MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ig1pdvds Structured version   Unicode version

Theorem ig1pdvds 21617
Description: The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
ig1pval.p  |-  P  =  (Poly1 `  R )
ig1pval.g  |-  G  =  (idlGen1p `
 R )
ig1pcl.u  |-  U  =  (LIdeal `  P )
ig1pdvds.d  |-  .||  =  (
||r `  P )
Assertion
Ref Expression
ig1pdvds  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  .|| 
X )

Proof of Theorem ig1pdvds
StepHypRef Expression
1 drngrng 16815 . . . . . . 7  |-  ( R  e.  DivRing  ->  R  e.  Ring )
2 ig1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
32ply1rng 17674 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
41, 3syl 16 . . . . . 6  |-  ( R  e.  DivRing  ->  P  e.  Ring )
543ad2ant1 1009 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  P  e.  Ring )
6 eqid 2437 . . . . . . . 8  |-  ( Base `  P )  =  (
Base `  P )
7 ig1pcl.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7lidlss 17265 . . . . . . 7  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
983ad2ant2 1010 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  I  C_  ( Base `  P
) )
10 ig1pval.g . . . . . . . 8  |-  G  =  (idlGen1p `
 R )
112, 10, 7ig1pcl 21616 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U )  ->  ( G `  I )  e.  I )
12113adant3 1008 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  I )
139, 12sseldd 3350 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  ( Base `  P
) )
14 ig1pdvds.d . . . . . 6  |-  .||  =  (
||r `  P )
15 eqid 2437 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
166, 14, 15dvdsr01 16733 . . . . 5  |-  ( ( P  e.  Ring  /\  ( G `  I )  e.  ( Base `  P
) )  ->  ( G `  I )  .||  ( 0g `  P
) )
175, 13, 16syl2anc 661 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  .||  ( 0g `  P
) )
1817adantr 465 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  -> 
( G `  I
)  .||  ( 0g `  P ) )
19 eleq2 2498 . . . . . 6  |-  ( I  =  { ( 0g
`  P ) }  ->  ( X  e.  I  <->  X  e.  { ( 0g `  P ) } ) )
2019biimpac 486 . . . . 5  |-  ( ( X  e.  I  /\  I  =  { ( 0g `  P ) } )  ->  X  e.  { ( 0g `  P
) } )
21203ad2antl3 1152 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  ->  X  e.  { ( 0g `  P ) } )
22 elsni 3895 . . . 4  |-  ( X  e.  { ( 0g
`  P ) }  ->  X  =  ( 0g `  P ) )
2321, 22syl 16 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  ->  X  =  ( 0g `  P ) )
2418, 23breqtrrd 4311 . 2  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  -> 
( G `  I
)  .||  X )
25 simpl1 991 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  R  e.  DivRing )
2625, 1syl 16 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  R  e.  Ring )
27 simpl2 992 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  I  e.  U )
2827, 8syl 16 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  I  C_  ( Base `  P
) )
29 simpl3 993 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  X  e.  I )
3028, 29sseldd 3350 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  X  e.  ( Base `  P
) )
31 simpr 461 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  I  =/=  { ( 0g `  P ) } )
32 eqid 2437 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
33 eqid 2437 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
342, 10, 15, 7, 32, 33ig1pval3 21615 . . . . . . . . . 10  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{ ( 0g `  P ) } )  ->  ( ( G `
 I )  e.  I  /\  ( G `
 I )  e.  (Monic1p `  R )  /\  ( ( deg1  `  R ) `  ( G `  I
) )  =  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  ) ) )
3525, 27, 31, 34syl3anc 1218 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( G `  I
)  e.  I  /\  ( G `  I )  e.  (Monic1p `  R )  /\  ( ( deg1  `  R ) `  ( G `  I
) )  =  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  ) ) )
3635simp2d 1001 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  (Monic1p `  R ) )
37 eqid 2437 . . . . . . . . 9  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
3837, 33mon1puc1p 21591 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  (Monic1p `  R ) )  ->  ( G `  I )  e.  (Unic1p `  R ) )
3926, 36, 38syl2anc 661 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  (Unic1p `  R ) )
40 eqid 2437 . . . . . . . 8  |-  (rem1p `  R
