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

Theorem ig1pdvds 20091
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 15834 . . . . . . 7  |-  ( R  e.  DivRing  ->  R  e.  Ring )
2 ig1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
32ply1rng 16634 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
41, 3syl 16 . . . . . 6  |-  ( R  e.  DivRing  ->  P  e.  Ring )
543ad2ant1 978 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  P  e.  Ring )
6 eqid 2435 . . . . . . . 8  |-  ( Base `  P )  =  (
Base `  P )
7 ig1pcl.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7lidlss 16272 . . . . . . 7  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
983ad2ant2 979 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  I  C_  ( Base `  P
) )
10 ig1pval.g . . . . . . . 8  |-  G  =  (idlGen1p `
 R )
112, 10, 7ig1pcl 20090 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U )  ->  ( G `  I )  e.  I )
12113adant3 977 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  I )
139, 12sseldd 3341 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  ( Base `  P
) )
14 ig1pdvds.d . . . . . 6  |-  .||  =  (
||r `  P )
15 eqid 2435 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
166, 14, 15dvdsr01 15752 . . . . 5  |-  ( ( P  e.  Ring  /\  ( G `  I )  e.  ( Base `  P
) )  ->  ( G `  I )  .||  ( 0g `  P
) )
175, 13, 16syl2anc 643 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  .||  ( 0g `  P
) )
1817adantr 452 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  -> 
( G `  I
)  .||  ( 0g `  P ) )
19 eleq2 2496 . . . . . 6  |-  ( I  =  { ( 0g
`  P ) }  ->  ( X  e.  I  <->  X  e.  { ( 0g `  P ) } ) )
2019biimpac 473 . . . . 5  |-  ( ( X  e.  I  /\  I  =  { ( 0g `  P ) } )  ->  X  e.  { ( 0g `  P
) } )
21203ad2antl3 1121 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  ->  X  e.  { ( 0g `  P ) } )
22 elsni 3830 . . . 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 4230 . 2  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  -> 
( G `  I
)  .||  X )
25 simpl1 960 . . . . . . . 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 961 . . . . . . . . 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 962 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  X  e.  I )
3028, 29sseldd 3341 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  X  e.  ( Base `  P
) )
31 simpr 448 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  I  =/=  { ( 0g `  P ) } )
32 eqid 2435 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
33 eqid 2435 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
342, 10, 15, 7, 32, 33ig1pval3 20089 . . . . . . . . . 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 1184 . . . . . . . . 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 970 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  (Monic1p `  R ) )
37 eqid 2435 . . . . . . . . 9  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
3837, 33mon1puc1p 20065 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  (Monic1p `  R ) )  ->  ( G `  I )  e.  (Unic1p `  R ) )
3926, 36, 38syl2anc 643 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  (Unic1p `  R ) )
40 eqid 2435 . . . . . . . 8  |-  (rem1p `  R
)  =  (rem1p `  R
)
4140, 2, 6, 37, 32r1pdeglt 20073 . . . . . . 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 1184 . . . . . 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 971 . . . . . 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 4228 . . . . 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 19996 . . . . . . 7  |-  ( deg1  `  R
) : ( Base `  P ) --> RR*
4635simp1d 969 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  I )
4728, 46sseldd 3341 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  ( Base `  P
) )
48 eqid 2435 . . . . . . . . . . 11  |-  (quot1p `  R
)  =  (quot1p `  R
)
49 eqid 2435 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
50 eqid 2435 . . . . . . . . . . 11  |-  ( -g `  P )  =  (
-g `  P )
5140, 2, 6, 48, 49, 50r1pval 20071 . . . . . . . . . 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 643 . . . . . . . . 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 20070 . . . . . . . . . . . 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 1184 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (quot1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
567, 6, 49lidlmcl 16280 . . . . . . . . . . 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 1185 . . . . . . . . . 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 16279 . . . . . . . . . 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 1185 . . . . . . . . 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 2509 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  I )
6128, 60sseldd 3341 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
62 ffvelrn 5860 . . . . . . 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 645 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  e.  RR* )
