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Theorem ig1pdvds 22443
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 17272 . . . . . . 7  |-  ( R  e.  DivRing  ->  R  e.  Ring )
2 ig1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
32ply1ring 18157 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
41, 3syl 16 . . . . . 6  |-  ( R  e.  DivRing  ->  P  e.  Ring )
543ad2ant1 1017 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  P  e.  Ring )
6 eqid 2467 . . . . . . . 8  |-  ( Base `  P )  =  (
Base `  P )
7 ig1pcl.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7lidlss 17725 . . . . . . 7  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
983ad2ant2 1018 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  I  C_  ( Base `  P
) )
10 ig1pval.g . . . . . . . 8  |-  G  =  (idlGen1p `
 R )
112, 10, 7ig1pcl 22442 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U )  ->  ( G `  I )  e.  I )
12113adant3 1016 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  I )
139, 12sseldd 3510 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  ->  ( G `  I )  e.  ( Base `  P
) )
14 ig1pdvds.d . . . . . 6  |-  .||  =  (
||r `  P )
15 eqid 2467 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
166, 14, 15dvdsr01 17174 . . . . 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 2540 . . . . . 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 1160 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  ->  X  e.  { ( 0g `  P ) } )
22 elsni 4058 . . . 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 4479 . 2  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =  {
( 0g `  P
) } )  -> 
( G `  I
)  .||  X )
25 simpl1 999 . . . . . . . 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 1000 . . . . . . . . 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 1001 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  X  e.  I )
3028, 29sseldd 3510 . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
33 eqid 2467 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
342, 10, 15, 7, 32, 33ig1pval3 22441 . . . . . . . . . 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 1228 . . . . . . . . 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 1009 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  (Monic1p `  R ) )
37 eqid 2467 . . . . . . . . 9  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
3837, 33mon1puc1p 22417 . . . . . . . 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 2467 . . . . . . . 8  |-  (rem1p `  R
)  =  (rem1p `  R
)
4140, 2, 6, 37, 32r1pdeglt 22425 . . . . . . 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 1228 . . . . . 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 1010 . . . . . 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 4477 . . . . 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 22347 . . . . . . 7  |-  ( deg1  `  R
) : ( Base `  P ) --> RR*
4635simp1d 1008 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  I )
4728, 46sseldd 3510 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( G `  I )  e.  ( Base `  P
) )
48 eqid 2467 . . . . . . . . . . 11  |-  (quot1p `  R
)  =  (quot1p `  R
)
49 eqid 2467 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
50 eqid 2467 . . . . . . . . . . 11  |-  ( -g `  P )  =  (
-g `  P )
5140, 2, 6, 48, 49, 50r1pval 22423 . . . . . . . . . 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 22422 . . . . . . . . . . . 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 1228 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (quot1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
567, 6, 49lidlmcl 17733 . . . . . . . . . . 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 1229 . . . . . . . . . 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 17732 . . . . . . . . . 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 1229 . . . . . . . . 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 2555 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  I )
6128, 60sseldd 3510 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  ( X (rem1p `  R ) ( G `  I ) )  e.  ( Base `  P ) )
62 ffvelrn 6030 . . . . . . 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 3645 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( ( Base `  P )  \  { ( 0g `  P ) } ) )
65 imass2 5378 . . . . . . . . . 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 22355 . . . . . . . . . . 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 11128 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
7068, 69syl6sseq 3555 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( ( Base `  P
)  \  { ( 0g `  P ) } ) )  C_  ( ZZ>=
`  0 ) )
7166, 70sstrd 3519 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  ( ZZ>= `  0
) )
72 uzssz 11113 . . . . . . . . 9  |-  ( ZZ>= ` 
0 )  C_  ZZ
73 zssre 10883 . . . . . . . . . 10  |-  ZZ  C_  RR
74 ressxr 9649 . . . . . . . . . 10  |-  RR  C_  RR*
7573, 74sstri 3518 . . . . . . . . 9  |-  ZZ  C_  RR*
7672, 75sstri 3518 . . . . . . . 8  |-  ( ZZ>= ` 
0 )  C_  RR*
7771, 76syl6ss 3521 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
( deg1  `
 R ) "
( I  \  {
( 0g `  P
) } ) ) 
C_  RR* )
787, 15lidl0cl 17728 . . . . . . . . . . . 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 4178 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  C_  I )
8131necomd 2738 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  { ( 0g `  P ) }  =/=  I )
82 pssdifn0 3893 . . . . . . . . . 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 5737 . . . . . . . . . . . 12  |-  ( ( deg1  `  R ) : (
Base `  P ) --> RR* 
->  ( deg1  `  R )  Fn  ( Base `  P
) )
8545, 84ax-mp 5 . . . . . . . . . . 11  |-  ( deg1  `  R
)  Fn  ( Base `  P )
8628ssdifssd 3647 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  (
I  \  { ( 0g `  P ) } )  C_  ( Base `  P ) )
87 fnimaeq0 5708 . . . . . . . . . . 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 2725 . . . . . . . . 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 11177 . . . . . . . 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 3510 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  X  e.  I )  /\  I  =/=  { ( 0g `  P ) } )  ->  sup ( ( ( deg1  `  R
) " ( I 
\  { ( 0g
`  P ) } ) ) ,  RR ,  `'  <  )  e. 
RR* )
94 xrltnle 9665 . . . . . 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 4158 . . . . . . . . 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 6149 . . . . . . . 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 1228 . . . . . . 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 11176 . . . . . . 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 2685 . . . 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 22428 . . . 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 1228 . . 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 2785 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478    C_ wss 3481   (/)c0 3790   {csn 4033   class class class wbr 4453   `'ccnv 5004   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   supcsup 7912   RRcr 9503   0cc0 9504   RR*cxr 9639    < clt 9640    <_ cle 9641   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   Basecbs 14506   .rcmulr 14572   0gc0g 14711   -gcsg 15926   Ringcrg 17068   ||rcdsr 17157   DivRingcdr 17265  LIdealclidl 17685  Poly1cpl1 18084   deg1 cdg1 22318  Monic1pcmn1 22392  Unic1pcuc1p 22393  quot1pcq1p 22394  rem1pcr1p 22395  idlGen1pcig1p 22396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-0g 14713  df-gsum 14714  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-mhm 15838  df-submnd 15839  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mulg 15931  df-subg 16069  df-ghm 16136  df-cntz 16226  df-cmn 16671  df-abl 16672  df-mgp 17012  df-ur 17024  df-ring 17070  df-cring 17071  df-oppr 17142  df-dvdsr 17160  df-unit 17161  df-invr 17191  df-drng 17267  df-subrg 17296  df-lmod 17383  df-lss 17448  df-sra 17687  df-rgmod 17688  df-lidl 17689  df-rlreg 17799  df-ascl 17831  df-psr 17873  df-mvr 17874  df-mpl 17875  df-opsr 17877  df-psr1 18087  df-vr1 18088  df-ply1 18089  df-coe1 18090  df-cnfld 18289  df-mdeg 22319  df-deg1 22320  df-mon1 22397  df-uc1p 22398  df-q1p 22399  df-r1p 22400  df-ig1p 22401
This theorem is referenced by:  ig1prsp  22444
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