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Theorem ply1remlem 23113
Description: A term of the form  x  -  N is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
ply1rem.p  |-  P  =  (Poly1 `  R )
ply1rem.b  |-  B  =  ( Base `  P
)
ply1rem.k  |-  K  =  ( Base `  R
)
ply1rem.x  |-  X  =  (var1 `  R )
ply1rem.m  |-  .-  =  ( -g `  P )
ply1rem.a  |-  A  =  (algSc `  P )
ply1rem.g  |-  G  =  ( X  .-  ( A `  N )
)
ply1rem.o  |-  O  =  (eval1 `  R )
ply1rem.1  |-  ( ph  ->  R  e. NzRing )
ply1rem.2  |-  ( ph  ->  R  e.  CRing )
ply1rem.3  |-  ( ph  ->  N  e.  K )
ply1rem.u  |-  U  =  (Monic1p `  R )
ply1rem.d  |-  D  =  ( deg1  `  R )
ply1rem.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
ply1remlem  |-  ( ph  ->  ( G  e.  U  /\  ( D `  G
)  =  1  /\  ( `' ( O `
 G ) " {  .0.  } )  =  { N } ) )

Proof of Theorem ply1remlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ply1rem.g . . . 4  |-  G  =  ( X  .-  ( A `  N )
)
2 ply1rem.1 . . . . . . . 8  |-  ( ph  ->  R  e. NzRing )
3 nzrring 18485 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
42, 3syl 17 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
5 ply1rem.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
65ply1ring 18841 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
74, 6syl 17 . . . . . 6  |-  ( ph  ->  P  e.  Ring )
8 ringgrp 17785 . . . . . 6  |-  ( P  e.  Ring  ->  P  e. 
Grp )
97, 8syl 17 . . . . 5  |-  ( ph  ->  P  e.  Grp )
10 ply1rem.x . . . . . . 7  |-  X  =  (var1 `  R )
11 ply1rem.b . . . . . . 7  |-  B  =  ( Base `  P
)
1210, 5, 11vr1cl 18810 . . . . . 6  |-  ( R  e.  Ring  ->  X  e.  B )
134, 12syl 17 . . . . 5  |-  ( ph  ->  X  e.  B )
14 ply1rem.a . . . . . . . 8  |-  A  =  (algSc `  P )
15 ply1rem.k . . . . . . . 8  |-  K  =  ( Base `  R
)
165, 14, 15, 11ply1sclf 18878 . . . . . . 7  |-  ( R  e.  Ring  ->  A : K
--> B )
174, 16syl 17 . . . . . 6  |-  ( ph  ->  A : K --> B )
18 ply1rem.3 . . . . . 6  |-  ( ph  ->  N  e.  K )
1917, 18ffvelrnd 6023 . . . . 5  |-  ( ph  ->  ( A `  N
)  e.  B )
20 ply1rem.m . . . . . 6  |-  .-  =  ( -g `  P )
2111, 20grpsubcl 16734 . . . . 5  |-  ( ( P  e.  Grp  /\  X  e.  B  /\  ( A `  N )  e.  B )  -> 
( X  .-  ( A `  N )
)  e.  B )
229, 13, 19, 21syl3anc 1268 . . . 4  |-  ( ph  ->  ( X  .-  ( A `  N )
)  e.  B )
231, 22syl5eqel 2533 . . 3  |-  ( ph  ->  G  e.  B )
241fveq2i 5868 . . . . . . 7  |-  ( D `
 G )  =  ( D `  ( X  .-  ( A `  N ) ) )
25 ply1rem.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
2625, 5, 11deg1xrcl 23031 . . . . . . . . . . 11  |-  ( ( A `  N )  e.  B  ->  ( D `  ( A `  N ) )  e. 
