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Theorem ply1remlem 20090
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 nzrrng 16337 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
5 ply1rem.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
65ply1rng 16647 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
74, 6syl 16 . . . . . 6  |-  ( ph  ->  P  e.  Ring )
8 rnggrp 15674 . . . . . 6  |-  ( P  e.  Ring  ->  P  e. 
Grp )
97, 8syl 16 . . . . 5  |-  ( ph  ->  P  e.  Grp )
10 ply1rem.x . . . . . . 7  |-  X  =  (var1 `  R )
11 ply1rem.b . . . . . . 7  |-  B  =  ( Base `  P
)
1210, 5, 11vr1cl 16616 . . . . . 6  |-  ( R  e.  Ring  ->  X  e.  B )
134, 12syl 16 . . . . 5  |-  ( ph  ->  X  e.  B )
14 ply1rem.a . . . . . . . 8  |-  A  =  (algSc `  P )
15 ply1rem.k . . . . . . . 8  |-  K  =  ( Base `  R
)
165, 14, 15, 11ply1sclf 16682 . . . . . . 7  |-  ( R  e.  Ring  ->  A : K
--> B )
174, 16syl 16 . . . . . 6  |-  ( ph  ->  A : K --> B )
18 ply1rem.3 . . . . . 6  |-  ( ph  ->  N  e.  K )
1917, 18ffvelrnd 5874 . . . . 5  |-  ( ph  ->  ( A `  N
)  e.  B )
20 ply1rem.m . . . . . 6  |-  .-  =  ( -g `  P )
2111, 20grpsubcl 14874 . . . . 5  |-  ( ( P  e.  Grp  /\  X  e.  B  /\  ( A `  N )  e.  B )  -> 
( X  .-  ( A `  N )
)  e.  B )
229, 13, 19, 21syl3anc 1185 . . . 4  |-  ( ph  ->  ( X  .-  ( A `  N )
)  e.  B )
231, 22syl5eqel 2522 . . 3  |-  ( ph  ->  G  e.  B )
241fveq2i 5734 . . . . . . 7  |-  ( D `
 G )  =  ( D `  ( X  .-  ( A `  N ) ) )
25 ply1rem.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
2625, 5, 11deg1xrcl 20010 . . . . . . . . . . 11  |-  ( ( A `  N )  e.  B  ->  ( D `  ( A `  N ) )  e. 
RR* )
2719, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  e.  RR* )
28 0xr 9136 . . . . . . . . . . 11  |-  0  e.  RR*
2928a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR* )
30 1re 9095 . . . . . . . . . . 11  |-  1  e.  RR
31 rexr 9135 . . . . . . . . . . 11  |-  ( 1  e.  RR  ->  1  e.  RR* )
3230, 31mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR* )
3325, 5, 15, 14deg1sclle 20040 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  ( D `  ( A `  N ) )  <_ 
0 )
344, 18, 33syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  <_  0 )
35 0lt1 9555 . . . . . . . . . . 11  |-  0  <  1
3635a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  <  1 )
3727, 29, 32, 34, 36xrlelttrd 10755 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( A `  N )
)  <  1 )
38 eqid 2438 . . . . . . . . . . . . . 14  |-  (mulGrp `  P )  =  (mulGrp `  P )
3938, 11mgpbas 15659 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  P ) )
40 eqid 2438 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4139, 40mulg1 14902 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4213, 41syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4342fveq2d 5735 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( D `
 X ) )
44 1nn0 10242 . . . . . . . . . . 11  |-  1  e.  NN0
4525, 5, 10, 38, 40deg1pw 20048 . . . . . . . . . . 11  |-  ( ( R  e. NzRing  /\  1  e.  NN0 )  ->  ( D `  ( 1
(.g `  (mulGrp `  P
) ) X ) )  =  1 )
462, 44, 45sylancl 645 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  1 )
4743, 46eqtr3d 2472 . . . . . . . . 9  |-  ( ph  ->  ( D `  X
)  =  1 )
4837, 47breqtrrd 4241 . . . . . . . 8  |-  ( ph  ->  ( D `  ( A `  N )
)  <  ( D `  X ) )
495, 25, 4, 11, 20, 13, 19, 48deg1sub 20036 . . . . . . 7  |-  ( ph  ->  ( D `  ( X  .-  ( A `  N ) ) )  =  ( D `  X ) )
5024, 49syl5eq 2482 . . . . . 6  |-  ( ph  ->  ( D `  G
)  =  ( D `
 X ) )
5150, 47eqtrd 2470 . . . . 5  |-  ( ph  ->  ( D `  G
)  =  1 )
5251, 44syl6eqel 2526 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
53 eqid 2438 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
5425, 5, 53, 11deg1nn0clb 20018 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
554, 23, 54syl2anc 644 . . . 4  |-  ( ph  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
5652, 55mpbird 225 . . 3  |-  ( ph  ->  G  =/=  ( 0g
`  P ) )
5751fveq2d 5735 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( (coe1 `  G ) `  1
) )
581fveq2i 5734 . . . . . 6  |-  (coe1 `  G
)  =  (coe1 `  ( X  .-  ( A `  N ) ) )
5958fveq1i 5732 . . . . 5  |-  ( (coe1 `  G ) `  1
)  =  ( (coe1 `  ( X  .-  ( A `  N )
) ) `  1
)
6044a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  NN0 )
61 eqid 2438 . . . . . . 7  |-  ( -g `  R )  =  (
-g `  R )
625, 11, 20, 61coe1subfv 16664 . . . . . 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 1188 . . . . 5  |-  ( ph  ->  ( (coe1 `  ( X  .-  ( A `  N ) ) ) `  1
)  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6459, 63syl5eq 2482 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) ` 
1 )  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6542oveq2d 6100 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  ( ( 1r `  R ) ( .s `  P
) X ) )
665ply1sca 16652 . . . . . . . . . . . . 13  |-  ( R  e. NzRing  ->  R  =  (Scalar `  P ) )
672, 66syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  P ) )
6867fveq2d 5735 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  P )
) )
6968oveq1d 6099 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) X )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) X ) )
705ply1lmod 16651 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  P  e. 
