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Theorem ply1remlem 19564
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 16029 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
42, 3syl 15 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
5 ply1rem.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
65ply1rng 16342 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
74, 6syl 15 . . . . . 6  |-  ( ph  ->  P  e.  Ring )
8 rnggrp 15362 . . . . . 6  |-  ( P  e.  Ring  ->  P  e. 
Grp )
97, 8syl 15 . . . . 5  |-  ( ph  ->  P  e.  Grp )
10 ply1rem.x . . . . . . 7  |-  X  =  (var1 `  R )
11 ply1rem.b . . . . . . 7  |-  B  =  ( Base `  P
)
1210, 5, 11vr1cl 16310 . . . . . 6  |-  ( R  e.  Ring  ->  X  e.  B )
134, 12syl 15 . . . . 5  |-  ( ph  ->  X  e.  B )
14 ply1rem.a . . . . . . . 8  |-  A  =  (algSc `  P )
15 ply1rem.k . . . . . . . 8  |-  K  =  ( Base `  R
)
165, 14, 15, 11ply1sclf 16377 . . . . . . 7  |-  ( R  e.  Ring  ->  A : K
--> B )
174, 16syl 15 . . . . . 6  |-  ( ph  ->  A : K --> B )
18 ply1rem.3 . . . . . 6  |-  ( ph  ->  N  e.  K )
19 ffvelrn 5679 . . . . . 6  |-  ( ( A : K --> B  /\  N  e.  K )  ->  ( A `  N
)  e.  B )
2017, 18, 19syl2anc 642 . . . . 5  |-  ( ph  ->  ( A `  N
)  e.  B )
21 ply1rem.m . . . . . 6  |-  .-  =  ( -g `  P )
2211, 21grpsubcl 14562 . . . . 5  |-  ( ( P  e.  Grp  /\  X  e.  B  /\  ( A `  N )  e.  B )  -> 
( X  .-  ( A `  N )
)  e.  B )
239, 13, 20, 22syl3anc 1182 . . . 4  |-  ( ph  ->  ( X  .-  ( A `  N )
)  e.  B )
241, 23syl5eqel 2380 . . 3  |-  ( ph  ->  G  e.  B )
251fveq2i 5544 . . . . . . 7  |-  ( D `
 G )  =  ( D `  ( X  .-  ( A `  N ) ) )
26 ply1rem.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
2726, 5, 11deg1xrcl 19484 . . . . . . . . . . 11  |-  ( ( A `  N )  e.  B  ->  ( D `  ( A `  N ) )  e. 
RR* )
2820, 27syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  e.  RR* )
29 0xr 8894 . . . . . . . . . . 11  |-  0  e.  RR*
3029a1i 10 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR* )
31 1re 8853 . . . . . . . . . . 11  |-  1  e.  RR
32 rexr 8893 . . . . . . . . . . 11  |-  ( 1  e.  RR  ->  1  e.  RR* )
3331, 32mp1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR* )
3426, 5, 15, 14deg1sclle 19514 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  ( D `  ( A `  N ) )  <_ 
0 )
354, 18, 34syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  <_  0 )
36 0lt1 9312 . . . . . . . . . . 11  |-  0  <  1
3736a1i 10 . . . . . . . . . 10  |-  ( ph  ->  0  <  1 )
3828, 30, 33, 35, 37xrlelttrd 10507 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( A `  N )
)  <  1 )
39 eqid 2296 . . . . . . . . . . . . . 14  |-  (mulGrp `  P )  =  (mulGrp `  P )
4039, 11mgpbas 15347 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  P ) )
41 eqid 2296 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4240, 41mulg1 14590 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4313, 42syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4443fveq2d 5545 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( D `
 X ) )
45 1nn0 9997 . . . . . . . . . . 11  |-  1  e.  NN0
4626, 5, 10, 39, 41deg1pw 19522 . . . . . . . . . . 11  |-  ( ( R  e. NzRing  /\  1  e.  NN0 )  ->  ( D `  ( 1
(.g `  (mulGrp `  P
) ) X ) )  =  1 )
472, 45, 46sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  1 )
4844, 47eqtr3d 2330 . . . . . . . . 9  |-  ( ph  ->  ( D `  X
