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Theorem ply1remlem 19548
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 16013 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
42, 3syl 15 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
5 ply1rem.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
65ply1rng 16326 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
74, 6syl 15 . . . . . 6  |-  ( ph  ->  P  e.  Ring )
8 rnggrp 15346 . . . . . 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 16294 . . . . . 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 16361 . . . . . . 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 5663 . . . . . 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 14546 . . . . 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 2367 . . 3  |-  ( ph  ->  G  e.  B )
251fveq2i 5528 . . . . . . 7  |-  ( D `
 G )  =  ( D `  ( X  .-  ( A `  N ) ) )
26 ply1rem.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
2726, 5, 11deg1xrcl 19468 . . . . . . . . . . 11  |-  ( ( A `  N )  e.  B  ->  ( D `  ( A `  N ) )  e. 
RR* )
2820, 27syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  e.  RR* )
29 0xr 8878 . . . . . . . . . . 11  |-  0  e.  RR*
3029a1i 10 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR* )
31 1re 8837 . . . . . . . . . . 11  |-  1  e.  RR
32 rexr 8877 . . . . . . . . . . 11  |-  ( 1  e.  RR  ->  1  e.  RR* )
3331, 32mp1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR* )
3426, 5, 15, 14deg1sclle 19498 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  ( D `  ( A `  N ) )  <_ 
0 )
354, 18, 34syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  <_  0 )
36 0lt1 9296 . . . . . . . . . . 11  |-  0  <  1
3736a1i 10 . . . . . . . . . 10  |-  ( ph  ->  0  <  1 )
3828, 30, 33, 35, 37xrlelttrd 10491 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( A `  N )
)  <  1 )
39 eqid 2283 . . . . . . . . . . . . . 14  |-  (mulGrp `  P )  =  (mulGrp `  P )
4039, 11mgpbas 15331 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  P ) )
41 eqid 2283 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4240, 41mulg1 14574 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4313, 42syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4443fveq2d 5529 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( D `
 X ) )
45 1nn0 9981 . . . . . . . . . . 11  |-  1  e.  NN0
4626, 5, 10, 39, 41deg1pw 19506 . . . . . . . . . . 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 2317 . . . . . . . . 9  |-  ( ph  ->  ( D `  X
)  =  1 )
4938, 48breqtrrd 4049 . . . . . . . 8  |-  ( ph  ->  ( D `  ( A `  N )
)  <  ( D `  X ) )
505, 26, 4, 11, 21, 13, 20, 49deg1sub 19494 . . . . . . 7  |-  ( ph  ->  ( D `  ( X  .-  ( A `  N ) ) )  =  ( D `  X ) )
5125, 50syl5eq 2327 . . . . . 6  |-  ( ph  ->  ( D `  G
)  =  ( D `
 X ) )
5251, 48eqtrd 2315 . . . . 5  |-  ( ph  ->  ( D `  G
)  =  1 )
5352, 45syl6eqel 2371 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
54 eqid 2283 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
5526, 5, 54, 11deg1nn0clb 19476 . . . . 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 5529 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( (coe1 `  G ) `  1
) )
591fveq2i 5528 . . . . . 6  |-  (coe1 `  G
)  =  (coe1 `  ( X  .-  ( A `  N ) ) )
6059fveq1i 5526 . . . . 5  |-  ( (coe1 `  G ) `  1
)  =  ( (coe1 `  ( X  .-  ( A `  N )
) ) `  1
)
6145a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  NN0 )
62 eqid 2283 . . . . . . 7  |-  ( -g `  R )  =  (
-g `  R )
635, 11, 21, 62coe1subfv 16343 . . . . . 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 2327 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) ` 
1 )  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6643oveq2d 5874 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  ( ( 1r `  R ) ( .s `  P
) X ) )
675ply1sca 16331 . . . . . . . . . . . . 13  |-  ( R  e. NzRing  ->  R  =  (Scalar `  P ) )
682, 67syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  P ) )
6968fveq2d 5529 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  P )
) )
7069oveq1d 5873 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) X )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) X ) )
715ply1lmod 16330 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  P  e. 