LMod )
714, 70syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  LMod )
72 eqid 2438 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
73 eqid 2438 . . . . . . . . . . . 12  |-  ( .s
`  P )  =  ( .s `  P
)
74 eqid 2438 . . . . . . . . . . . 12  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
7511, 72, 73, 74lmodvs1 15983 . . . . . . . . . . 11  |-  ( ( P  e.  LMod  /\  X  e.  B )  ->  (
( 1r `  (Scalar `  P ) ) ( .s `  P ) X )  =  X )
7671, 13, 75syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  (Scalar `  P ) ) ( .s `  P
) X )  =  X )
7765, 69, 763eqtrd 2474 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  X )
7877fveq2d 5735 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) )  =  (coe1 `  X ) )
7978fveq1d 5733 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  X ) ` 
1 ) )
80 eqid 2438 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
8115, 80rngidcl 15689 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  K )
824, 81syl 16 . . . . . . . 8  |-  ( ph  ->  ( 1r `  R
)  e.  K )
83 ply1rem.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
8483, 15, 5, 10, 73, 38, 40coe1tmfv1 16671 . . . . . . . 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 1185 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
8679, 85eqtr3d 2472 . . . . . 6  |-  ( ph  ->  ( (coe1 `  X ) ` 
1 )  =  ( 1r `  R ) )
87 eqid 2438 . . . . . . . . . 10  |-  ( 0g
`  R )  =  ( 0g `  R
)
885, 14, 15, 87coe1scl 16683 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) )
894, 18, 88syl2anc 644 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) )
9089fveq1d 5733 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) `  1
) )
91 ax-1ne0 9064 . . . . . . . . . . 11  |-  1  =/=  0
92 neeq1 2611 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  =/=  0  <->  1  =/=  0 ) )
9391, 92mpbiri 226 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =/=  0 )
94 ifnefalse 3749 . . . . . . . . . 10  |-  ( x  =/=  0  ->  if ( x  =  0 ,  N ,  ( 0g
`  R ) )  =  ( 0g `  R ) )
9593, 94syl 16 . . . . . . . . 9  |-  ( x  =  1  ->  if ( x  =  0 ,  N ,  ( 0g
`  R ) )  =  ( 0g `  R ) )
96 eqid 2438 . . . . . . . . 9  |-  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )
97 fvex 5745 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
9895, 96, 97fvmpt 5809 . . . . . . . 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 2486 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( 0g `  R ) )
10186, 100oveq12d 6102 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) ) )
102 rnggrp 15674 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1034, 102syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
10415, 87, 61grpsubid1 14879 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  K )  -> 
( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
105103, 82, 104syl2anc 644 . . . . 5  |-  ( ph  ->  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
106101, 105eqtrd 2470 . . . 4  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( 1r `  R
) )
10757, 64, 1063eqtrd 2474 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( 1r
`  R ) )
108 ply1rem.u . . . 4  |-  U  =  (Monic1p `  R )
1095, 11, 53, 25, 108, 80ismon1p 20070 . . 3  |-  ( G  e.  U  <->  ( G  e.  B  /\  G  =/=  ( 0g `  P
)  /\  ( (coe1 `  G ) `  ( D `  G )
)  =  ( 1r
`  R ) ) )
11023, 56, 107, 109syl3anbrc 1139 . 2  |-  ( ph  ->  G  e.  U )
1111fveq2i 5734 . . . . . . . . . 10  |-  ( O `
 G )  =  ( O `  ( X  .-  ( A `  N ) ) )
112111fveq1i 5732 . . . . . . . . 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 453 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  CRing )
116 simpr 449 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
117113, 10, 15, 5, 11, 115, 116evl1vard 19958 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( X  e.  B  /\  ( ( O `  X ) `  x
)  =  x ) )
11818adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  K )  ->  N  e.  K )
119113, 5, 15, 14, 11, 115, 118, 116evl1scad 19956 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  (
( A `  N
)  e.  B  /\  ( ( O `  ( A `  N ) ) `  x )  =  N ) )
120113, 5, 15, 11, 115, 116, 117, 119, 20, 61evl1subd 19960 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( X  .-  ( A `  N )
)  e.  B  /\  ( ( O `  ( X  .-  ( A `
 N ) ) ) `  x )  =  ( x (
-g `  R ) N ) ) )