)  =  (rem1p `  R
)
4140, 2, 6, 37, 32r1pdeglt 21599 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  P
)  /\  ( G `  I )  e.  (Unic1p `  R ) )  -> 
( ( deg1  `  R ) `  ( X (rem1p `  R
) ( G `  I ) ) )  <  ( ( deg1  `  R
) `  ( G `  I ) ) )
4226, 30, 39, 41syl3anc 1218 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  <  (
( deg1  `
 R ) `  ( G `  I ) ) )
4335simp3d 1002 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) `  ( G `  I ) )  =  sup (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  ) )
4442, 43breqtrd 4309 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  <  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  ) )
4532, 2, 6deg1xrf 21521 . . . . . . 7  |-  ( deg1  `  R
) : ( Base `  P ) --> RR*
4635simp1d 1000 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  I )
4728, 46sseldd 3350 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  ( Base `  P
) )
48 eqid 2437 . . . . . . . . . . 11  |-  (quot1p `  R
)  =  (quot1p `  R
)
49 eqid 2437 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
50 eqid 2437 . . . . . . . . . . 11  |-  ( -g `  P )  =  (
-g `  P )
5140, 2, 6, 48, 49, 50r1pval 21597 . . . . . . . . . 10  |-  ( ( X  e.  ( Base `  P )  /\  ( G `  I )  e.  ( Base `  P
) )  ->  ( X (rem1p `  R ) ( G `  I ) )  =  ( X ( -g `  P
) ( ( X (quot1p `  R ) ( G `  I ) ) ( .r `  P ) ( G `
 I ) ) ) )
5230, 47, 51syl2anc 661 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  =  ( X ( -g `  P
) ( ( X (quot1p `  R ) ( G `  I ) ) ( .r `  P ) ( G `
 I ) ) ) )
5326, 3syl 16 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  P  e.  Ring )
5448, 2, 6, 37q1pcl 21596 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  P
)  /\  ( G `  I )  e.  (Unic1p `  R ) )  -> 
( X (quot1p `  R
) ( G `  I ) )  e.  ( Base `  P
) )
5526, 30, 39, 54syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (quot1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
567, 6, 49lidlmcl 17273 . . . . . . . . . . 11  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( ( X (quot1p `  R ) ( G `  I ) )  e.  ( Base `  P )  /\  ( G `  I )  e.  I ) )  -> 
( ( X (quot1p `  R ) ( G `
 I ) ) ( .r `  P
) ( G `  I ) )  e.  I )
5753, 27, 55, 46, 56syl22anc 1219 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( X (quot1p `  R
) ( G `  I ) ) ( .r `  P ) ( G `  I
) )  e.  I
)
587, 50lidlsubcl 17272 . . . . . . . . . 10  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( X  e.  I  /\  ( ( X (quot1p `  R ) ( G `  I ) ) ( .r `  P ) ( G `
 I ) )  e.  I ) )  ->  ( X (
-g `  P )
( ( X (quot1p `  R ) ( G `
 I ) ) ( .r `  P
) ( G `  I ) ) )  e.  I )
5953, 27, 29, 57, 58syl22anc 1219 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X ( -g `  P
) ( ( X (quot1p `  R ) ( G `  I ) ) ( .r `  P ) ( G `
 I ) ) )  e.  I )
6052, 59eqeltrd 2511 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  I )
6128, 60sseldd 3350 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
62 ffvelrn 5834 . . . . . . 7  |-  ( ( ( deg1  `  R ) : ( Base `  P
) --> RR*  /\  ( X (rem1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )  -> 
( ( deg1  `  R ) `  ( X (rem1p `  R
) ( G `  I ) ) )  e.  RR* )
6345, 61, 62sylancr 663 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  e.  RR* )
6428ssdifd 3485 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( ( Base `  P )  \  { ( 0g `  P ) } ) )
65 imass2 5197 . . . . . . . . . 10  |-  ( ( I  \  { ( 0g `  P ) } )  C_  (
( Base `  P )  \  { ( 0g `  P ) } )  ->  ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) )  C_  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) ) )
6664, 65syl 16 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  ( ( deg1  `  R
) " ( (
Base `  P )  \  { ( 0g `  P ) } ) ) )
6732, 2, 15, 6deg1n0ima 21529 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( ( deg1  `  R ) " (
( Base `  P )  \  { ( 0g `  P ) } ) )  C_  NN0 )
6826, 67syl 16 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) )  C_  NN0 )
69 nn0uz 10887 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
7068, 69syl6sseq 3395 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) )  C_  ( ZZ>=
`  0 ) )
7166, 70sstrd 3359 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  ( ZZ>= `  0
