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Theorem hbtlem7 26652
Description: Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypotheses
Ref Expression
hbtlem.p  |-  P  =  (Poly1 `  R )
hbtlem.u  |-  U  =  (LIdeal `  P )
hbtlem.s  |-  S  =  (ldgIdlSeq `  R )
hbtlem7.t  |-  T  =  (LIdeal `  R )
Assertion
Ref Expression
hbtlem7  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  ( S `  I ) : NN0 --> T )

Proof of Theorem hbtlem7
Dummy variables  i 
j  x  y  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . . . . . 9  |-  ( ( ( ( deg1  `  R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) )  -> 
y  =  ( (coe1 `  j ) `  x
) )
21reximi 2726 . . . . . . . 8  |-  ( E. j  e.  I  ( ( ( deg1  `  R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) )  ->  E. j  e.  I 
y  =  ( (coe1 `  j ) `  x
) )
32ss2abi 3321 . . . . . . 7  |-  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) }  C_  { y  |  E. j  e.  I  y  =  ( (coe1 `  j ) `  x ) }
4 abrexexg 5850 . . . . . . 7  |-  ( I  e.  U  ->  { y  |  E. j  e.  I  y  =  ( (coe1 `  j ) `  x ) }  e.  _V )
5 ssexg 4241 . . . . . . 7  |-  ( ( { y  |  E. j  e.  I  (
( ( deg1  `  R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) } 
C_  { y  |  E. j  e.  I 
y  =  ( (coe1 `  j ) `  x
) }  /\  {
y  |  E. j  e.  I  y  =  ( (coe1 `  j ) `  x ) }  e.  _V )  ->  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) }  e.  _V )
63, 4, 5sylancr 644 . . . . . 6  |-  ( I  e.  U  ->  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) }  e.  _V )
76ralrimivw 2703 . . . . 5  |-  ( I  e.  U  ->  A. x  e.  NN0  { y  |  E. j  e.  I 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) }  e.  _V )
87adantl 452 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  A. x  e.  NN0  { y  |  E. j  e.  I 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) }  e.  _V )
9 eqid 2358 . . . . 5  |-  ( x  e.  NN0  |->  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } )  =  ( x  e. 
NN0  |->  { y  |  E. j  e.  I 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) } )
109fnmpt 5452 . . . 4  |-  ( A. x  e.  NN0  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) }  e.  _V  ->  ( x  e. 
NN0  |->  { y  |  E. j  e.  I 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) } )  Fn  NN0 )
118, 10syl 15 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  (
x  e.  NN0  |->  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } )  Fn  NN0 )
12 hbtlem.s . . . . . . 7  |-  S  =  (ldgIdlSeq `  R )
13 elex 2872 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
_V )
14 fveq2 5608 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
15 hbtlem.p . . . . . . . . . . . . 13  |-  P  =  (Poly1 `  R )
1614, 15syl6eqr 2408 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
1716fveq2d 5612 . . . . . . . . . . 11  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
18 hbtlem.u . . . . . . . . . . 11  |-  U  =  (LIdeal `  P )
1917, 18syl6eqr 2408 . . . . . . . . . 10  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
20 fveq2 5608 . . . . . . . . . . . . . . . 16  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
2120fveq1d 5610 . . . . . . . . . . . . . . 15  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  j )  =  ( ( deg1  `  R ) `  j ) )
2221breq1d 4114 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  j )  <_  x  <->  ( ( deg1  `  R ) `  j )  <_  x
) )
2322anbi1d 685 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (
( ( ( deg1  `  r
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) )  <->  ( (
( deg1  `
 R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) ) )
2423rexbidv 2640 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( E. j  e.  i 
( ( ( deg1  `  r
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) )  <->  E. j  e.  i  ( (
( deg1  `
 R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) ) )
2524abbidv 2472 . . . . . . . . . . 11  |-  ( r  =  R  ->  { y  |  E. j  e.  i  ( ( ( deg1  `  r ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) }  =  { y  |  E. j  e.  i  (
( ( deg1  `  R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) } )
2625mpteq2dv 4188 . . . . . . . . . 10  |-  ( r  =  R  ->  (
x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  r ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } )  =  ( x  e. 
NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) } ) )
2719, 26mpteq12dv 4179 . . . . . . . . 9  |-  ( r  =  R  ->  (
i  e.  (LIdeal `  (Poly1 `  r ) )  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  r ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) )  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) ) )
28 df-ldgis 26649 . . . . . . . . 9  |- ldgIdlSeq  =  ( r  e.  _V  |->  ( i  e.  (LIdeal `  (Poly1 `  r ) )  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  r ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) ) )
29 fvex 5622 . . . . . . . . . . 11  |-  (LIdeal `  P )  e.  _V
3018, 29eqeltri 2428 . . . . . . . . . 10  |-  U  e. 
