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Theorem ply1remlem 21637
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 17346 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
5 ply1rem.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
65ply1rng 17706 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
74, 6syl 16 . . . . . 6  |-  ( ph  ->  P  e.  Ring )
8 rnggrp 16653 . . . . . 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 17674 . . . . . 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 17741 . . . . . . 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 5847 . . . . 5  |-  ( ph  ->  ( A `  N
)  e.  B )
20 ply1rem.m . . . . . 6  |-  .-  =  ( -g `  P )
2111, 20grpsubcl 15609 . . . . 5  |-  ( ( P  e.  Grp  /\  X  e.  B  /\  ( A `  N )  e.  B )  -> 
( X  .-  ( A `  N )
)  e.  B )
229, 13, 19, 21syl3anc 1218 . . . 4  |-  ( ph  ->  ( X  .-  ( A `  N )
)  e.  B )
231, 22syl5eqel 2527 . . 3  |-  ( ph  ->  G  e.  B )
241fveq2i 5697 . . . . . . 7  |-  ( D `
 G )  =  ( D `  ( X  .-  ( A `  N ) ) )
25 ply1rem.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
2625, 5, 11deg1xrcl 21556 . . . . . . . . . . 11  |-  ( ( A `  N )  e.  B  ->  ( D `  ( A `  N ) )  e. 
RR* )
2719, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  e.  RR* )
28 0xr 9433 . . . . . . . . . . 11  |-  0  e.  RR*
2928a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR* )
30 1re 9388 . . . . . . . . . . 11  |-  1  e.  RR
31 rexr 9432 . . . . . . . . . . 11  |-  ( 1  e.  RR  ->  1  e.  RR* )
3230, 31mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR* )
3325, 5, 15, 14deg1sclle 21587 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  ( D `  ( A `  N ) )  <_ 
0 )
344, 18, 33syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  <_  0 )
35 0lt1 9865 . . . . . . . . . . 11  |-  0  <  1
3635a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  <  1 )
3727, 29, 32, 34, 36xrlelttrd 11137 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( A `  N )
)  <  1 )
38 eqid 2443 . . . . . . . . . . . . . 14  |-  (mulGrp `  P )  =  (mulGrp `  P )
3938, 11mgpbas 16600 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  P ) )
40 eqid 2443 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4139, 40mulg1 15637 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4213, 41syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4342fveq2d 5698 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( D `
 X ) )
44 1nn0 10598 . . . . . . . . . . 11  |-  1  e.  NN0
4525, 5, 10, 38, 40deg1pw 21595 . . . . . . . . . . 11  |-  ( ( R  e. NzRing  /\  1  e.  NN0 )  ->  ( D `  ( 1
(.g `  (mulGrp `  P
) ) X ) )  =  1 )
462, 44, 45sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  1 )
4743, 46eqtr3d 2477 . . . . . . . . 9  |-  ( ph  ->  ( D `  X
)  =  1 )
4837, 47breqtrrd 4321 . . . . . . . 8  |-  ( ph  ->  ( D `  ( A `  N )
)  <  ( D `  X ) )
495, 25, 4, 11, 20, 13, 19, 48deg1sub 21583 . . . . . . 7  |-  ( ph  ->  ( D `  ( X  .-  ( A `  N ) ) )  =  ( D `  X ) )
5024, 49syl5eq 2487 . . . . . 6  |-  ( ph  ->  ( D `  G
)  =  ( D `
 X ) )
5150, 47eqtrd 2475 . . . . 5  |-  ( ph  ->  ( D `  G
)  =  1 )
5251, 44syl6eqel 2531 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
53 eqid 2443 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
5425, 5, 53, 11deg1nn0clb 21564 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
554, 23, 54syl2anc 661 . . . 4  |-  ( ph  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
5652, 55mpbird 232 . . 3  |-  ( ph  ->  G  =/=  ( 0g
`  P ) )
5751fveq2d 5698 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( (coe1 `  G ) `  1
) )
581fveq2i 5697 . . . . . 6  |-  (coe1 `  G
)  =  (coe1 `  ( X  .-  ( A `  N ) ) )
5958fveq1i 5695 . . . . 5  |-  ( (coe1 `  G ) `  1
)  =  ( (coe1 `  ( X  .-  ( A `  N )
) ) `  1
)
6044a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  NN0 )
61 eqid 2443 . . . . . . 7  |-  ( -g `  R )  =  (
-g `  R )
625, 11, 20, 61coe1subfv 17723 . . . . . 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 1221 . . . . 5  |-  ( ph  ->  ( (coe1 `  ( X  .-  ( A `  N ) ) ) `  1
)  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6459, 63syl5eq 2487 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) ` 
1 )  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6542oveq2d 6110 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  ( ( 1r `  R ) ( .s `  P
) X ) )
665ply1sca 17711 . . . . . . . . . . . . 13  |-  ( R  e. NzRing  ->  R  =  (Scalar `  P ) )
672, 66syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  P ) )
6867fveq2d 5698 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  P )
) )
6968oveq1d 6109 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) X )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) X ) )
705ply1lmod 17710 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  P  e. 
