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Theorem abelthlem7a 19829
Description: Lemma for abelth 19833. (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 3311 . . 3  |-  ( X  e.  ( S  \  { 1 } )  ->  X  e.  S
)
31, 2syl 15 . 2  |-  ( ph  ->  X  e.  S )
4 oveq2 5882 . . . . 5  |-  ( z  =  X  ->  (
1  -  z )  =  ( 1  -  X ) )
54fveq2d 5545 . . . 4  |-  ( z  =  X  ->  ( abs `  ( 1  -  z ) )  =  ( abs `  (
1  -  X ) ) )
6 fveq2 5541 . . . . . 6  |-  ( z  =  X  ->  ( abs `  z )  =  ( abs `  X
) )
76oveq2d 5890 . . . . 5  |-  ( z  =  X  ->  (
1  -  ( abs `  z ) )  =  ( 1  -  ( abs `  X ) ) )
87oveq2d 5890 . . . 4  |-  ( z  =  X  ->  ( M  x.  ( 1  -  ( abs `  z
) ) )  =  ( M  x.  (
1  -  ( abs `  X ) ) ) )
95, 8breq12d 4052 . . 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 2938 . 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 1632    e. wcel 1696   {crab 2560    \ cdif 3162   {csn 3653   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   NN0cn0 9981    seq cseq 11062   ^cexp 11120   abscabs 11735    ~~> cli 11974   sum_csu 12174
This theorem is referenced by:  abelthlem7  19830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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