RR* )
2719, 26syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  e.  RR* )
28 0xr 9687 . . . . . . . . . . 11  |-  0  e.  RR*
2928a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR* )
30 1re 9642 . . . . . . . . . . 11  |-  1  e.  RR
31 rexr 9686 . . . . . . . . . . 11  |-  ( 1  e.  RR  ->  1  e.  RR* )
3230, 31mp1i 13 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR* )
3325, 5, 15, 14deg1sclle 23061 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  ( D `  ( A `  N ) )  <_ 
0 )
344, 18, 33syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  <_  0 )
35 0lt1 10136 . . . . . . . . . . 11  |-  0  <  1
3635a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  <  1 )
3727, 29, 32, 34, 36xrlelttrd 11457 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( A `  N )
)  <  1 )
38 eqid 2451 . . . . . . . . . . . . . 14  |-  (mulGrp `  P )  =  (mulGrp `  P )
3938, 11mgpbas 17729 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  P ) )
40 eqid 2451 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4139, 40mulg1 16765 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4213, 41syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4342fveq2d 5869 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( D `
 X ) )
44 1nn0 10885 . . . . . . . . . . 11  |-  1  e.  NN0
4525, 5, 10, 38, 40deg1pw 23069 . . . . . . . . . . 11  |-  ( ( R  e. NzRing  /\  1  e.  NN0 )  ->  ( D `  ( 1
(.g `  (mulGrp `  P
) ) X ) )  =  1 )
462, 44, 45sylancl 668 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  1 )
4743, 46eqtr3d 2487 . . . . . . . . 9  |-  ( ph  ->  ( D `  X
)  =  1 )
4837, 47breqtrrd 4429 . . . . . . . 8  |-  ( ph  ->  ( D `  ( A `  N )
)  <  ( D `  X ) )
495, 25, 4, 11, 20, 13, 19, 48deg1sub 23057 . . . . . . 7  |-  ( ph  ->  ( D `  ( X  .-  ( A `  N ) ) )  =  ( D `  X ) )
5024, 49syl5eq 2497 . . . . . 6  |-  ( ph  ->  ( D `  G
)  =  ( D `
 X ) )
5150, 47eqtrd 2485 . . . . 5  |-  ( ph  ->  ( D `  G
)  =  1 )
5251, 44syl6eqel 2537 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
53 eqid 2451 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
5425, 5, 53, 11deg1nn0clb 23039 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
554, 23, 54syl2anc 667 . . . 4  |-  ( ph  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
5652, 55mpbird 236 . . 3  |-  ( ph  ->  G  =/=  ( 0g
`  P ) )
5751fveq2d 5869 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( (coe1 `  G ) `  1
) )
581fveq2i 5868 . . . . . 6  |-  (coe1 `  G
)  =  (coe1 `  ( X  .-  ( A `  N ) ) )
5958fveq1i 5866 . . . . 5  |-  ( (coe1 `  G ) `  1
)  =  ( (coe1 `  ( X  .-  ( A `  N )
) ) `  1
)
6044a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  NN0 )
61 eqid 2451 . . . . . . 7  |-  ( -g `  R )  =  (
-g `  R )
625, 11, 20, 61coe1subfv 18859 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  X  e.  B  /\  ( A `  N )  e.  B )  /\  1  e.  NN0 )  -> 
( (coe1 `  ( X  .-  ( A `  N ) ) ) `  1
)  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
634, 13, 19, 60, 62syl31anc 1271 . . . . 5  |-  ( ph  ->  ( (coe1 `  ( X  .-  ( A `  N ) ) ) `  1
)  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6459, 63syl5eq 2497 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) ` 
1 )  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6542oveq2d 6306 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  ( ( 1r `  R ) ( .s `  P
) X ) )
665ply1sca 18846 . . . . . . . . . . . . 13  |-  ( R  e. NzRing  ->  R  =  (Scalar `  P ) )
672, 66syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  P ) )
6867fveq2d 5869 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  P )
) )
6968oveq1d 6305 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) X )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) X ) )
705ply1lmod 18845 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  P  e. 
LMod )
714, 70syl 17 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  LMod )
72 eqid 2451 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
73 eqid 2451 . . . . . . . . . . . 12  |-  ( .s
`  P )  =  ( .s `  P
)
74 eqid 2451 . . . . . . . . . . . 12  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
7511, 72, 73, 74lmodvs1 18119 . . . . . . . . . . 11  |-  ( ( P  e.  LMod  /\  X  e.  B )  ->  (
( 1r `  (Scalar `  P ) ) ( .s `  P ) X )  =  X )
7671, 13, 75syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  (Scalar `  P ) ) ( .s `  P
) X )  =  X )
7765, 69, 763eqtrd 2489 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  X )
7877fveq2d 5869 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) )  =  (coe1 `  X ) )
7978fveq1d 5867 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  X ) ` 
1 ) )
80 eqid 2451 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
8115, 80ringidcl 17801 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  K )
824, 81syl 17 . . . . . . . 8  |-  ( ph  ->  ( 1r `  R
)  e.  K )
83 ply1rem.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
8483, 15, 5, 10, 73, 38, 40coe1tmfv1 18867 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  K  /\  1  e. 
