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

Theorem ig1pdvds 19562
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 15519 . . . . . . 7  |-  ( R  e.  DivRing  ->  R  e.  Ring )
2 ig1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
32ply1rng 16326 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
41, 3syl 15 . . . . . 6  |-  ( R  e.  DivRing  ->  P  e.  Ring )
543ad2ant1 976 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  P  e.  Ring )
6 eqid 2283 . . . . . . . 8  |-  ( Base `  P )  =  (
Base `  P )
7 ig1pcl.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7lidlss 15961 . . . . . . 7  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
983ad2ant2 977 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  I  C_  ( Base `  P
) )
10 ig1pval.g . . . . . . . 8  |-  G  =  (idlGen1p `
 R )
112, 10, 7ig1pcl 19561 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U )  ->  ( G `  I )  e.  I )
12113adant3 975 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  I )
139, 12sseldd 3181 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  ( Base `  P
) )
14 ig1pdvds.d . . . . . 6  |-  .||  =  (
||r `  P )
15 eqid 2283 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
166, 14, 15dvdsr01 15437 . . . . 5  |-  ( ( P  e.  Ring  /\  ( G `  I )  e.  ( Base `  P
) )  ->  ( G `  I )  .||  ( 0g `  P
) )
175, 13, 16syl2anc 642 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  .||  ( 0g `  P
) )
1817adantr 451 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  -> 
( G `  I
)  .||  ( 0g `  P ) )
19 eleq2 2344 . . . . . 6  |-  ( I  =  { ( 0g
`  P ) }  ->  ( X  e.  I  <->  X  e.  { ( 0g `  P ) } ) )
2019biimpac 472 . . . . 5  |-  ( ( X  e.  I  /\  I  =  { ( 0g `  P ) } )  ->  X  e.  { ( 0g `  P
) } )
21203ad2antl3 1119 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  ->  X  e.  { ( 0g `  P ) } )
22 elsni 3664 . . . 4  |-  ( X  e.  { ( 0g
`  P ) }  ->  X  =  ( 0g `  P ) )
2321, 22syl 15 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  ->  X  =  ( 0g `  P ) )
2418, 23breqtrrd 4049 . 2  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  -> 
( G `  I
)  .||  X )
25 simpl1 958 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  R  e.  DivRing )
2625, 1syl 15 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  R  e.  Ring )
27 simpl2 959 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  I  e.  U )
2827, 8syl 15 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  I  C_  ( Base `  P
) )
29 simpl3 960 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  X  e.  I )
3028, 29sseldd 3181 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  X  e.  ( Base `  P
) )
31 simpr 447 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  I  =/=  { ( 0g `  P ) } )
32 eqid 2283 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
33 eqid 2283 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
342, 10, 15, 7, 32, 33ig1pval3 19560 . . . . . . . . . 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 1182 . . . . . . . . 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 968 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  (Monic1p `  R ) )
37 eqid 2283 . . . . . . . . 9  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
3837, 33mon1puc1p 19536 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( G `  I )  e.  (Monic1p `  R ) )  ->  ( G `  I )  e.  (Unic1p `  R ) )
3926, 36, 38syl2anc 642 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  (Unic1p `  R ) )
40 eqid 2283 . . . . . . . 8  |-  (rem1p `  R
)  =  (rem1p `  R
)
4140, 2, 6, 37, 32r1pdeglt 19544 . . . . . . 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 1182 . . . . . 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 969 . . . . . 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 4047 . . . . 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 19467 . . . . . . 7  |-  ( deg1  `  R
) : ( Base `  P ) --> RR*
4635simp1d 967 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  I )
4728, 46sseldd 3181 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  ( Base `  P
) )
48 eqid 2283 . . . . . . . . . . 11  |-  (quot1p `  R
)  =  (quot1p `  R
)
49 eqid 2283 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
50 eqid 2283 . . . . . . . . . . 11  |-  ( -g `  P )  =  (
-g `  P )
5140, 2, 6, 48, 49, 50r1pval 19542 . . . . . . . . . 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 642 . . . . . . . . 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 15 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  P  e.  Ring )
5448, 2, 6, 37q1pcl 19541 . . . . . . . . . . . 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 1182 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (quot1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
567, 6, 49lidlmcl 15969 . . . . . . . . . . 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 1183 . . . . . . . . . 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 15968 . . . . . . . . . 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 1183 . . . . . . . . 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 2357 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  I )
6128, 60sseldd 3181 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
62 ffvelrn 5663 . . . . . . 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 644 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) `  ( X (rem1p `  R ) ( G `  I ) ) )  e.  RR* )
64 ssdif 3311 . . . . . . . . . . 11  |-  ( I 
C_  ( Base `  P
)  ->  ( I  \  { ( 0g `  P ) } ) 
C_  ( ( Base `  P )  \  {
( 0g `  P
) } ) )
6528, 64syl 15 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( ( Base `  P )  \  { ( 0g `  P ) } ) )
66 imass2 5049 . . . . . . . . . 10  |-  ( ( I  \  { ( 0g `  P ) } )  C_  (
