Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hbtlem1 Unicode version

Theorem hbtlem1 27199
Description: Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
hbtlem.p  |-  P  =  (Poly1 `  R )
hbtlem.u  |-  U  =  (LIdeal `  P )
hbtlem.s  |-  S  =  (ldgIdlSeq `  R )
hbtlem.d  |-  D  =  ( deg1  `  R )
Assertion
Ref Expression
hbtlem1  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  -> 
( ( S `  I ) `  X
)  =  { j  |  E. k  e.  I  ( ( D `
 k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) } )
Distinct variable groups:    j, I,
k    R, j, k    j, X, k
Allowed substitution hints:    D( j, k)    P( j, k)    S( j, k)    U( j, k)    V( j, k)

Proof of Theorem hbtlem1
Dummy variables  i 
r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem.s . . . . . 6  |-  S  =  (ldgIdlSeq `  R )
2 elex 2928 . . . . . . 7  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5691 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 hbtlem.p . . . . . . . . . . . 12  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2458 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5695 . . . . . . . . . 10  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 hbtlem.u . . . . . . . . . 10  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2458 . . . . . . . . 9  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
9 fveq2 5691 . . . . . . . . . . . . . . . 16  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
10 hbtlem.d . . . . . . . . . . . . . . . 16  |-  D  =  ( deg1  `  R )
119, 10syl6eqr 2458 . . . . . . . . . . . . . . 15  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1211fveq1d 5693 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  k )  =  ( D `  k ) )
1312breq1d 4186 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  k )  <_  x  <->  ( D `  k )  <_  x ) )
1413anbi1d 686 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (
( ( ( deg1  `  r
) `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) )  <->  ( ( D `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) ) )
1514rexbidv 2691 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( E. k  e.  i 
( ( ( deg1  `  r
) `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) )  <->  E. k  e.  i  ( ( D `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) ) )
1615abbidv 2522 . . . . . . . . . 10  |-  ( r  =  R  ->  { j  |  E. k  e.  i  ( ( ( deg1  `  r ) `  k
)  <_  x  /\  j  =  ( (coe1 `  k ) `  x
) ) }  =  { j  |  E. k  e.  i  (
( D `  k
)  <_  x  /\  j  =  ( (coe1 `  k ) `  x
) ) } )
1716mpteq2dv 4260 . . . . . . . . 9  |-  ( r  =  R  ->  (
x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( ( deg1  `  r ) `  k
)  <_  x  /\  j  =  ( (coe1 `  k ) `  x
) ) } )  =  ( x  e. 
NN0  |->  { j  |  E. k  e.  i  ( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) )
188, 17mpteq12dv 4251 . . . . . . . 8  |-  ( r  =  R  ->  (
i  e.  (LIdeal `  (Poly1 `  r ) )  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( ( deg1  `  r ) `  k
)  <_  x  /\  j  =  ( (coe1 `  k ) `  x
) ) } ) )  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) ) )
19 df-ldgis 27198 . . . . . . . 8  |- ldgIdlSeq  =  ( r  e.  _V  |->  ( i  e.  (LIdeal `  (Poly1 `  r ) )  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( ( deg1  `  r ) `  k
)  <_  x  /\  j  =  ( (coe1 `  k ) `  x
) ) } ) ) )
20 fvex 5705 . . . . . . . . . 10  |-  (LIdeal `  P )  e.  _V
217, 20eqeltri 2478 . . . . . . . . 9  |-  U  e. 
_V
2221mptex 5929 . . . . . . . 8  |-  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) )  e.  _V
2318, 19, 22fvmpt 5769 . . . . . . 7  |-  ( R  e.  _V  ->  (ldgIdlSeq `  R )  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) ) )
242, 23syl 16 . . . . . 6  |-  ( R  e.  V  ->  (ldgIdlSeq `  R )  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) ) )
251, 24syl5eq 2452 . . . . 5  |-  ( R  e.  V  ->  S  =  ( i  e.  U  |->  ( x  e. 
NN0  |->  { j  |  E. k  e.  i  ( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) ) )
2625fveq1d 5693 . . . 4  |-  ( R  e.  V  ->  ( S `  I )  =  ( ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) ) `  I
) )
2726fveq1d 5693 . . 3  |-  ( R  e.  V  ->  (
( S `  I
) `  X )  =  ( ( ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) ) `  I
) `  X )
)
28273ad2ant1 978 . 2  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  -> 
( ( S `  I ) `  X
)  =  ( ( ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  (
( D `  k
)  <_  x  /\  j  =  ( (coe1 `  k ) `  x
) ) } ) ) `  I ) `
 X ) )
29 rexeq 2869 . . . . . . 7  |-  ( i  =  I  ->  ( E. k  e.  i 
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) )  <->  E. k  e.  I  ( ( D `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) ) )
3029abbidv 2522 . . . . . 6  |-  ( i  =  I  ->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) }  =  { j  |  E. k  e.  I 
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } )
3130mpteq2dv 4260 . . . . 5  |-  ( i  =  I  ->  (
x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } )  =  ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) )
32 eqid 2408 . . . . 5  |-  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) )  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) )
33 nn0ex 10187 . . . . . 6  |-  NN0  e.  _V
3433mptex 5929 . . . . 5  |-  ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } )  e.  _V
3531, 32, 34fvmpt 5769 . . . 4  |-  ( I  e.  U  ->  (
( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  (
( D `  k
)  <_  x  /\  j  =  ( (coe1 `  k ) `  x
) ) } ) ) `  I )  =  ( x  e. 
