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Theorem ply1remlem 20077
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 16324 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
5 ply1rem.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
65ply1rng 16634 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
74, 6syl 16 . . . . . 6  |-  ( ph  ->  P  e.  Ring )
8 rnggrp 15661 . . . . . 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 16603 . . . . . 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 16669 . . . . . . 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 5863 . . . . 5  |-  ( ph  ->  ( A `  N
)  e.  B )
20 ply1rem.m . . . . . 6  |-  .-  =  ( -g `  P )
2111, 20grpsubcl 14861 . . . . 5  |-  ( ( P  e.  Grp  /\  X  e.  B  /\  ( A `  N )  e.  B )  -> 
( X  .-  ( A `  N )
)  e.  B )
229, 13, 19, 21syl3anc 1184 . . . 4  |-  ( ph  ->  ( X  .-  ( A `  N )
)  e.  B )
231, 22syl5eqel 2519 . . 3  |-  ( ph  ->  G  e.  B )
241fveq2i 5723 . . . . . . 7  |-  ( D `
 G )  =  ( D `  ( X  .-  ( A `  N ) ) )
25 ply1rem.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
2625, 5, 11deg1xrcl 19997 . . . . . . . . . . 11  |-  ( ( A `  N )  e.  B  ->  ( D `  ( A `  N ) )  e. 
RR* )
2719, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  e.  RR* )
28 0xr 9123 . . . . . . . . . . 11  |-  0  e.  RR*
2928a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR* )
30 1re 9082 . . . . . . . . . . 11  |-  1  e.  RR
31 rexr 9122 . . . . . . . . . . 11  |-  ( 1  e.  RR  ->  1  e.  RR* )
3230, 31mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR* )
3325, 5, 15, 14deg1sclle 20027 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  ( D `  ( A `  N ) )  <_ 
0 )
344, 18, 33syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  <_  0 )
35 0lt1 9542 . . . . . . . . . . 11  |-  0  <  1
3635a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  <  1 )
3727, 29, 32, 34, 36xrlelttrd 10742 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( A `  N )
)  <  1 )
38 eqid 2435 . . . . . . . . . . . . . 14  |-  (mulGrp `  P )  =  (mulGrp `  P )
3938, 11mgpbas 15646 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  P ) )
40 eqid 2435 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4139, 40mulg1 14889 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4213, 41syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4342fveq2d 5724 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( D `
 X ) )
44 1nn0 10229 . . . . . . . . . . 11  |-  1  e.  NN0
4525, 5, 10, 38, 40deg1pw 20035 . . . . . . . . . . 11  |-  ( ( R  e. NzRing  /\  1  e.  NN0 )  ->  ( D `  ( 1
(.g `  (mulGrp `  P
) ) X ) )  =  1 )
462, 44, 45sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  1 )
4743, 46eqtr3d 2469 . . . . . . . . 9  |-  ( ph  ->  ( D `  X
)  =  1 )
4837, 47breqtrrd 4230 . . . . . . . 8  |-  ( ph  ->  ( D `  ( A `  N )
)  <  ( D `  X ) )
495, 25, 4, 11, 20, 13, 19, 48deg1sub 20023 . . . . . . 7  |-  ( ph  ->  ( D `  ( X  .-  ( A `  N ) ) )  =  ( D `  X ) )
5024, 49syl5eq 2479 . . . . . 6  |-  ( ph  ->  ( D `  G
)  =  ( D `
 X ) )
5150, 47eqtrd 2467 . . . . 5  |-  ( ph  ->  ( D `  G
)  =  1 )
5251, 44syl6eqel 2523 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
53 eqid 2435 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
5425, 5, 53, 11deg1nn0clb 20005 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
554, 23, 54syl2anc 643 . . . 4  |-  ( ph  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
5652, 55mpbird 224 . . 3  |-  ( ph  ->  G  =/=  ( 0g
`  P ) )
5751fveq2d 5724 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( (coe1 `  G ) `  1
) )
581fveq2i 5723 . . . . . 6  |-  (coe1 `  G
)  =  (coe1 `  ( X  .-  ( A `  N ) ) )
5958fveq1i 5721 . . . . 5  |-  ( (coe1 `  G ) `  1
)  =  ( (coe1 `  ( X  .-  ( A `  N )
) ) `  1
)
6044a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  NN0 )
61 eqid 2435 . . . . . . 7  |-  ( -g `  R )  =  (
-g `  R )
625, 11, 20, 61coe1subfv 16651 . . . . . 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 1187 . . . . 5  |-  ( ph  ->  ( (coe1 `  ( X  .-  ( A `  N ) ) ) `  1
)  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6459, 63syl5eq 2479 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) ` 
1 )  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6542oveq2d 6089 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  ( ( 1r `  R ) ( .s `  P
) X ) )
665ply1sca 16639 . . . . . . . . . . . . 13  |-  ( R  e. NzRing  ->  R  =  (Scalar `  P ) )
672, 66syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  P ) )
6867fveq2d 5724 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  P )
) )
6968oveq1d 6088 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) X )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) X ) )
705ply1lmod 16638 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  P  e. 