)  =  1 )
4938, 48breqtrrd 4065 . . . . . . . 8  |-  ( ph  ->  ( D `  ( A `  N )
)  <  ( D `  X ) )
505, 26, 4, 11, 21, 13, 20, 49deg1sub 19510 . . . . . . 7  |-  ( ph  ->  ( D `  ( X  .-  ( A `  N ) ) )  =  ( D `  X ) )
5125, 50syl5eq 2340 . . . . . 6  |-  ( ph  ->  ( D `  G
)  =  ( D `
 X ) )
5251, 48eqtrd 2328 . . . . 5  |-  ( ph  ->  ( D `  G
)  =  1 )
5352, 45syl6eqel 2384 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
54 eqid 2296 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
5526, 5, 54, 11deg1nn0clb 19492 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
564, 24, 55syl2anc 642 . . . 4  |-  ( ph  ->  ( G  =/=  ( 0g `  P )  <->  ( D `  G )  e.  NN0 ) )
5753, 56mpbird 223 . . 3  |-  ( ph  ->  G  =/=  ( 0g
`  P ) )
5852fveq2d 5545 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( (coe1 `  G ) `  1
) )
591fveq2i 5544 . . . . . 6  |-  (coe1 `  G
)  =  (coe1 `  ( X  .-  ( A `  N ) ) )
6059fveq1i 5542 . . . . 5  |-  ( (coe1 `  G ) `  1
)  =  ( (coe1 `  ( X  .-  ( A `  N )
) ) `  1
)
6145a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  NN0 )
62 eqid 2296 . . . . . . 7  |-  ( -g `  R )  =  (
-g `  R )
635, 11, 21, 62coe1subfv 16359 . . . . . 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
) ) )
644, 13, 20, 61, 63syl31anc 1185 . . . . 5  |-  ( ph  ->  ( (coe1 `  ( X  .-  ( A `  N ) ) ) `  1
)  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6560, 64syl5eq 2340 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) ` 
1 )  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6643oveq2d 5890 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  ( ( 1r `  R ) ( .s `  P
) X ) )
675ply1sca 16347 . . . . . . . . . . . . 13  |-  ( R  e. NzRing  ->  R  =  (Scalar `  P ) )
682, 67syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  P ) )
6968fveq2d 5545 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  P )
) )
7069oveq1d 5889 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) X )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) X ) )
715ply1lmod 16346 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  P  e. 
LMod )
724, 71syl 15 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  LMod )
73 eqid 2296 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
74 eqid 2296 . . . . . . . . . . . 12  |-  ( .s
`  P )  =  ( .s `  P
)
75 eqid 2296 . . . . . . . . . . . 12  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
7611, 73, 74, 75lmodvs1 15674 . . . . . . . . . . 11  |-  ( ( P  e.  LMod  /\  X  e.  B )  ->  (
( 1r `  (Scalar `  P ) ) ( .s `  P ) X )  =  X )
7772, 13, 76syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  (Scalar `  P ) ) ( .s `  P
) X )  =  X )
7866, 70, 773eqtrd 2332 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  X )
7978fveq2d 5545 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) )  =  (coe1 `  X ) )
8079fveq1d 5543 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  X ) ` 
1 ) )
81 eqid 2296 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
8215, 81rngidcl 15377 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  K )
834, 82syl 15 . . . . . . . 8  |-  ( ph  ->  ( 1r `  R
)  e.  K )
84 ply1rem.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
8584, 15, 5, 10, 74, 39, 41coe1tmfv1 16366 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  K  /\  1  e. 
