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Theorem abelthlem7a 20213
Description: Lemma for abelth 20217. (Contributed by Mario Carneiro, 8-May-2015.)
Hypotheses
Ref Expression
abelth.1  |-  ( ph  ->  A : NN0 --> CC )
abelth.2  |-  ( ph  ->  seq  0 (  +  ,  A )  e. 
dom 
~~>  )
abelth.3  |-  ( ph  ->  M  e.  RR )
abelth.4  |-  ( ph  ->  0  <_  M )
abelth.5  |-  S  =  { z  e.  CC  |  ( abs `  (
1  -  z ) )  <_  ( M  x.  ( 1  -  ( abs `  z ) ) ) }
abelth.6  |-  F  =  ( x  e.  S  |-> 
sum_ n  e.  NN0  ( ( A `  n )  x.  (
x ^ n ) ) )
abelth.7  |-  ( ph  ->  seq  0 (  +  ,  A )  ~~>  0 )
abelthlem6.1  |-  ( ph  ->  X  e.  ( S 
\  { 1 } ) )
Assertion
Ref Expression
abelthlem7a  |-  ( ph  ->  ( X  e.  CC  /\  ( abs `  (
1  -  X ) )  <_  ( M  x.  ( 1  -  ( abs `  X ) ) ) ) )
Distinct variable groups:    x, n, z, M    n, X, x, z    A, n, x, z    ph, n, x    S, n, x
Allowed substitution hints:    ph( z)    S( z)    F( x, z, n)

Proof of Theorem abelthlem7a
StepHypRef Expression
1 abelthlem6.1 . . 3  |-  ( ph  ->  X  e.  ( S 
\  { 1 } ) )
21eldifad 3268 . 2  |-  ( ph  ->  X  e.  S )
3 oveq2 6021 . . . . 5  |-  ( z  =  X  ->  (
1  -  z )  =  ( 1  -  X ) )
43fveq2d 5665 . . . 4  |-  ( z  =  X  ->  ( abs `  ( 1  -  z ) )  =  ( abs `  (
1  -  X ) ) )
5 fveq2 5661 . . . . . 6  |-  ( z  =  X  ->  ( abs `  z )  =  ( abs `  X
) )
65oveq2d 6029 . . . . 5  |-  ( z  =  X  ->  (
1  -  ( abs `  z ) )  =  ( 1  -  ( abs `  X ) ) )
76oveq2d 6029 . . . 4  |-  ( z  =  X  ->  ( M  x.  ( 1  -  ( abs `  z
) ) )  =  ( M  x.  (
1  -  ( abs `  X ) ) ) )
84, 7breq12d 4159 . . 3  |-  ( z  =  X  ->  (
( abs `  (
1  -  z ) )  <_  ( M  x.  ( 1  -  ( abs `  z ) ) )  <->  ( abs `  (
1  -  X ) )  <_  ( M  x.  ( 1  -  ( abs `  X ) ) ) ) )
9 abelth.5 . . 3  |-  S  =  { z  e.  CC  |  ( abs `  (
1  -  z ) )  <_  ( M  x.  ( 1  -  ( abs `  z ) ) ) }
108, 9elrab2 3030 . 2  |-  ( X  e.  S  <->  ( X  e.  CC  /\  ( abs `  ( 1  -  X
) )  <_  ( M  x.  ( 1  -  ( abs `  X
) ) ) ) )
112, 10sylib 189 1  |-  ( ph  ->  ( X  e.  CC  /\  ( abs `  (
1  -  X ) )  <_  ( M  x.  ( 1  -  ( abs `  X ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2646    \ cdif 3253   {csn 3750   class class class wbr 4146    e. cmpt 4200   dom cdm 4811   -->wf 5383   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    <_ cle 9047    - cmin 9216   NN0cn0 10146    seq cseq 11243   ^cexp 11302   abscabs 11959    ~~> cli 12198   sum_csu 12399
This theorem is referenced by:  abelthlem7  20214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-ov 6016
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