6428ssdifd 3475 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( ( Base `  P )  \  { ( 0g `  P ) } ) )
65 imass2 5232 . . . . . . . . . 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 20004 . . . . . . . . . . 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 10512 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
7068, 69syl6sseq 3386 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) )  C_  ( ZZ>=
`  0 ) )
7166, 70sstrd 3350 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  ( ZZ>= `  0
) )
72 uzssz 10497 . . . . . . . . 9  |-  ( ZZ>= ` 
0 )  C_  ZZ
73 zssre 10281 . . . . . . . . . 10  |-  ZZ  C_  RR
74 ressxr 9121 . . . . . . . . . 10  |-  RR  C_  RR*
7573, 74sstri 3349 . . . . . . . . 9  |-  ZZ  C_  RR*
7672, 75sstri 3349 . . . . . . . 8  |-  ( ZZ>= ` 
0 )  C_  RR*
7771, 76syl6ss 3352 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  RR* )
787, 15lidl0cl 16275 . . . . . . . . . . . 12  |-  ( ( P  e.  Ring  /\  I  e.  U )  ->  ( 0g `  P )  e.  I )
7953, 27, 78syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( 0g `  P )  e.  I )
8079snssd 3935 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  C_  I )
8131necomd 2681 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  =/=  I )
82 pssdifn0 3681 . . . . . . . . . 10  |-  ( ( { ( 0g `  P ) }  C_  I  /\  { ( 0g
`  P ) }  =/=  I )  -> 
( I  \  {
( 0g `  P
) } )  =/=  (/) )
8380, 81, 82syl2anc 643 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  =/=  (/) )
84 ffn 5583 . . . . . . . . . . . 12  |-  ( ( deg1  `  R ) : (
Base `  P ) --> RR* 
->  ( deg1  `  R )  Fn  ( Base `  P
) )
8545, 84ax-mp 8 . . . . . . . . . . 11  |-  ( deg1  `  R
)  Fn  ( Base `  P )
8628ssdifssd 3477 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( Base `  P ) )
87 fnimaeq0 5558 . . . . . . . . . . 11  |-  ( ( ( deg1  `  R )  Fn  ( Base `  P
)  /\  ( I  \  { ( 0g `  P ) } ) 
C_  ( Base `  P
) )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =  (/)  <->  ( I  \  { ( 0g `  P ) } )  =  (/) ) )
8885, 86, 87sylancr 645 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =  (/)  <->  ( I  \  { ( 0g `  P ) } )  =  (/) ) )
8988necon3bid 2633 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =/=  (/)  <->  ( I  \  { ( 0g `  P ) } )  =/=  (/) ) )
9083, 89mpbird 224 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) )  =/=  (/) )
91 infmssuzcl 10551 . . . . . . . 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 643 . . . . . . 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 3341 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  e. 
RR* )
94 xrltnle 9136 . . . . . 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 643 . . . . 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 202 . . . 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 452 . . . . . . 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 452 . . . . . . . 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 452 . . . . . . . . 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 448 . . . . . . . . 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 3919 . . . . . . . . 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 646 . . . . . . . 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 5968 . . . . . . . 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 1184 . . . . . . 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 10550 . . . . . . 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 643 . . . . . 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 424 . . . . 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 2666 . . . 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 20076 . . . 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 1184 . . 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 224 . 2  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  .|| 
X )
11424, 113pm2.61dane 2676 1  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  .|| 
X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   RRcr 8981   0cc0 8982   RR*cxr 9111    < clt 9112    <_ cle 9113   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   Basecbs 13461   .rcmulr 13522   0gc0g 13715   -gcsg 14680   Ringcrg 15652   ||rcdsr 15735   DivRingcdr 15827  LIdealclidl 16234  Poly1cpl1 16563   deg1 cdg1 19969  Monic1pcmn1 20040  Unic1pcuc1p 20041  quot1pcq1p 20042  rem1pcr1p 20043  idlGen1pcig1p 20044
This theorem is referenced by:  ig1prsp  20092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-subrg 15858  df-lmod 15944  df-lss 16001  df-sra 16236  df-rgmod 16237  df-lidl 16238  df-rlreg 16335  df-ascl 16366  df-psr 16409  df-mvr 16410  df-mpl 16411  df-opsr 16417  df-psr1 16568  df-vr1 16569  df-ply1 16570  df-coe1 16573  df-cnfld 16696  df-mdeg 19970  df-deg1 19971  df-mon1 20045  df-uc1p 20046  df-q1p 20047  df-r1p 20048  df-ig1p 20049
  Copyright terms: Public domain W3C validator