LMod )
714, 70syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  LMod )
72 eqid 2435 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
73 eqid 2435 . . . . . . . . . . . 12  |-  ( .s
`  P )  =  ( .s `  P
)
74 eqid 2435 . . . . . . . . . . . 12  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
7511, 72, 73, 74lmodvs1 15970 . . . . . . . . . . 11  |-  ( ( P  e.  LMod  /\  X  e.  B )  ->  (
( 1r `  (Scalar `  P ) ) ( .s `  P ) X )  =  X )
7671, 13, 75syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  (Scalar `  P ) ) ( .s `  P
) X )  =  X )
7765, 69, 763eqtrd 2471 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  X )
7877fveq2d 5724 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) )  =  (coe1 `  X ) )
7978fveq1d 5722 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  X ) ` 
1 ) )
80 eqid 2435 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
8115, 80rngidcl 15676 . . . . . . . . 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 16658 . . . . . . . 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 1184 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
8679, 85eqtr3d 2469 . . . . . 6  |-  ( ph  ->  ( (coe1 `  X ) ` 
1 )  =  ( 1r `  R ) )
87 eqid 2435 . . . . . . . . . 10  |-  ( 0g
`  R )  =  ( 0g `  R
)
885, 14, 15, 87coe1scl 16670 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) )
894, 18, 88syl2anc 643 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) )
9089fveq1d 5722 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) `  1
) )
91 ax-1ne0 9051 . . . . . . . . . . 11  |-  1  =/=  0
92 neeq1 2606 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  =/=  0  <->  1  =/=  0 ) )
9391, 92mpbiri 225 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =/=  0 )
94 ifnefalse 3739 . . . . . . . . . 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 2435 . . . . . . . . 9  |-  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )
97 fvex 5734 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
9895, 96, 97fvmpt 5798 . . . . . . . 8  |-  ( 1  e.  NN0  ->  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) `  1 )  =  ( 0g `  R ) )
9944, 98ax-mp 8 . . . . . . 7  |-  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) `  1 )  =  ( 0g `  R )
10090, 99syl6eq 2483 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( 0g `  R ) )
10186, 100oveq12d 6091 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) ) )
102 rnggrp 15661 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1034, 102syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
10415, 87, 61grpsubid1 14866 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  K )  -> 
( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
105103, 82, 104syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
106101, 105eqtrd 2467 . . . 4  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( 1r `  R
) )
10757, 64, 1063eqtrd 2471 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( 1r
`  R ) )
108 ply1rem.u . . . 4  |-  U  =  (Monic1p `  R )
1095, 11, 53, 25, 108, 80ismon1p 20057 . . 3  |-  ( G  e.  U  <->  ( G  e.  B  /\  G  =/=  ( 0g `  P
)  /\  ( (coe1 `  G ) `  ( D `  G )
)  =  ( 1r
`  R ) ) )
11023, 56, 107, 109syl3anbrc 1138 . 2  |-  ( ph  ->  G  e.  U )
1111fveq2i 5723 . . . . . . . . . 10  |-  ( O `
 G )  =  ( O `  ( X  .-  ( A `  N ) ) )
112111fveq1i 5721 . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  CRing )
116 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
117113, 10, 15, 5, 11, 115, 116evl1vard 19945 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( X  e.  B  /\  ( ( O `  X ) `  x
)  =  x ) )
11818adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  K )  ->  N  e.  K )
119113, 5, 15, 14, 11, 115, 118, 116evl1scad 19943 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  (
( A `  N
)  e.  B  /\  ( ( O `  ( A `  N ) ) `  x )  =  N ) )
