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Theorem hbtlem1 27327
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 2796 . . . . . . 7  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5525 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 hbtlem.p . . . . . . . . . . . 12  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2333 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5529 . . . . . . . . . 10  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 hbtlem.u . . . . . . . . . 10  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2333 . . . . . . . . 9  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
9 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
10 hbtlem.d . . . . . . . . . . . . . . . 16  |-  D  =  ( deg1  `  R )
119, 10syl6eqr 2333 . . . . . . . . . . . . . . 15  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1211fveq1d 5527 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  k )  =  ( D `  k ) )
1312breq1d 4033 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  k )  <_  x  <->  ( D `  k )  <_  x ) )
1413anbi1d 685 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (
( ( ( deg1  `  r
) `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) )  <->  ( ( D `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) ) )
1514rexbidv 2564 . . . . . . . . . . 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 2397 . . . . . . . . . 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 4107 . . . . . . . . 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 4098 . . . . . . . 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 27326 . . . . . . . 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 5539 . . . . . . . . . 10  |-  (LIdeal `  P )  e.  _V
217, 20eqeltri 2353 . . . . . . . . 9  |-  U  e. 
_V
2221mptex 5746 . . . . . . . 8  |-  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) )  e.  _V
2318, 19, 22fvmpt 5602 . . . . . . 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 15 . . . . . 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 2327 . . . . 5  |-  ( R  e.  V  ->  S  =  ( i  e.  U  |->  ( x  e. 
NN0  |->  { j  |  E. k  e.  i  ( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) ) )
2625fveq1d 5527 . . . 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 5527 . . 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 976 . 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 2737 . . . . . . 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 2397 . . . . . 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 4107 . . . . 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 2283 . . . . 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 9971 . . . . . 6  |-  NN0  e.  _V
3433mptex 5746 . . . . 5  |-  ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } )  e.  _V
3531, 32, 34fvmpt 5602 . . . 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 5527 . . 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 977 . 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 957 . . 3  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  X  e.  NN0 )
39 simpr 447 . . . . . 6  |-  ( ( ( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) )  ->  j  =  ( (coe1 `  k
) `  X )
)
4039reximi 2650 . . . . 5  |-  ( E. k  e.  I  ( ( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) )  ->  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) )
4140ss2abi 3245 . . . 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 5764 . . . . 5  |-  ( I  e.  U  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) }  e.  _V )
43423ad2ant2 977 . . . 4  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k
) `  X ) }  e.  _V )
44 ssexg 4160 . . . 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 644 . . 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 4027 . . . . . . 7  |-  ( x  =  X  ->  (
( D `  k
)  <_  x  <->  ( D `  k )  <_  X
) )
47 fveq2 5525 . . . . . . . 8  |-  ( x  =  X  ->  (
(coe1 `  k ) `  x )  =  ( (coe1 `  k ) `  X ) )
4847eqeq2d 2294 . . . . . . 7  |-  ( x  =  X  ->  (
j  =  ( (coe1 `  k ) `  x
)  <->  j  =  ( (coe1 `  k ) `  X ) ) )
4946, 48anbi12d 691 . . . . . 6  |-  ( x  =  X  ->  (
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) )  <->  ( ( D `  k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) ) )
5049rexbidv 2564 . . . . 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 2397 . . . 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 2283 . . . 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 5600 . . 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 642 . 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 2319 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255    <_ cle 8868   NN0cn0 9965  LIdealclidl 15923  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440  ldgIdlSeqcldgis 27325
This theorem is referenced by:  hbtlem2  27328  hbtlem4  27330  hbtlem3  27331  hbtlem5  27332  hbtlem6  27333
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-n0 9966  df-ldgis 27326
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