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Theorem abelthlem7a 19813
Description: Lemma for abelth 19817. (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 } ) )
2 eldifi 3298 . . 3  |-  ( X  e.  ( S  \  { 1 } )  ->  X  e.  S
)
31, 2syl 15 . 2  |-  ( ph  ->  X  e.  S )
4 oveq2 5866 . . . . 5  |-  ( z  =  X  ->  (
1  -  z )  =  ( 1  -  X ) )
54fveq2d 5529 . . . 4  |-  ( z  =  X  ->  ( abs `  ( 1  -  z ) )  =  ( abs `  (
1  -  X ) ) )
6 fveq2 5525 . . . . . 6  |-  ( z  =  X  ->  ( abs `  z )  =  ( abs `  X
) )
76oveq2d 5874 . . . . 5  |-  ( z  =  X  ->  (
1  -  ( abs `  z ) )  =  ( 1  -  ( abs `  X ) ) )
87oveq2d 5874 . . . 4  |-  ( z  =  X  ->  ( M  x.  ( 1  -  ( abs `  z
) ) )  =  ( M  x.  (
1  -  ( abs `  X ) ) ) )
95, 8breq12d 4036 . . 3  |-  ( z  =  X  ->  (
( abs `  (
1  -  z ) )  <_  ( M  x.  ( 1  -  ( abs `  z ) ) )  <->  ( abs `  (
1  -  X ) )  <_  ( M  x.  ( 1  -  ( abs `  X ) ) ) ) )
10 abelth.5 . . 3  |-  S  =  { z  e.  CC  |  ( abs `  (
1  -  z ) )  <_  ( M  x.  ( 1  -  ( abs `  z ) ) ) }
119, 10elrab2 2925 . 2  |-  ( X  e.  S  <->  ( X  e.  CC  /\  ( abs `  ( 1  -  X
) )  <_  ( M  x.  ( 1  -  ( abs `  X
) ) ) ) )
123, 11sylib 188 1  |-  ( ph  ->  ( X  e.  CC  /\  ( abs `  (
1  -  X ) )  <_  ( M  x.  ( 1  -  ( abs `  X ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    \ cdif 3149   {csn 3640   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037   NN0cn0 9965    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  abelthlem7  19814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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