121120simprd 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( X  .-  ( A `  N ) ) ) `
 x )  =  ( x ( -g `  R ) N ) )
122112, 121syl5eq 2482 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  =  ( x (
-g `  R ) N ) )
123122eqeq1d 2446 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  ( x
( -g `  R ) N )  =  .0.  ) )
124103adantr 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  Grp )
12515, 83, 61grpsubeq0 14880 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  K  /\  N  e.  K )  ->  ( ( x (
-g `  R ) N )  =  .0.  <->  x  =  N ) )
126124, 116, 118, 125syl3anc 1185 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( x ( -g `  R ) N )  =  .0.  <->  x  =  N ) )
127123, 126bitrd 246 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  =  N ) )
128 elsn 3831 . . . . . 6  |-  ( x  e.  { N }  <->  x  =  N )
129127, 128syl6bbr 256 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  e.  { N } ) )
130129pm5.32da 624 . . . 4  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  )  <->  ( x  e.  K  /\  x  e.  { N } ) ) )
131 eqid 2438 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
132 eqid 2438 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
133 fvex 5745 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
13415, 133eqeltri 2508 . . . . . . . 8  |-  K  e. 
_V
135134a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
136113, 5, 131, 15evl1rhm 19954 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
137114, 136syl 16 . . . . . . . . 9  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
13811, 132rhmf 15832 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
139137, 138syl 16 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
140139, 23ffvelrnd 5874 . . . . . . 7  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
141131, 15, 132, 2, 135, 140pwselbas 13716 . . . . . 6  |-  ( ph  ->  ( O `  G
) : K --> K )
142 ffn 5594 . . . . . 6  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
143141, 142syl 16 . . . . 5  |-  ( ph  ->  ( O `  G
)  Fn  K )
144 fniniseg 5854 . . . . 5  |-  ( ( O `  G )  Fn  K  ->  (
x  e.  ( `' ( O `  G
) " {  .0.  } )  <->  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  ) ) )
145143, 144syl 16 . . . 4  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  .0.  ) ) )
14618snssd 3945 . . . . . 6  |-  ( ph  ->  { N }  C_  K )
147146sseld 3349 . . . . 5  |-  ( ph  ->  ( x  e.  { N }  ->  x  e.  K ) )
148147pm4.71rd 618 . . . 4  |-  ( ph  ->  ( x  e.  { N }  <->  ( x  e.  K  /\  x  e. 
{ N } ) ) )
149130, 145, 1483bitr4d 278 . . 3  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  x  e.  { N } ) )
150149eqrdv 2436 . 2  |-  ( ph  ->  ( `' ( O `
 G ) " {  .0.  } )  =  { N } )
151110, 51, 1503jca 1135 1  |-  ( ph  ->  ( G  e.  U  /\  ( D `  G
)  =  1  /\  ( `' ( O `
 G ) " {  .0.  } )  =  { N } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   ifcif 3741   {csn 3816   class class class wbr 4215    e. cmpt 4269   `'ccnv 4880   "cima 4884    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995   1c1 8996   RR*cxr 9124    < clt 9125    <_ cle 9126   NN0cn0 10226   Basecbs 13474  Scalarcsca 13537   .scvsca 13538    ^s cpws 13675   0gc0g 13728   Grpcgrp 14690   -gcsg 14693  .gcmg 14694  mulGrpcmgp 15653   Ringcrg 15665   CRingccrg 15666   1rcur 15667   RingHom crh 15822   LModclmod 15955  NzRingcnzr 16333  algSccascl 16376  var1cv1 16575  Poly1cpl1 16576  eval1ce1 16578  coe1cco1 16579   deg1 cdg1 19982  Monic1pcmn1 20053
This theorem is referenced by:  ply1rem  20091  facth1  20092  fta1glem1  20093  fta1glem2  20094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-ofr 6309  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-prds 13676  df-pws 13678  df-0g 13732  df-gsum 13733  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-ghm 15009  df-cntz 15121  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-rnghom 15824  df-subrg 15871  df-lmod 15957  df-lss 16014  df-lsp 16053  df-nzr 16334  df-rlreg 16348  df-assa 16377  df-asp 16378  df-ascl 16379  df-psr 16422  df-mvr 16423  df-mpl 16424  df-evls 16425  df-evl 16426  df-opsr 16430  df-psr1 16581  df-vr1 16582  df-ply1 16583  df-evl1 16585  df-coe1 16586  df-cnfld 16709  df-mdeg 19983  df-deg1 19984  df-mon1 20058
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