LMod )
714, 70syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  LMod )
72 eqid 2443 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
73 eqid 2443 . . . . . . . . . . . 12  |-  ( .s
`  P )  =  ( .s `  P
)
74 eqid 2443 . . . . . . . . . . . 12  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
7511, 72, 73, 74lmodvs1 16979 . . . . . . . . . . 11  |-  ( ( P  e.  LMod  /\  X  e.  B )  ->  (
( 1r `  (Scalar `  P ) ) ( .s `  P ) X )  =  X )
7671, 13, 75syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  (Scalar `  P ) ) ( .s `  P
) X )  =  X )
7765, 69, 763eqtrd 2479 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  X )
7877fveq2d 5698 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) )  =  (coe1 `  X ) )
7978fveq1d 5696 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  X ) ` 
1 ) )
80 eqid 2443 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
8115, 80rngidcl 16668 . . . . . . . . 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 17730 . . . . . . . 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 1218 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
8679, 85eqtr3d 2477 . . . . . 6  |-  ( ph  ->  ( (coe1 `  X ) ` 
1 )  =  ( 1r `  R ) )
87 eqid 2443 . . . . . . . . . 10  |-  ( 0g
`  R )  =  ( 0g `  R
)
885, 14, 15, 87coe1scl 17742 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) )
894, 18, 88syl2anc 661 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) )
9089fveq1d 5696 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) `  1
) )
91 ax-1ne0 9354 . . . . . . . . . . 11  |-  1  =/=  0
92 neeq1 2619 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  =/=  0  <->  1  =/=  0 ) )
9391, 92mpbiri 233 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =/=  0 )
94 ifnefalse 3804 . . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )
97 fvex 5704 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
9895, 96, 97fvmpt 5777 . . . . . . . 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 2491 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( 0g `  R ) )
10186, 100oveq12d 6112 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) ) )
102 rnggrp 16653 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1034, 102syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
10415, 87, 61grpsubid1 15614 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  K )  -> 
( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
105103, 82, 104syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
106101, 105eqtrd 2475 . . . 4  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( 1r `  R
) )
10757, 64, 1063eqtrd 2479 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( 1r
`  R ) )
108 ply1rem.u . . . 4  |-  U  =  (Monic1p `  R )
1095, 11, 53, 25, 108, 80ismon1p 21617 . . 3  |-  ( G  e.  U  <->  ( G  e.  B  /\  G  =/=  ( 0g `  P
)  /\  ( (coe1 `  G ) `  ( D `  G )
)  =  ( 1r
`  R ) ) )
11023, 56, 107, 109syl3anbrc 1172 . 2  |-  ( ph  ->  G  e.  U )
1111fveq2i 5697 . . . . . . . . . 10  |-  ( O `
 G )  =  ( O `  ( X  .-  ( A `  N ) ) )
112111fveq1i 5695 . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  CRing )
116 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
117113, 10, 15, 5, 11, 115, 116evl1vard 17774 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( X  e.  B  /\  ( ( O `  X ) `  x
)  =  x ) )
11818adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  K )  ->  N  e.  K )
119113, 5, 15, 14, 11, 115, 118, 116evl1scad 17772 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  (
( A `  N
)  e.  B  /\  ( ( O `  ( A `  N ) ) `  x )  =  N ) )
120113, 5, 15, 11, 115, 116, 117, 119, 20, 61evl1subd 17779 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( X  .-  ( A `  N )
)  e.  B  /\  ( ( O `  ( X  .-  ( A `
 N ) ) ) `  x )  =  ( x (
-g `  R ) N ) ) )
121120simprd 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( X  .-  ( A `  N ) ) ) `
 x )  =  ( x ( -g `  R ) N ) )
122112, 121syl5eq 2487 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  =  ( x (
-g `  R ) N ) )
123122eqeq1d 2451 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  ( x
( -g `  R ) N )  =  .0.  ) )
124103adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  Grp )
12515, 83, 61grpsubeq0 15615 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  K  /\  N  e.  K )  ->  ( ( x (
-g `  R ) N )  =  .0.  <->  x  =  N ) )
126124, 116, 118, 125syl3anc 1218 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( x ( -g `  R ) N )  =  .0.  <->  x  =  N ) )
127123, 126bitrd 253 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  =  N ) )
128 elsn 3894 . . . . . 6  |-  ( x  e.  { N }  <->  x  =  N )
129127, 128syl6bbr 263 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  e.  { N } ) )
130129pm5.32da 641 . . . 4  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  )  <->  ( x  e.  K  /\  x  e.  { N } ) ) )
131 eqid 2443 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
132 eqid 2443 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
133 fvex 5704 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
13415, 133eqeltri 2513 . . . . . . . 8  |-  K  e. 
