x.  B ) ) )
14 0red 7639 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  e.  RR )
15 ltmininf 10845 . . . . . . 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 1184 . . . . . 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 9328 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  e.  RR+ )
195rphalfcld 9343 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  /  2 )  e.  RR+ )
2018, 19rpmulcld 9347 . . 3  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )  e.  RR+ )
211, 20syl5eqel 2186 . 2  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  T  e.  RR+ )
223adantr 272 . . . . . . . . 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 501 . . . . . . . . . . 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 3878 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
w #  0  <->  z #  0
) )
2524elrab 2793 . . . . . . . . . . 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 7658 . . . . . . . . 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 272 . . . . . . . . . 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 8280 . . . . . . . . 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 8451 . . . . . . . 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 7655 . . . . . . . . . . 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 5721 . . . . . . . . . 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 7654 . . . . . . . . . . 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 8438 . . . . . . . . . 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 2134 . . . . . . . . 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 7655 . . . . . . . . . . 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 7659 . . . . . . . . . . 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 5724 . . . . . . . . . 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 8438 . . . . . . . . . 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 2134 . . . . . . . . 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 5724 . . . . . . . 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 2132 . . . . . . 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 5357 . . . . . 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 7944 . . . . . . 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 10807 . . . . . 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 2134 . . . . 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 10793 . . . . . . 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 272 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  e.  RR+ )
5150rpred 9330 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  e.  RR )
5228abscld 10793 . . . . . . . 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 9330 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  B  e.  RR )
5453adantr 272 . . . . . . . 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 7668 . . . . . . 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 10805 . . . . . . . 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 502 . . . . . . . 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 3895 . . . . . . 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 272 . . . . . . . . . 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 9330 . . . . . . . . 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 272 . . . . . . . . . 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 9330 . . . . . . . . 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 7668 . . . . . . . 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 7637 . . . . . . . . . . 11  |-  1  e.  RR
65 min2inf 10843 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  ( ( abs `  A )  x.  B ) )
6664, 60, 65sylancr 408 . . . . . . . . . 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 272 . . . . . . . . . . 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 9374 . . . . . . . . . 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, 69syl5eqbr 3908 . . . . . . . 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 10793 . . . . . . . . . 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 10793 . . . . . . . . . . . . . 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 7666 . . . . . . . . . . . . 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 8817 . . . . . . . . . . . 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 8010 . . . . . . . . . . . . . 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 7944 . . . . . . . . . . . . . . . 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 10793 . . . . . . . . . . . . . . 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 2176 . . . . . . . . . . . . . 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 10809 . . . . . . . . . . . . . 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 10842 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  1 )
8164, 60, 80sylancr 408 . . . . . . . . . . . . . . . . . 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 7653 . . . . . . . . . . . . . . . . . . 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 9374 . . . . . . . . . . . . . . . . . 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, 84syl5eqbr 3908 . . . . . . . . . . . . . . . 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 7666 . . . . . . . . . . . . . . . . 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 7656 . . . . . . . . . . . . . . . 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 3899 . . . . . . . . . . . . . . 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 8052 . . . . . . . . . . . . . 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 7758 . . . . . . . . . . . . 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 8171 . . . . . . . . . . . . 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 3895 . . . . . . . . . . 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 8166 . . . . . . . . . . 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 9387 . . . . . . . . 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 10806 . . . . . . . . . . 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 5721 . . . . . . . . . 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 7666 . . . . . . . . . . 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 7666 . . . . . . . . . . 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 7786 . . . . . . . . . 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 2132 . . . . . . . . 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 3901 . . . . . . . 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 7758 . . . . . . 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 7759 . . . . . 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 10800 . . . . . . 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 9382 . . . . . 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 3895 . . . 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 370 . . 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 2464 . 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 3879 . . 3  |-  ( y  =  T  ->  (
( abs `  (
z  -  A ) )  <  y  <->  ( abs `  ( z  -  A
) )  <  T
) )
113112rspceaimv 2751 . 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 406 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 930    = wceq 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   {crab 2379   {cpr 3475   class class class wbr 3875   ` cfv 5059  (class class class)co 5706  infcinf 6785   CCcc 7498   RRcr 7499   0cc0 7500   1c1 7501    + caddc 7503    x. cmul 7505    < clt 7672    <_ cle 7673    - cmin 7804   # cap 8209    / cdiv 8293   2c2 8629   RR+crp 9291   abscabs 10609
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611
This theorem is referenced by:  cdivcncfap  12499
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