_V
3130mptex 5832 . . . . . . . . 9  |-  ( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) )  e.  _V
3227, 28, 31fvmpt 5685 . . . . . . . 8  |-  ( R  e.  _V  ->  (ldgIdlSeq `  R )  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) ) )
3313, 32syl 15 . . . . . . 7  |-  ( R  e.  Ring  ->  (ldgIdlSeq `  R
)  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) ) )
3412, 33syl5eq 2402 . . . . . 6  |-  ( R  e.  Ring  ->  S  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  (
( ( deg1  `  R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) } ) ) )
3534fveq1d 5610 . . . . 5  |-  ( R  e.  Ring  ->  ( S `
 I )  =  ( ( i  e.  U  |->  ( x  e. 
NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) } ) ) `  I
) )
36 rexeq 2813 . . . . . . . 8  |-  ( i  =  I  ->  ( E. j  e.  i 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) )  <->  E. j  e.  I  ( (
( deg1  `
 R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) ) )
3736abbidv 2472 . . . . . . 7  |-  ( i  =  I  ->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) }  =  { y  |  E. j  e.  I  (
( ( deg1  `  R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) } )
3837mpteq2dv 4188 . . . . . 6  |-  ( i  =  I  ->  (
x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } )  =  ( x  e. 
NN0  |->  { y  |  E. j  e.  I 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) } ) )
39 eqid 2358 . . . . . 6  |-  ( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) )  =  ( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) )
40 nn0ex 10063 . . . . . . 7  |-  NN0  e.  _V
4140mptex 5832 . . . . . 6  |-  ( x  e.  NN0  |->  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } )  e.  _V
4238, 39, 41fvmpt 5685 . . . . 5  |-  ( I  e.  U  ->  (
( i  e.  U  |->  ( x  e.  NN0  |->  { y  |  E. j  e.  i  (
( ( deg1  `  R ) `  j )  <_  x  /\  y  =  (
(coe1 `  j ) `  x ) ) } ) ) `  I
)  =  ( x  e.  NN0  |->  { y  |  E. j  e.  I  ( ( ( deg1  `  R ) `  j
)  <_  x  /\  y  =  ( (coe1 `  j ) `  x
) ) } ) )
4335, 42sylan9eq 2410 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  ( S `  I )  =  ( x  e. 
NN0  |->  { y  |  E. j  e.  I 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) } ) )
4443fneq1d 5417 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  (
( S `  I
)  Fn  NN0  <->  ( x  e.  NN0  |->  { y  |  E. j  e.  I 
( ( ( deg1  `  R
) `  j )  <_  x  /\  y  =  ( (coe1 `  j ) `  x ) ) } )  Fn  NN0 )
)
4511, 44mpbird 223 . 2  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  ( S `  I )  Fn  NN0 )
46 hbtlem7.t . . . . 5  |-  T  =  (LIdeal `  R )
4715, 18, 12, 46hbtlem2 26651 . . . 4  |-  ( ( R  e.  Ring  /\  I  e.  U  /\  x  e.  NN0 )  ->  (
( S `  I
) `  x )  e.  T )
48473expa 1151 . . 3  |-  ( ( ( R  e.  Ring  /\  I  e.  U )  /\  x  e.  NN0 )  ->  ( ( S `
 I ) `  x )  e.  T
)
4948ralrimiva 2702 . 2  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  A. x  e.  NN0  ( ( S `
 I ) `  x )  e.  T
)
50 ffnfv 5768 . 2  |-  ( ( S `  I ) : NN0 --> T  <->  ( ( S `  I )  Fn  NN0  /\  A. x  e.  NN0  ( ( S `
 I ) `  x )  e.  T
) )
5145, 49, 50sylanbrc 645 1  |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  ( S `  I ) : NN0 --> T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619   E.wrex 2620   _Vcvv 2864    C_ wss 3228   class class class wbr 4104    e. cmpt 4158    Fn wfn 5332   -->wf 5333   ` cfv 5337    <_ cle 8958   NN0cn0 10057   Ringcrg 15436  LIdealclidl 16022  Poly1cpl1 16351  coe1cco1 16354   deg1 cdg1 19544  ldgIdlSeqcldgis 26648
This theorem is referenced by:  hbt  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-ofr 6166  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-0g 13503  df-gsum 13504  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-ghm 14780  df-cntz 14892  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-cring 15440  df-ur 15441  df-subrg 15642  df-lmod 15728  df-lss 15789  df-sra 16024  df-rgmod 16025  df-lidl 16026  df-ascl 16154  df-psr 16197  df-mvr 16198  df-mpl 16199  df-opsr 16205  df-psr1 16356  df-vr1 16357  df-ply1 16358  df-coe1 16361  df-cnfld 16483  df-mdeg 19545  df-deg1 19546  df-ldgis 26649
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