NN0 )  ->  (
(coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
864, 83, 61, 85syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
8780, 86eqtr3d 2330 . . . . . 6  |-  ( ph  ->  ( (coe1 `  X ) ` 
1 )  =  ( 1r `  R ) )
88 eqid 2296 . . . . . . . . . 10  |-  ( 0g
`  R )  =  ( 0g `  R
)
895, 14, 15, 88coe1scl 16378 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) )
904, 18, 89syl2anc 642 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( A `  N ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) )
9190fveq1d 5543 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) `  1
) )
92 ax-1ne0 8822 . . . . . . . . . . 11  |-  1  =/=  0
93 neeq1 2467 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  =/=  0  <->  1  =/=  0 ) )
9492, 93mpbiri 224 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =/=  0 )
95 ifnefalse 3586 . . . . . . . . . 10  |-  ( x  =/=  0  ->  if ( x  =  0 ,  N ,  ( 0g
`  R ) )  =  ( 0g `  R ) )
9694, 95syl 15 . . . . . . . . 9  |-  ( x  =  1  ->  if ( x  =  0 ,  N ,  ( 0g
`  R ) )  =  ( 0g `  R ) )
97 eqid 2296 . . . . . . . . 9  |-  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )
98 fvex 5555 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
9996, 97, 98fvmpt 5618 . . . . . . . 8  |-  ( 1  e.  NN0  ->  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) `  1 )  =  ( 0g `  R ) )
10045, 99ax-mp 8 . . . . . . 7  |-  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) ) `  1 )  =  ( 0g `  R )
10191, 100syl6eq 2344 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( 0g `  R ) )
10287, 101oveq12d 5892 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) ) )
103 rnggrp 15362 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1044, 103syl 15 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
10515, 88, 62grpsubid1 14567 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( 1r `  R )  e.  K )  -> 
( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
106104, 83, 105syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) )  =  ( 1r `  R ) )
107102, 106eqtrd 2328 . . . 4  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( 1r `  R
) )
10858, 65, 1073eqtrd 2332 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( 1r
`  R ) )
109 ply1rem.u . . . 4  |-  U  =  (Monic1p `  R )
1105, 11, 54, 26, 109, 81ismon1p 19544 . . 3  |-  ( G  e.  U  <->  ( G  e.  B  /\  G  =/=  ( 0g `  P
)  /\  ( (coe1 `  G ) `  ( D `  G )
)  =  ( 1r
`  R ) ) )
11124, 57, 108, 110syl3anbrc 1136 . 2  |-  ( ph  ->  G  e.  U )
1121fveq2i 5544 . . . . . . . . . 10  |-  ( O `
 G )  =  ( O `  ( X  .-  ( A `  N ) ) )
113112fveq1i 5542 . . . . . . . . 9  |-  ( ( O `  G ) `
 x )  =  ( ( O `  ( X  .-  ( A `
 N ) ) ) `  x )
114 ply1rem.o . . . . . . . . . . 11  |-  O  =  (eval1 `  R )
115 ply1rem.2 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  CRing )
116115adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  CRing )
117 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
118114, 10, 15, 5, 11, 116, 117evl1vard 19432 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( X  e.  B  /\  ( ( O `  X ) `  x
)  =  x ) )
11918adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  K )  ->  N  e.  K )
120114, 5, 15, 14, 11, 116, 119, 117evl1scad 19430 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  (
( A `  N
)  e.  B  /\  ( ( O `  ( A `  N ) ) `  x )  =  N ) )
121114, 5, 15, 11, 116, 117, 118, 120, 21, 62evl1subd 19434 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( X  .-  ( A `  N )
)  e.  B  /\  ( ( O `  ( X  .-  ( A `
 N ) ) ) `  x )  =  ( x (
-g `  R ) N ) ) )
122121simprd 449 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( X  .-  ( A `  N ) ) ) `
 x )  =  ( x ( -g `  R ) N ) )
123113, 122syl5eq 2340 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  =  ( x (
-g `  R ) N ) )
124123eqeq1d 2304 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  ( x
( -g `  R ) N )  =  .0.  ) )
125104adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  R  e.  Grp )
12615, 84, 62grpsubeq0 14568 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  K  /\  N  e.  K )  ->  ( ( x (
-g `  R ) N )  =  .0.  <->  x  =  N ) )
127125, 117, 119, 126syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( x ( -g `  R ) N )  =  .0.  <->  x  =  N ) )
128124, 127bitrd 244 . . . . . 6  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  =  N ) )
129 elsn 3668 . . . . . 6  |-  ( x  e.  { N }  <->  x  =  N )
130128, 129syl6bbr 254 . . . . 5  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  G ) `  x
)  =  .0.  <->  x  e.  { N } ) )
131130pm5.32da 622 . . . 4  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  )  <->  ( x  e.  K  /\  x  e.  { N } ) ) )
132 eqid 2296 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
133 eqid 2296 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
134 fvex 5555 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
13515, 134eqeltri 2366 . . . . . . . 8  |-  K  e. 
