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Theorem reccn2ap 11430
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 2189. (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 8020 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  1  e.  RR )
3 simp1 999 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  A  e.  CC )
4 simp2 1000 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  A #  0 )
53, 4absrpclapd 11306 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( abs `  A )  e.  RR+ )
6 simp3 1001 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  B  e.  RR+ )
75, 6rpmulcld 9765 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  x.  B )  e.  RR+ )
87rpred 9748 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  x.  B )  e.  RR )
9 mincl 11348 . . . . . 6  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  e.  RR )
102, 8, 9syl2anc 411 . . . . 5  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  e.  RR )
117rpgt0d 9751 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  < 
( ( abs `  A
)  x.  B ) )
12 0lt1 8132 . . . . . . 7  |-  0  <  1
1311, 12jctil 312 . . . . . 6  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( 0  <  1  /\  0  <  ( ( abs `  A
)  x.  B ) ) )
14 0red 8006 . . . . . . 7  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  e.  RR )
15 ltmininf 11352 . . . . . . 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 1249 . . . . . 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 167 . . . . 5  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  0  < inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  ) )
1810, 17elrpd 9745 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  -> inf ( { 1 ,  ( ( abs `  A )  x.  B ) } ,  RR ,  <  )  e.  RR+ )
195rphalfcld 9761 . . . 4  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  ( ( abs `  A )  /  2 )  e.  RR+ )
2018, 19rpmulcld 9765 . . 3  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  (inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  x.  ( ( abs `  A )  /  2 ) )  e.  RR+ )
211, 20eqeltrid 2276 . 2  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  T  e.  RR+ )
223adantr 276 . . . . . . . . 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 529 . . . . . . . . . . 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 4028 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
w #  0  <->  z #  0
) )
2524elrab 2912 . . . . . . . . . . 11  |-  ( z  e.  { w  e.  CC  |  w #  0 }  <->  ( z  e.  CC  /\  z #  0 ) )
2623, 25sylib 122 . . . . . . . . . 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 112 . . . . . . . . 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 8026 . . . . . . . . 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 276 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  A #  0 )
3026simprd 114 . . . . . . . . . 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 8663 . . . . . . . . 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 8835 . . . . . . . 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 )
) ) )
3322mulridd 8022 . . . . . . . . . . 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 5921 . . . . . . . . . 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 8021 . . . . . . . . . . 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 8822 . . . . . . . . . 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 2224 . . . . . . . . 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
) )
3827mulridd 8022 . . . . . . . . . . 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 8027 . . . . . . . . . . 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 5924 . . . . . . . . . 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 8822 . . . . . . . . . 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 2224 . . . . . . . . 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 5924 . . . . . . . 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 2222 . . . . . . 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 5546 . . . . . 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 8316 . . . . . . 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 11313 . . . . . 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 2224 . . . . 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 11299 . . . . . . 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 276 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  e.  RR+ )
5150rpred 9748 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  /\  (
z  e.  { w  e.  CC  |  w #  0 }  /\  ( abs `  ( z  -  A
) )  <  T
) )  ->  T  e.  RR )
5228abscld 11299 . . . . . . . 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 9748 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A #  0  /\  B  e.  RR+ )  ->  B  e.  RR )
5453adantr 276 . . . . . . . 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 8036 . . . . . . 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 11311 . . . . . . . 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 531 . . . . . . . 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 4047 . . . . . . 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 276 . . . . . . . . . 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 9748 . . . . . . . . 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 276 . . . . . . . . . 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 9748 . . . . . . . . 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 8036 . . . . . . . 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 8004 . . . . . . . . . . 11  |-  1  e.  RR
65 min2inf 11350 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  ( ( abs `  A )  x.  B ) )