NN0 )  ->  (
(coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
854, 82, 60, 84syl3anc 1268 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
8679, 85eqtr3d 2487 . . . . . 6  |-  ( ph  ->  ( (coe1 `  X ) ` 
1 )  =  ( 1r `  R ) )
87 eqid 2451 . . . . . . . . . 10  |-  ( 0g
`  R )  =  ( 0g `  R
)
885, 14, 15, 87coe1scl 18880 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) )
894, 18, 88syl2anc 667 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) )
9089fveq1d 5867 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) `  1
) )
91 ax-1ne0 9608 . . . . . . . . . . 11  |-  1  =/=  0
92 neeq1 2686 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  =/=  0  <->  1  =/=  0 ) )
9391, 92mpbiri 237 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =/=  0 )
94 ifnefalse 3893 . . . . . . . . . 10  |-  ( x  =/=  0  ->  if ( x  =  0 ,  N ,  ( 0g
`  R ) )  =  ( 0g `  R ) )
9593, 94syl 17 . . . . . . . . 9  |-  ( x  =  1  ->  if ( x  =  0 ,  N ,  ( 0g
`  R ) )  =  ( 0g `  R ) )
96 eqid 2451 . . . . . . . . 9  |-  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )
97 fvex 5875 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
9895, 96, 97fvmpt 5948 . . . . . . . 8  |-  ( 1  e.  NN0  ->  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) `  1 )  =  ( 0g `  R ) )
9944, 98ax-mp 5 . . . . . . 7  |-  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) `  1 )  =  ( 0g `  R )
10090, 99syl6eq 2501 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( 0g `  R ) )
10186, 100oveq12d 6308 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) ) )
102 ringgrp 17785 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1034, 102syl 17 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
10415, 87, 61grpsubid1 16739 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  K )  -> 
( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
105103, 82, 104syl2anc 667 . . . . 5  |-  ( ph  ->  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
106101, 105eqtrd 2485 . . . 4  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( 1r `  R
) )
10757, 64, 1063eqtrd 2489 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( 1r
`  R ) )
108 ply1rem.u . . . 4  |-  U  =  (Monic1p `  R )
1095, 11, 53, 25, 108, 80ismon1p 23093 . . 3  |-  ( G  e.  U  <->  ( G  e.  B  /\  G  =/=  ( 0g `  P
)  /\  ( (coe1 `  G ) `  ( D `  G )
)  =  ( 1r
`  R ) ) )
11023, 56, 107, 109syl3anbrc 1192 . 2  |-  ( ph  ->  G  e.  U )
1111fveq2i 5868 . . . . . . . . . 10  |-  ( O `
 G )  =  ( O `  ( X  .-  ( A `  N ) ) )
112111fveq1i 5866 . . . . . . . . 9  |-  ( ( O `  G ) `
 x )  =  ( ( O `  ( X  .-  ( A `
 N ) ) ) `  x )
113 ply1rem.o . . . . . . . . . . 11  |-  O  =  (eval1 `  R )
114 ply1rem.2 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  CRing )
115114adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  CRing )
116 simpr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
117113, 10, 15, 5, 11, 115, 116evl1vard 18925 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( X  e.  B  /\  ( ( O `  X ) `  x
)  =  x ) )
11818adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  K )  ->  N  e.  K )
119113, 5, 15, 14, 11, 115, 118, 116evl1scad 18923 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  (
( A `  N
)  e.  B  /\  ( ( O `  ( A `  N ) ) `  x )  =  N ) )
120113, 5, 15, 11, 115, 116, 117, 119, 20, 61evl1subd 18930 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( X  .-  ( A `  N )
)  e.  B  /\  ( ( O `  ( X  .-  ( A `
 N ) ) ) `  x )  =  ( x (
-g `  R ) N ) ) )
121120simprd 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( X  .-  ( A `  N ) ) ) `
 x )  =  ( x ( -g `  R ) N ) )
122112, 121syl5eq 2497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  =  ( x (
-g `  R ) N ) )
123122eqeq1d 2453 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  ( x
( -g `  R ) N )  =  .0.  ) )
124103adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  Grp )
12515, 83, 61grpsubeq0 16740 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  K  /\  N  e.  K )  ->  ( ( x (
-g `  R ) N )  =  .0.  <->  x  =  N ) )
126124, 116, 118, 125syl3anc 1268 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( x ( -g `  R ) N )  =  .0.  <->  x  =  N ) )
127123, 126bitrd 257 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  =  N ) )
128 elsn 3982 . . . . . 6  |-  ( x  e.  { N }  <->  x  =  N )
129127, 128syl6bbr 267 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  e.  { N } ) )
130129pm5.32da 647 . . . 4  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  )  <->  ( x  e.  K  /\  x  e.  { N } ) ) )
131 eqid 2451 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
132 eqid 2451 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
133 fvex 5875 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
13415, 133eqeltri 2525 . . . . . . . 8  |-  K  e. 
