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Theorem hbtlem1 29498
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 2996 . . . . . . 7  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5706 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 hbtlem.p . . . . . . . . . . . 12  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2493 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5710 . . . . . . . . . 10  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 hbtlem.u . . . . . . . . . 10  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2493 . . . . . . . . 9  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
9 fveq2 5706 . . . . . . . . . . . . . . . 16  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
10 hbtlem.d . . . . . . . . . . . . . . . 16  |-  D  =  ( deg1  `  R )
119, 10syl6eqr 2493 . . . . . . . . . . . . . . 15  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1211fveq1d 5708 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  k )  =  ( D `  k ) )
1312breq1d 4317 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  k )  <_  x  <->  ( D `  k )  <_  x ) )
1413anbi1d 704 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (
( ( ( deg1  `  r
) `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) )  <->  ( ( D `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) ) )
1514rexbidv 2751 . . . . . . . . . . 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 2563 . . . . . . . . . 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 4394 . . . . . . . . 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 4385 . . . . . . . 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 29497 . . . . . . . 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 5716 . . . . . . . . . 10  |-  (LIdeal `  P )  e.  _V
217, 20eqeltri 2513 . . . . . . . . 9  |-  U  e. 
_V
2221mptex 5963 . . . . . . . 8  |-  ( i  e.  U  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } ) )  e.  _V
2318, 19, 22fvmpt 5789 . . . . . . 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 2487 . . . . 5  |-  ( R  e.  V  ->  S  =  ( i  e.  U  |->  ( x  e. 
NN0  |->  { j  |  E. k  e.  i  ( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) ) } ) ) )
2625fveq1d 5708 . . . 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 5708 . . 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 1009 . 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 2933 . . . . . . 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 2563 . . . . . 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 4394 . . . . 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 2443 . . . . 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 10600 . . . . . 6  |-  NN0  e.  _V
3433mptex 5963 . . . . 5  |-  ( x  e.  NN0  |->  { j  |  E. k  e.  I  ( ( D `
 k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) } )  e.  _V
3531, 32, 34fvmpt 5789 . . . 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 5708 . . 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 1010 . 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 990 . . 3  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  X  e.  NN0 )
39 simpr 461 . . . . . 6  |-  ( ( ( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) )  ->  j  =  ( (coe1 `  k
) `  X )
)
4039reximi 2838 . . . . 5  |-  ( E. k  e.  I  ( ( D `  k
)  <_  X  /\  j  =  ( (coe1 `  k ) `  X
) )  ->  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) )
4140ss2abi 3439 . . . 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 6567 . . . . 5  |-  ( I  e.  U  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k ) `  X ) }  e.  _V )
43423ad2ant2 1010 . . . 4  |-  ( ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  { j  |  E. k  e.  I  j  =  ( (coe1 `  k
) `  X ) }  e.  _V )
44 ssexg 4453 . . . 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 663 . . 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 4311 . . . . . . 7  |-  ( x  =  X  ->  (
( D `  k
)  <_  x  <->  ( D `  k )  <_  X
) )
47 fveq2 5706 . . . . . . . 8  |-  ( x  =  X  ->  (
(coe1 `  k ) `  x )  =  ( (coe1 `  k ) `  X ) )
4847eqeq2d 2454 . . . . . . 7  |-  ( x  =  X  ->  (
j  =  ( (coe1 `  k ) `  x
)  <->  j  =  ( (coe1 `  k ) `  X ) ) )
4946, 48anbi12d 710 . . . . . 6  |-  ( x  =  X  ->  (
( ( D `  k )  <_  x  /\  j  =  (
(coe1 `  k ) `  x ) )  <->  ( ( D `  k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) ) )
5049rexbidv 2751 . . . . 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 2563 . . . 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 2443 . . . 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 5787 . . 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 661 . 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 2479 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2731   _Vcvv 2987    C_ wss 3343   class class class wbr 4307    e. cmpt 4365   ` cfv 5433    <_ cle 9434   NN0cn0 10594  LIdealclidl 17266  Poly1cpl1 17648  coe1cco1 17649   deg1 cdg1 21538  ldgIdlSeqcldgis 29496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-i2m1 9365  ax-1ne0 9366  ax-rrecex 9369  ax-cnre 9370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-om 6492  df-recs 6847  df-rdg 6881  df-nn 10338  df-n0 10595  df-ldgis 29497
This theorem is referenced by:  hbtlem2  29499  hbtlem4  29501  hbtlem3  29502  hbtlem5  29503  hbtlem6  29504
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