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Theorem abelthlem7a 20345
Description: Lemma for abelth 20349. (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 3324 . 2  |-  ( ph  ->  X  e.  S )
3 oveq2 6081 . . . . 5  |-  ( z  =  X  ->  (
1  -  z )  =  ( 1  -  X ) )
43fveq2d 5724 . . . 4  |-  ( z  =  X  ->  ( abs `  ( 1  -  z ) )  =  ( abs `  (
1  -  X ) ) )
5 fveq2 5720 . . . . . 6  |-  ( z  =  X  ->  ( abs `  z )  =  ( abs `  X
) )
65oveq2d 6089 . . . . 5  |-  ( z  =  X  ->  (
1  -  ( abs `  z ) )  =  ( 1  -  ( abs `  X ) ) )
76oveq2d 6089 . . . 4  |-  ( z  =  X  ->  ( M  x.  ( 1  -  ( abs `  z
) ) )  =  ( M  x.  (
1  -  ( abs `  X ) ) ) )
84, 7breq12d 4217 . . 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 3086 . 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 1652    e. wcel 1725   {crab 2701    \ cdif 3309   {csn 3806   class class class wbr 4204    e. cmpt 4258   dom cdm 4870   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    <_ cle 9113    - cmin 9283   NN0cn0 10213    seq cseq 11315   ^cexp 11374   abscabs 12031    ~~> cli 12270   sum_csu 12471
This theorem is referenced by:  abelthlem7  20346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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