_V
135134a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
136113, 5, 131, 15evl1rhm 18920 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
137114, 136syl 17 . . . . . . . . 9  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
13811, 132rhmf 17954 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
139137, 138syl 17 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
140139, 23ffvelrnd 6023 . . . . . . 7  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
141131, 15, 132, 2, 135, 140pwselbas 15387 . . . . . 6  |-  ( ph  ->  ( O `  G
) : K --> K )
142 ffn 5728 . . . . . 6  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
143141, 142syl 17 . . . . 5  |-  ( ph  ->  ( O `  G
)  Fn  K )
144 fniniseg 6003 . . . . 5  |-  ( ( O `  G )  Fn  K  ->  (
x  e.  ( `' ( O `  G
) " {  .0.  } )  <->  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  ) ) )
145143, 144syl 17 . . . 4  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  .0.  ) ) )
14618snssd 4117 . . . . . 6  |-  ( ph  ->  { N }  C_  K )
147146sseld 3431 . . . . 5  |-  ( ph  ->  ( x  e.  { N }  ->  x  e.  K ) )
148147pm4.71rd 641 . . . 4  |-  ( ph  ->  ( x  e.  { N }  <->  ( x  e.  K  /\  x  e. 
{ N } ) ) )
149130, 145, 1483bitr4d 289 . . 3  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  x  e.  { N } ) )
150149eqrdv 2449 . 2  |-  ( ph  ->  ( `' ( O `
 G ) " {  .0.  } )  =  { N } )
151110, 51, 1503jca 1188 1  |-  ( ph  ->  ( G  e.  U  /\  ( D `  G
)  =  1  /\  ( `' ( O `
 G ) " {  .0.  } )  =  { N } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045   ifcif 3881   {csn 3968   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   RR*cxr 9674    < clt 9675    <_ cle 9676   NN0cn0 10869   Basecbs 15121  Scalarcsca 15193   .scvsca 15194   0gc0g 15338    ^s cpws 15345   Grpcgrp 16669   -gcsg 16671  .gcmg 16672  mulGrpcmgp 17723   1rcur 17735   Ringcrg 17780   CRingccrg 17781   RingHom crh 17940   LModclmod 18091  NzRingcnzr 18481  algSccascl 18535  var1cv1 18769  Poly1cpl1 18770  coe1cco1 18771  eval1ce1 18903   deg1 cdg1 23003  Monic1pcmn1 23074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-0g 15340  df-gsum 15341  df-prds 15346  df-pws 15348  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-srg 17740  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-rnghom 17943  df-subrg 18006  df-lmod 18093  df-lss 18156  df-lsp 18195  df-nzr 18482  df-rlreg 18507  df-assa 18536  df-asp 18537  df-ascl 18538  df-psr 18580  df-mvr 18581  df-mpl 18582  df-opsr 18584  df-evls 18729  df-evl 18730  df-psr1 18773  df-vr1 18774  df-ply1 18775  df-coe1 18776  df-evl1 18905  df-cnfld 18971  df-mdeg 23004  df-deg1 23005  df-mon1 23080
This theorem is referenced by:  ply1rem  23114  facth1  23115  fta1glem1  23116  fta1glem2  23117
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