) )
72 uzssz 10872 . . . . . . . . 9  |-  ( ZZ>= ` 
0 )  C_  ZZ
73 zssre 10645 . . . . . . . . . 10  |-  ZZ  C_  RR
74 ressxr 9419 . . . . . . . . . 10  |-  RR  C_  RR*
7573, 74sstri 3358 . . . . . . . . 9  |-  ZZ  C_  RR*
7672, 75sstri 3358 . . . . . . . 8  |-  ( ZZ>= ` 
0 )  C_  RR*
7771, 76syl6ss 3361 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  RR* )
787, 15lidl0cl 17268 . . . . . . . . . . . 12  |-  ( ( P  e.  Ring  /\  I  e.  U )  ->  ( 0g `  P )  e.  I )
7953, 27, 78syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( 0g `  P )  e.  I )
8079snssd 4011 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  C_  I )
8131necomd 2689 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  =/=  I )
82 pssdifn0 3733 . . . . . . . . . 10  |-  ( ( { ( 0g `  P ) }  C_  I  /\  { ( 0g
`  P ) }  =/=  I )  -> 
( I  \  {
( 0g `  P
) } )  =/=  (/) )
8380, 81, 82syl2anc 661 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  =/=  (/) )
84 ffn 5552 . . . . . . . . . . . 12  |-  ( ( deg1  `  R ) : (
Base `  P ) --> RR* 
->  ( deg1  `  R )  Fn  ( Base `  P
) )
8545, 84ax-mp 5 . . . . . . . . . . 11  |-  ( deg1  `  R
)  Fn  ( Base `  P )
8628ssdifssd 3487 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( Base `  P ) )
87 fnimaeq0 5525 . . . . . . . . . . 11  |-  ( ( ( deg1  `  R )  Fn  ( Base `  P
)  /\  ( I  \  { ( 0g `  P ) } ) 
C_  ( Base `  P
) )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =  (/)  <->  ( I  \  { ( 0g `  P ) } )  =  (/) ) )
8885, 86, 87sylancr 663 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =  (/)  <->  ( I  \  { ( 0g `  P ) } )  =  (/) ) )
8988necon3bid 2637 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =/=  (/)  <->  ( I  \  { ( 0g `  P ) } )  =/=  (/) ) )
9083, 89mpbird 232 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) )  =/=  (/) )
91 infmssuzcl 10930 . . . . . . . 8  |-  ( ( ( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  C_  ( ZZ>= ` 
0 )  /\  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) )  =/=  (/) )  ->  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  e.  ( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) )
9271, 90, 91syl2anc 661 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  e.  ( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) )
9377, 92sseldd 3350 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  e. 
RR* )
94 xrltnle 9435 . . . . . 6  |-  ( ( ( ( deg1  `  R ) `  ( X (rem1p `  R
) ( G `  I ) ) )  e.  RR*  /\  sup (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  e.  RR* )  ->  ( ( ( deg1  `  R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  <  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  <->  -.  sup (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  <_  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) ) ) )
9563, 93, 94syl2anc 661 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( ( deg1  `  R ) `  ( X (rem1p `  R
) ( G `  I ) ) )  <  sup ( ( ( deg1  `  R ) " (
I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  <->  -.  sup (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  <_  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) ) ) )
9644, 95mpbid 210 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  -.  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  <_ 
( ( deg1  `  R ) `  ( X (rem1p `  R
) ( G `  I ) ) ) )
9771adantr 465 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) )  C_  ( ZZ>=
`  0 ) )
9885a1i 11 . . . . . . . 8  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  ( deg1  `  R )  Fn  ( Base `  P
) )
9986adantr 465 . . . . . . . 8  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  ( I  \  { ( 0g `  P ) } ) 
C_  ( Base `  P
) )
10060adantr 465 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  ( X (rem1p `  R ) ( G `
 I ) )  e.  I )
101 simpr 461 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  ( X (rem1p `  R ) ( G `
 I ) )  =/=  ( 0g `  P ) )
102 eldifsn 3993 . . . . . . . . 9  |-  ( ( X (rem1p `  R ) ( G `  I ) )  e.  ( I 
\  { ( 0g
`  P ) } )  <->  ( ( X (rem1p `  R ) ( G `  I ) )  e.  I  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) ) )
103100, 101, 102sylanbrc 664 . . . . . . . 8  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  ( X (rem1p `  R ) ( G `