120113, 5, 15, 11, 115, 116, 117, 119, 20, 61evl1subd 19947 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( X  .-  ( A `  N )
)  e.  B  /\  ( ( O `  ( X  .-  ( A `
 N ) ) ) `  x )  =  ( x (
-g `  R ) N ) ) )
121120simprd 450 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( X  .-  ( A `  N ) ) ) `
 x )  =  ( x ( -g `  R ) N ) )
122112, 121syl5eq 2479 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  =  ( x (
-g `  R ) N ) )
123122eqeq1d 2443 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  ( x
( -g `  R ) N )  =  .0.  ) )
124103adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  Grp )
12515, 83, 61grpsubeq0 14867 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  K  /\  N  e.  K )  ->  ( ( x (
-g `  R ) N )  =  .0.  <->  x  =  N ) )
126124, 116, 118, 125syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( x ( -g `  R ) N )  =  .0.  <->  x  =  N ) )
127123, 126bitrd 245 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  =  N ) )
128 elsn 3821 . . . . . 6  |-  ( x  e.  { N }  <->  x  =  N )
129127, 128syl6bbr 255 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  e.  { N } ) )
130129pm5.32da 623 . . . 4  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  )  <->  ( x  e.  K  /\  x  e.  { N } ) ) )
131 eqid 2435 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
132 eqid 2435 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
133 fvex 5734 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
13415, 133eqeltri 2505 . . . . . . . 8  |-  K  e. 
_V
135134a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
136113, 5, 131, 15evl1rhm 19941 . . . . . . . . . 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 15819 . . . . . . . . 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 5863 . . . . . . 7  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
141131, 15, 132, 2, 135, 140pwselbas 13703 . . . . . 6  |-  ( ph  ->  ( O `  G
) : K --> K )
142 ffn 5583 . . . . . 6  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
143141, 142syl 16 . . . . 5  |-  ( ph  ->  ( O `  G
)  Fn  K )
144 fniniseg 5843 . . . . 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 3935 . . . . . 6  |-  ( ph  ->  { N }  C_  K )
147146sseld 3339 . . . . 5  |-  ( ph  ->  ( x  e.  { N }  ->  x  e.  K ) )
148147pm4.71rd 617 . . . 4  |-  ( ph  ->  ( x  e.  { N }  <->  ( x  e.  K  /\  x  e. 
{ N } ) ) )
149130, 145, 1483bitr4d 277 . . 3  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  x  e.  { N } ) )
150149eqrdv 2433 . 2  |-  ( ph  ->  ( `' ( O `
 G ) " {  .0.  } )  =  { N } )
151110, 51, 1503jca 1134 1  |-  ( ph  ->  ( G  e.  U  /\  ( D `  G
)  =  1  /\  ( `' ( O `
 G ) " {  .0.  } )  =  { N } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   ifcif 3731   {csn 3806   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983   RR*cxr 9111    < clt 9112    <_ cle 9113   NN0cn0 10213   Basecbs 13461  Scalarcsca 13524   .scvsca 13525    ^s cpws 13662   0gc0g 13715   Grpcgrp 14677   -gcsg 14680  .gcmg 14681  mulGrpcmgp 15640   Ringcrg 15652   CRingccrg 15653   1rcur 15654   RingHom crh 15809   LModclmod 15942  NzRingcnzr 16320  algSccascl 16363  var1cv1 16562  Poly1cpl1 16563  eval1ce1 16565  coe1cco1 16566   deg1 cdg1 19969  Monic1pcmn1 20040
This theorem is referenced by:  ply1rem  20078  facth1  20079  fta1glem1  20080  fta1glem2  20081
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-hom 13545  df-cco 13546  df-prds 13663  df-pws 13665  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-rnghom 15811  df-subrg 15858  df-lmod 15944  df-lss 16001  df-lsp 16040  df-nzr 16321  df-rlreg 16335  df-assa 16364  df-asp 16365  df-ascl 16366  df-psr 16409  df-mvr 16410  df-mpl 16411  df-evls 16412  df-evl 16413  df-opsr 16417  df-psr1 16568  df-vr1 16569  df-ply1 16570  df-evl1 16572  df-coe1 16573  df-cnfld 16696  df-mdeg 19970  df-deg1 19971  df-mon1 20045
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