Theorem abelthlem7a 24191

Description: Lemma for abelth 24195. (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

Step	Hyp	Ref	Expression
1		abelthlem6.1	. . 3 |- ( ph -> X e. ( S \ { 1 } ) )
2	1	eldifad 3586	. 2 |- ( ph -> X e. S )
3		oveq2 6658	. . . . 5 |- ( z = X -> ( 1 - z ) = ( 1 - X ) )
4	3	fveq2d 6195	. . . 4 |- ( z = X -> ( abs ` ( 1 - z ) ) = ( abs ` ( 1 - X ) ) )
5		fveq2 6191	. . . . . 6 |- ( z = X -> ( abs ` z ) = ( abs ` X ) )
6	5	oveq2d 6666	. . . . 5 |- ( z = X -> ( 1 - ( abs ` z ) ) = ( 1 - ( abs ` X ) ) )
7	6	oveq2d 6666	. . . 4 |- ( z = X -> ( M x. ( 1 - ( abs ` z ) ) ) = ( M x. ( 1 - ( abs ` X ) ) ) )
8	4, 7	breq12d 4666	. . 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 ) ) ) }
10	8, 9	elrab2 3366	. 2 |- ( X e. S <-> ( X e. CC /\ ( abs ` ( 1 - X ) ) <_ ( M x. ( 1 - ( abs ` X ) ) ) ) )
11	2, 10	sylib 208	1 |- ( ph -> ( X e. CC /\ ( abs ` ( 1 - X ) ) <_ ( M x. ( 1 - ( abs ` X ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NN0cn0 11292   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  abelthlem7  24192