_V
135134a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
136113, 5, 131, 15evl1rhm 17769 . . . . . . . . . 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 16819 . . . . . . . . 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 5847 . . . . . . 7  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
141131, 15, 132, 2, 135, 140pwselbas 14430 . . . . . 6  |-  ( ph  ->  ( O `  G
) : K --> K )
142 ffn 5562 . . . . . 6  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
143141, 142syl 16 . . . . 5  |-  ( ph  ->  ( O `  G
)  Fn  K )
144 fniniseg 5827 . . . . 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 4021 . . . . . 6  |-  ( ph  ->  { N }  C_  K )
147146sseld 3358 . . . . 5  |-  ( ph  ->  ( x  e.  { N }  ->  x  e.  K ) )
148147pm4.71rd 635 . . . 4  |-  ( ph  ->  ( x  e.  { N }  <->  ( x  e.  K  /\  x  e. 
{ N } ) ) )
149130, 145, 1483bitr4d 285 . . 3  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  x  e.  { N } ) )
150149eqrdv 2441 . 2  |-  ( ph  ->  ( `' ( O `
 G ) " {  .0.  } )  =  { N } )
151110, 51, 1503jca 1168 1  |-  ( ph  ->  ( G  e.  U  /\  ( D `  G
)  =  1  /\  ( `' ( O `
 G ) " {  .0.  } )  =  { N } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2609   _Vcvv 2975   ifcif 3794   {csn 3880   class class class wbr 4295    e. cmpt 4353   `'ccnv 4842   "cima 4846    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6094   RRcr 9284   0cc0 9285   1c1 9286   RR*cxr 9420    < clt 9421    <_ cle 9422   NN0cn0 10582   Basecbs 14177  Scalarcsca 14244   .scvsca 14245   0gc0g 14381    ^s cpws 14388   Grpcgrp 15413   -gcsg 15416  .gcmg 15417  mulGrpcmgp 16594   1rcur 16606   Ringcrg 16648   CRingccrg 16649   RingHom crh 16807   LModclmod 16951  NzRingcnzr 17342  algSccascl 17386  var1cv1 17635  Poly1cpl1 17636  coe1cco1 17637  eval1ce1 17752   deg1 cdg1 21526  Monic1pcmn1 21600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-ofr 6324  df-om 6480  df-1st 6580  df-2nd 6581  df-supp 6694  df-tpos 6748  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fsupp 7624  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-fz 11441  df-fzo 11552  df-seq 11810  df-hash 12107  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-hom 14265  df-cco 14266  df-0g 14383  df-gsum 14384  df-prds 14389  df-pws 14391  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-mhm 15467  df-submnd 15468  df-grp 15548  df-minusg 15549  df-sbg 15550  df-mulg 15551  df-subg 15681  df-ghm 15748  df-cntz 15838  df-cmn 16282  df-abl 16283  df-mgp 16595  df-ur 16607  df-srg 16611  df-rng 16650  df-cring 16651  df-oppr 16718  df-dvdsr 16736  df-unit 16737  df-invr 16767  df-rnghom 16809  df-subrg 16866  df-lmod 16953  df-lss 17017  df-lsp 17056  df-nzr 17343  df-rlreg 17357  df-assa 17387  df-asp 17388  df-ascl 17389  df-psr 17426  df-mvr 17427  df-mpl 17428  df-opsr 17430  df-evls 17591  df-evl 17592  df-psr1 17639  df-vr1 17640  df-ply1 17641  df-coe1 17642  df-evl1 17754  df-cnfld 17822  df-mdeg 21527  df-deg1 21528  df-mon1 21605
This theorem is referenced by:  ply1rem  21638  facth1  21639  fta1glem1  21640  fta1glem2  21641
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