NN0  |->  { j  |  E. k  e.  I 
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) )
3635fveq1d 5693 . . 3  |-  ( I  e.  U  ->  (
( ( i  e.  U  |->  ( x  e. 
NN0  |->  { j  |  E. k  e.  i  ( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) ) `  I
) `  X )  =  ( ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) `  X ) )
37363ad2ant2 979 . 2  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  -> 
( ( ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) ) `  I
) `  X )  =  ( ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) `  X ) )
38 simp3 959 . . 3  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  X  e.  NN0 )
39 simpr 448 . . . . . 6  |-  ( ( ( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) )  ->  j  =  ( (coe1 `  k
) `  X )
)
4039reximi 2777 . . . . 5  |-  ( E. k  e.  I  ( ( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) )  ->  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) )
4140ss2abi 3379 . . . 4  |-  { j  |  E. k  e.  I  ( ( D `
 k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) } 
C_  { j  |  E. k  e.  I 
j  =  ( (coe1 `  k ) `  X
) }
42 abrexexg 5947 . . . . 5  |-  ( I  e.  U  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) }  e.  _V )
43423ad2ant2 979 . . . 4  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k
) `  X ) }  e.  _V )
44 ssexg 4313 . . . 4  |-  ( ( { j  |  E. k  e.  I  (
( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) ) }  C_  { j  |  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) }  /\  { j  |  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) }  e.  _V )  ->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) }  e.  _V )
4541, 43, 44sylancr 645 . . 3  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  { j  |  E. k  e.  I  (
( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) ) }  e.  _V )
46 breq2 4180 . . . . . . 7  |-  ( x  =  X  ->  (
( D `  k
)  <_  x  <->  ( D `  k )  <_  X
) )
47 fveq2 5691 . . . . . . . 8  |-  ( x  =  X  ->  (
(coe1 `  k ) `  x )  =  ( (coe1 `  k ) `  X ) )
4847eqeq2d 2419 . . . . . . 7  |-  ( x  =  X  ->  (
j  =  ( (coe1 `  k ) `  x
)  <->  j  =  ( (coe1 `  k ) `  X ) ) )
4946, 48anbi12d 692 . . . . . 6  |-  ( x  =  X  ->  (
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) )  <->  ( ( D `  k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) ) )
5049rexbidv 2691 . . . . 5  |-  ( x  =  X  ->  ( E. k  e.  I 
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) )  <->  E. k  e.  I  ( ( D `  k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) ) )
5150abbidv 2522 . . . 4  |-  ( x  =  X  ->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) }  =  { j  |  E. k  e.  I 
( ( D `  k )  <_  X  /\  j  =  (
(coe1 `  k ) `  X ) ) } )
52 eqid 2408 . . . 4  |-  ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } )  =  ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } )
5351, 52fvmptg 5767 . . 3  |-  ( ( X  e.  NN0  /\  { j  |  E. k  e.  I  ( ( D `  k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) }  e.  _V )  -> 
( ( x  e. 
NN0  |->  { j  |  E. k  e.  I 
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) `  X )  =  { j  |  E. k  e.  I 
( ( D `  k )  <_  X  /\  j  =  (
(coe1 `  k ) `  X ) ) } )
5438, 45, 53syl2anc 643 . 2  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  -> 
( ( x  e. 
NN0  |->  { j  |  E. k  e.  I 
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) `  X )  =  { j  |  E. k  e.  I 
( ( D `  k )  <_  X  /\  j  =  (
(coe1 `  k ) `  X ) ) } )
5528, 37, 543eqtrd 2444 1  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  -> 
( ( S `  I ) `  X
)  =  { j  |  E. k  e.  I  ( ( D `
 k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2394   E.wrex 2671   _Vcvv 2920    C_ wss 3284   class class class wbr 4176    e. cmpt 4230   ` cfv 5417    <_ cle 9081   NN0cn0 10181  LIdealclidl 16201  Poly1cpl1 16530  coe1cco1 16533   deg1 cdg1 19934  ldgIdlSeqcldgis 27197
This theorem is referenced by:  hbtlem2  27200  hbtlem4  27202  hbtlem3  27203  hbtlem5  27204  hbtlem6  27205
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-i2m1 9018  ax-1ne0 9019  ax-rrecex 9022  ax-cnre 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-recs 6596  df-rdg 6631  df-nn 9961  df-n0 10182  df-ldgis 27198
  Copyright terms: Public domain W3C validator