 I ) )  e.  ( I  \  { ( 0g `  P ) } ) )
104 fnfvima 5948 . . . . . . . 8  |-  ( ( ( deg1  `  R )  Fn  ( Base `  P
)  /\  ( I  \  { ( 0g `  P ) } ) 
C_  ( Base `  P
)  /\  ( X
(rem1p `
 R ) ( G `  I ) )  e.  ( I 
\  { ( 0g
`  P ) } ) )  ->  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  e.  ( ( deg1  `  R ) "
( I  \  {
( 0g `  P
) } ) ) )
10598, 99, 103, 104syl3anc 1218 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  ( ( deg1  `  R
) `  ( X
(rem1p `
 R ) ( G `  I ) ) )  e.  ( ( deg1  `  R ) "
( I  \  {
( 0g `  P
) } ) ) )
106 infmssuzle 10929 . . . . . . 7  |-  ( ( ( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  C_  ( ZZ>= ` 
0 )  /\  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  e.  ( ( deg1  `  R ) "
( I  \  {
( 0g `  P
) } ) ) )  ->  sup (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  <_  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) ) )
10797, 105, 106syl2anc 661 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  X  e.  I
)  /\  I  =/=  { ( 0g `  P
) } )  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) )  ->  sup ( ( ( deg1  `  R ) " (
I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  <_ 
( ( deg1  `  R ) `  ( X (rem1p `  R
) ( G `  I ) ) ) )
108107ex 434 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( X (rem1p `  R
) ( G `  I ) )  =/=  ( 0g `  P
)  ->  sup (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  <_  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) ) ) )
109108necon1bd 2673 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( -.  sup ( ( ( deg1  `  R ) " (
I  \  { ( 0g `  P ) } ) ) ,  RR ,  `'  <  )  <_ 
( ( deg1  `  R ) `  ( X (rem1p `  R
) ( G `  I ) ) )  ->  ( X (rem1p `  R ) ( G `
 I ) )  =  ( 0g `  P ) ) )
11096, 109mpd 15 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  =  ( 0g
`  P ) )
1112, 14, 6, 37, 15, 40dvdsr1p 21602 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  P
)  /\  ( G `  I )  e.  (Unic1p `  R ) )  -> 
( ( G `  I )  .||  X  <->  ( X
(rem1p `
 R ) ( G `  I ) )  =  ( 0g
`  P ) ) )
11226, 30, 39, 111syl3anc 1218 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( G `  I
)  .||  X  <->  ( X
(rem1p `
 R ) ( G `  I ) )  =  ( 0g
`  P ) ) )
113110, 112mpbird 232 . 2  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  .|| 
X )
11424, 113pm2.61dane 2683 1  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  .|| 
X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2600    \ cdif 3318    C_ wss 3321   (/)c0 3630   {csn 3870   class class class wbr 4285   `'ccnv 4831   "cima 4835    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   supcsup 7682   RRcr 9273   0cc0 9274   RR*cxr 9409    < clt 9410    <_ cle 9411   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   Basecbs 14166   .rcmulr 14231   0gc0g 14370   -gcsg 15405   Ringcrg 16631   ||rcdsr 16716   DivRingcdr 16808  LIdealclidl 17225  Poly1cpl1 17605   deg1 cdg1 21492  Monic1pcmn1 21566  Unic1pcuc1p 21567  quot1pcq1p 21568  rem1pcr1p 21569  idlGen1pcig1p 21570
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
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 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-iin 4167  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-ofr 6316  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-fzo 11541  df-seq 11799  df-hash 12096  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-0g 14372  df-gsum 14373  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15534  df-minusg 15535  df-sbg 15536  df-mulg 15537  df-subg 15667  df-ghm 15734  df-cntz 15824  df-cmn 16268  df-abl 16269  df-mgp 16578  df-ur 16590  df-rng 16633  df-cring 16634  df-oppr 16701  df-dvdsr 16719  df-unit 16720  df-invr 16750  df-drng 16810  df-subrg 16839  df-lmod 16926  df-lss 16988  df-sra 17227  df-rgmod 17228  df-lidl 17229  df-rlreg 17328  df-ascl 17360  df-psr 17397  df-mvr 17398  df-mpl 17399  df-opsr 17401  df-psr1 17608  df-vr1 17609  df-ply1 17610  df-coe1 17611  df-cnfld 17788  df-mdeg 21493  df-deg1 21494  df-mon1 21571  df-uc1p 21572  df-q1p 21573  df-r1p 21574  df-ig1p 21575
This theorem is referenced by:  ig1prsp  21618
  Copyright terms: Public domain W3C validator