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Theorem hbtlem1 26918
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 2881 . . . . . . 7  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5632 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 hbtlem.p . . . . . . . . . . . 12  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2416 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5636 . . . . . . . . . 10  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 hbtlem.u . . . . . . . . . 10  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2416 . . . . . . . . 9  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
9 fveq2 5632 . . . . . . . . . . . . . . . 16  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
10 hbtlem.d . . . . . . . . . . . . . . . 16  |-  D  =  ( deg1  `  R )
119, 10syl6eqr 2416 . . . . . . . . . . . . . . 15  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1211fveq1d 5634 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  k )  =  ( D `  k ) )
1312breq1d 4135 . . . . . . . . . . . . 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 2649 . . . . . . . . . . 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 2480 . . . . . . . . . 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 4209 . . . . . . . . 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 4200 . . . . . . . 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 26917 . . . . . . . 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 5646 . . . . . . . . . 10  |-  (LIdeal `  P )  e.  _V
217, 20eqeltri 2436 . . . . . . . . 9  |-  U  e. 
_V
2221mptex 5866 . . . . . . . 8  |-  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) )  e.  _V
2318, 19, 22fvmpt 5709 . . . . . . 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 2410 . . . . 5  |-  ( R  e.  V  ->  S  =  ( i  e.  U  |->  ( x  e. 
NN0  |->  { j  |  E. k  e.  i  ( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) ) )
2625fveq1d 5634 . . . 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 5634 . . 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 977 . 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 2822 . . . . . . 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 2480 . . . . . 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 4209 . . . . 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 2366 . . . . 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 10120 . . . . . 6  |-  NN0  e.  _V
3433mptex 5866 . . . . 5  |-  ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } )  e.  _V
3531, 32, 34fvmpt 5709 . . . 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 5634 . . 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 978 . 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 958 . . 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 2735 . . . . 5  |-  ( E. k  e.  I  ( ( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) )  ->  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) )
4140ss2abi 3331 . . . 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 5884 . . . . 5  |-  ( I  e.  U  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) }  e.  _V )
43423ad2ant2 978 . . . 4  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k
) `  X ) }  e.  _V )
44 ssexg 4262 . . . 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 4129 . . . . . . 7  |-  ( x  =  X  ->  (
( D `  k
)  <_  x  <->  ( D `  k )  <_  X
) )
47 fveq2 5632 . . . . . . . 8  |-  ( x  =  X  ->  (
(coe1 `  k ) `  x )  =  ( (coe1 `  k ) `  X ) )
4847eqeq2d 2377 . . . . . . 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 2649 . . . . 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 2480 . . . 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 2366 . . . 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 5707 . . 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 2402 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 935    = wceq 1647    e. wcel 1715   {cab 2352   E.wrex 2629   _Vcvv 2873    C_ wss 3238   class class class wbr 4125    e. cmpt 4179   ` cfv 5358    <_ cle 9015   NN0cn0 10114  LIdealclidl 16133  Poly1cpl1 16462  coe1cco1 16465   deg1 cdg1 19655  ldgIdlSeqcldgis 26916
This theorem is referenced by:  hbtlem2  26919  hbtlem4  26921  hbtlem3  26922  hbtlem5  26923  hbtlem6  26924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-i2m1 8952  ax-1ne0 8953  ax-rrecex 8956  ax-cnre 8957
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-recs 6530  df-rdg 6565  df-nn 9894  df-n0 10115  df-ldgis 26917
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