_V
136135a1i 10 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
137114, 5, 132, 15evl1rhm 19428 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
138115, 137syl 15 . . . . . . . . 9  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
13911, 133rhmf 15520 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
140138, 139syl 15 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
141 ffvelrn 5679 . . . . . . . 8  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  G  e.  B )  ->  ( O `  G )  e.  ( Base `  ( R  ^s  K ) ) )
142140, 24, 141syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
143132, 15, 133, 2, 136, 142pwselbas 13404 . . . . . 6  |-  ( ph  ->  ( O `  G
) : K --> K )
144 ffn 5405 . . . . . 6  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
145143, 144syl 15 . . . . 5  |-  ( ph  ->  ( O `  G
)  Fn  K )
146 fniniseg 5662 . . . . 5  |-  ( ( O `  G )  Fn  K  ->  (
x  e.  ( `' ( O `  G
) " {  .0.  } )  <->  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  .0.  ) ) )
147145, 146syl 15 . . . 4  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  .0.  ) ) )
14818snssd 3776 . . . . . 6  |-  ( ph  ->  { N }  C_  K )
149148sseld 3192 . . . . 5  |-  ( ph  ->  ( x  e.  { N }  ->  x  e.  K ) )
150149pm4.71rd 616 . . . 4  |-  ( ph  ->  ( x  e.  { N }  <->  ( x  e.  K  /\  x  e. 
{ N } ) ) )
151131, 147, 1503bitr4d 276 . . 3  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " {  .0.  } )  <->  x  e.  { N } ) )
152151eqrdv 2294 . 2  |-  ( ph  ->  ( `' ( O `
 G ) " {  .0.  } )  =  { N } )
153111, 52, 1523jca 1132 1  |-  ( ph  ->  ( G  e.  U  /\  ( D `  G
)  =  1  /\  ( `' ( O `
 G ) " {  .0.  } )  =  { N } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   ifcif 3578   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754   RR*cxr 8882    < clt 8883    <_ cle 8884   NN0cn0 9981   Basecbs 13164  Scalarcsca 13227   .scvsca 13228    ^s cpws 13363   0gc0g 13416   Grpcgrp 14378   -gcsg 14381  .gcmg 14382  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354   1rcur 15355   RingHom crh 15510   LModclmod 15643  NzRingcnzr 16025  algSccascl 16068  var1cv1 16267  Poly1cpl1 16268  eval1ce1 16270  coe1cco1 16271   deg1 cdg1 19456  Monic1pcmn1 19527
This theorem is referenced by:  ply1rem  19565  facth1  19566  fta1glem1  19567  fta1glem2  19568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-nzr 16026  df-rlreg 16040  df-assa 16069  df-asp 16070  df-ascl 16071  df-psr 16114  df-mvr 16115  df-mpl 16116  df-evls 16117  df-evl 16118  df-opsr 16122  df-psr1 16273  df-vr1 16274  df-ply1 16275  df-evl1 16277  df-coe1 16278  df-cnfld 16394  df-mdeg 19457  df-deg1 19458  df-mon1 19532
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