( Base `  P )  \  { ( 0g `  P ) } )  ->  ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) )  C_  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) ) )
6765, 66syl 15 . . . . . . . . 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 ) } ) ) )
6832, 2, 15, 6deg1n0ima 19475 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( ( deg1  `  R ) " (
( Base `  P )  \  { ( 0g `  P ) } ) )  C_  NN0 )
6926, 68syl 15 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) )  C_  NN0 )
70 nn0uz 10262 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
7169, 70syl6sseq 3224 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) )  C_  ( ZZ>=
`  0 ) )
7267, 71sstrd 3189 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  ( ZZ>= `  0
) )
73 uzssz 10247 . . . . . . . . 9  |-  ( ZZ>= ` 
0 )  C_  ZZ
74 zssre 10031 . . . . . . . . . 10  |-  ZZ  C_  RR
75 ressxr 8876 . . . . . . . . . 10  |-  RR  C_  RR*
7674, 75sstri 3188 . . . . . . . . 9  |-  ZZ  C_  RR*
7773, 76sstri 3188 . . . . . . . 8  |-  ( ZZ>= ` 
0 )  C_  RR*
7872, 77syl6ss 3191 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  RR* )
797, 15lidl0cl 15964 . . . . . . . . . . . 12  |-  ( ( P  e.  Ring  /\  I  e.  U )  ->  ( 0g `  P )  e.  I )
8053, 27, 79syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( 0g `  P )  e.  I )
8180snssd 3760 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  C_  I )
8231necomd 2529 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  =/=  I )
83 pssdifn0 3515 . . . . . . . . . 10  |-  ( ( { ( 0g `  P ) }  C_  I  /\  { ( 0g
`  P ) }  =/=  I )  -> 
( I  \  {
( 0g `  P
) } )  =/=  (/) )
8481, 82, 83syl2anc 642 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  =/=  (/) )
85 ffn 5389 . . . . . . . . . . . 12  |-  ( ( deg1  `  R ) : (
Base `  P ) --> RR* 
->  ( deg1  `  R )  Fn  ( Base `  P
) )
8645, 85ax-mp 8 . . . . . . . . . . 11  |-  ( deg1  `  R
)  Fn  ( Base `  P )
87 difss 3303 . . . . . . . . . . . 12  |-  ( I 
\  { ( 0g
`  P ) } )  C_  I
8887, 28syl5ss 3190 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( Base `  P ) )
89 fnimaeq0 5365 . . . . . . . . . . 11  |-  ( ( ( deg1  `  R )  Fn  ( Base `  P
)  /\  ( I  \  { ( 0g `  P ) } ) 
C_  ( Base `  P
) )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =  (/)  <->  ( I  \  { ( 0g `  P ) } )  =  (/) ) )
9086, 88, 89sylancr 644 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =  (/)  <->  ( I  \  { ( 0g `  P ) } )  =  (/) ) )
9190necon3bid 2481 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( ( deg1  `  R ) " ( I  \  { ( 0g `  P ) } ) )  =/=  (/)  <->  ( I  \  { ( 0g `  P ) } )  =/=  (/) ) )
9284, 91mpbird 223 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) )  =/=  (/) )
93 infmssuzcl 10301 . . . . . . . 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 ) } ) ) )
9472, 92, 93syl2anc 642 . . . . . . 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 ) } ) ) )
9578, 94sseldd 3181 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  e. 
RR* )
96 xrltnle 8891 . . . . . 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 ) ) ) ) )
9763, 95, 96syl2anc 642 . . . . 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 ) ) ) ) )
9844, 97mpbid 201 . . . 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 ) ) ) )
9972adantr 451 . . . . . . 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 ) )
10086a1i 10 . . . . . . . 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
) )
10188adantr 451 . . . . . . . 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
) )
10260adantr 451 . . . . . . . . 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 )
103 simpr 447 . . . . . . . . 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 ) )
104 eldifsn 3749 . . . . . . . . 9  |-  ( ( X (rem1p `  R ) ( G `  I ) )  e.  ( I 
\  { ( 0g
`  P ) } )  <->  ( ( X (rem1p `  R ) ( G `  I ) )  e.  I  /\  ( X (rem1p `  R ) ( G `  I ) )  =/=  ( 0g
`  P ) ) )
105102, 103, 104sylanbrc 645 . . . . . . . 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 ) } ) )
106 fnfvima 5756 . . . . . . . 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
) } ) ) )
107100, 101, 105, 106syl3anc 1182 . . . . . . 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
) } ) ) )
108 infmssuzle 10300 . . . . . . 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 ) ) ) )
10999, 107, 108syl2anc 642 . . . . . 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 ) ) ) )
110109ex 423 . . . . 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 ) ) ) ) )
111110necon1bd 2514 . . . 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 ) ) )
11298, 111mpd 14 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  =  ( 0g
`  P ) )
1132, 14, 6, 37, 15, 40dvdsr1p 19547 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  P
)  /\  ( G `  I )  e.  (Unic1p `  R ) )  -> 
( ( G `  I )  .||  X  <->  ( X
(rem1p `
 R ) ( G `  I ) )  =  ( 0g
`  P ) ) )
11426, 30, 39, 113syl3anc 1182 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( G `  I
)  .||  X  <->  ( X
(rem1p `
 R ) ( G `  I ) )  =  ( 0g
`  P ) ) )
115112, 114mpbird 223 . 2  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  .|| 
X )
11624, 115pm2.61dane 2524 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   RR*cxr 8866    < clt 8867    <_ cle 8868   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   Basecbs 13148   .rcmulr 13209   0gc0g 13400   -gcsg 14365   Ringcrg 15337   ||rcdsr 15420   DivRingcdr 15512  LIdealclidl 15923  Poly1cpl1 16252   deg1 cdg1 19440  Monic1pcmn1 19511  Unic1pcuc1p 19512  quot1pcq1p 19513  rem1pcr1p 19514  idlGen1pcig1p 19515
This theorem is referenced by:  ig1prsp  19563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rlreg 16024  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442  df-mon1 19516  df-uc1p 19517  df-q1p 19518  df-r1p 19519  df-ig1p 19520
  Copyright terms: Public domain W3C validator