6664, 60, 65sylancr 414 . . . . . . . . . 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 276 . . . . . . . . . . 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 9792 . . . . . . . . . 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 147 . . . . . . . . 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 4060 . . . . . . . 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 11299 . . . . . . . . . 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 11299 . . . . . . . . . . . . . 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 8034 . . . . . . . . . . . . 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 9214 . . . . . . . . . . . 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 8386 . . . . . . . . . . . . . 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 8316 . . . . . . . . . . . . . . . 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 11299 . . . . . . . . . . . . . . 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 2266 . . . . . . . . . . . . . 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 11315 . . . . . . . . . . . . . 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 11349 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  RR  /\  ( ( abs `  A
)  x.  B )  e.  RR )  -> inf ( { 1 ,  ( ( abs `  A
)  x.  B ) } ,  RR ,  <  )  <_  1 )
8164, 60, 80sylancr 414 . . . . . . . . . . . . . . . . . 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 8020 . . . . . . . . . . . . . . . . . . 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 9792 . . . . . . . . . . . . . . . . . 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 147 . . . . . . . . . . . . . . . . 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 4060 . . . . . . . . . . . . . . . 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 8034 . . . . . . . . . . . . . . . . 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 8024 . . . . . . . . . . . . . . . 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 4051 . . . . . . . . . . . . . . 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 8428 . . . . . . . . . . . . . 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 8130 . . . . . . . . . . . . 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 8548 . . . . . . . . . . . . 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 147 . . . . . . . . . . . 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 4047 . . . . . . . . . . 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 8543 . . . . . . . . . . 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 167 . . . . . . . . . 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 9805 . . . . . . . . 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 11312 . . . . . . . . . . 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 5921 . . . . . . . . . 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 8034 . . . . . . . . . . 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 8034 . . . . . . . . . . 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 8158 . . . . . . . . . 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 2222 . . . . . . . . 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 4053 . . . . . . . 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 8130 . . . . . . 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 8131 . . . . . 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 11306 . . . . . . 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 9800 . . . . . 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 167 . . . . 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 4047 . . . 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 375 . . 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 2563 . 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 4029 . . 3  |-  ( y  =  T  ->  (
( abs `  (
z  -  A ) )  <  y  <->  ( abs `  ( z  -  A
) )  <  T
) )
113112rspceaimv 2868 . 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 411 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 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   {cpr 3615   class class class wbr 4025   ` cfv 5242  (class class class)co 5906  infcinf 7028   CCcc 7856   RRcr 7857   0cc0 7858   1c1 7859    + caddc 7861    x. cmul 7863    < clt 8040    <_ cle 8041    - cmin 8176   # cap 8586    / cdiv 8677   2c2 9019   RR+crp 9705   abscabs 11115
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612  ax-cnex 7949  ax-resscn 7950  ax-1cn 7951  ax-1re 7952  ax-icn 7953  ax-addcl 7954  ax-addrcl 7955  ax-mulcl 7956  ax-mulrcl 7957  ax-addcom 7958  ax-mulcom 7959  ax-addass 7960  ax-mulass 7961  ax-distr 7962  ax-i2m1 7963  ax-0lt1 7964  ax-1rid 7965  ax-0id 7966  ax-rnegex 7967  ax-precex 7968  ax-cnre 7969  ax-pre-ltirr 7970  ax-pre-ltwlin 7971  ax-pre-lttrn 7972  ax-pre-apti 7973  ax-pre-ltadd 7974  ax-pre-mulgt0 7975  ax-pre-mulext 7976  ax-arch 7977  ax-caucvg 7978
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-if 3554  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-ilim 4394  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-isom 5251  df-riota 5861  df-ov 5909  df-oprab 5910  df-mpo 5911  df-1st 6180  df-2nd 6181  df-recs 6345  df-frec 6431  df-sup 7029  df-inf 7030  df-pnf 8042  df-mnf 8043  df-xr 8044  df-ltxr 8045  df-le 8046  df-sub 8178  df-neg 8179  df-reap 8580  df-ap 8587  df-div 8678  df-inn 8969  df-2 9027  df-3 9028  df-4 9029  df-n0 9227  df-z 9304  df-uz 9579  df-rp 9706  df-seqfrec 10505  df-exp 10584  df-cj 10960  df-re 10961  df-im 10962  df-rsqrt 11116  df-abs 11117
This theorem is referenced by:  divcnap  14680  cdivcncfap  14715
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