LMod )
724, 71syl 15 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  LMod )
73 eqid 2283 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
74 eqid 2283 . . . . . . . . . . . 12  |-  ( .s
`  P )  =  ( .s `  P
)
75 eqid 2283 . . . . . . . . . . . 12  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
7611, 73, 74, 75lmodvs1 15658 . . . . . . . . . . 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 2319 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  X )
7978fveq2d 5529 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) )  =  (coe1 `  X ) )
8079fveq1d 5527 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  X ) ` 
1 ) )
81 eqid 2283 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
8215, 81rngidcl 15361 . . . . . . . . 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 16350 . . . . . . . 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 2317 . . . . . 6  |-  ( ph  ->  ( (coe1 `  X ) ` 
1 )  =  ( 1r `  R ) )
88 eqid 2283 . . . . . . . . . 10  |-  ( 0g
`  R )  =  ( 0g `  R
)
895, 14, 15, 88coe1scl 16362 . . . . . . . . 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 5527 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) `  1
) )
92 ax-1ne0 8806 . . . . . . . . . . 11  |-  1  =/=  0
93 neeq1 2454 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  =/=  0  <->  1  =/=  0 ) )
9492, 93mpbiri 224 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =/=  0 )
95 ifnefalse 3573 . . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g
`  R ) ) )
98 fvex 5539 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
9996, 97, 98fvmpt 5602 . . . . . . . 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 2331 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( 0g `  R ) )
10287, 101oveq12d 5876 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) ) )
103 rnggrp 15346 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1044, 103syl 15 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
10515, 88, 62grpsubid1 14551 . . . . . 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 2315 . . . 4  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( 1r `  R
) )
10858, 65, 1073eqtrd 2319 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( 1r
`  R ) )
109 ply1rem.u . . . 4  |-  U  =  (Monic1p `  R )
1105, 11, 54, 26, 109, 81ismon1p 19528 . . 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 5528 . . . . . . . . . 10  |-  ( O `
 G )  =  ( O `  ( X  .-  ( A `  N ) ) )
113112fveq1i 5526 . . . . . . . . 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 19416 . . . . . . . . . . 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 19414 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  (
( A `  N
)  e.  B  /\  ( ( O `  ( A `  N ) ) `  x )  =  N ) )
121114, 5, 15, 11, 116, 117, 118, 120, 21, 62evl1subd 19418 . . . . . . . . . 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 2327 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  =  ( x (
-g `  R ) N ) )
124123eqeq1d 2291 . . . . . . 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 14552 . . . . . . . 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 3655 . . . . . 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 2283 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
133 eqid 2283 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
134 fvex 5539 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
13515, 134eqeltri 2353 . . . . . . . 8  |-  K  e. 
_V
136135a1i 10 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
137114, 5, 132, 15evl1rhm 19412 . . . . . . . . . 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 15504 . . . . . . . . 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 5663 . . . . . . . 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 13388 . . . . . 6  |-  ( ph  ->  ( O `  G
) : K --> K )
144 ffn 5389 . . . . . 6  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
145143, 144syl 15 . . . . 5  |-  ( ph  ->  ( O `  G
)  Fn  K )
146 fniniseg 5646 . . . . 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 3760 . . . . . 6  |-  ( ph  ->  { N }  C_  K )
149148sseld 3179 . . . . 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 2281 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868   NN0cn0 9965   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    ^s cpws 13347   0gc0g 13400   Grpcgrp 14362   -gcsg 14365  .gcmg 14366  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   1rcur 15339   RingHom crh 15494   LModclmod 15627  NzRingcnzr 16009  algSccascl 16052  var1cv1 16251  Poly1cpl1 16252  eval1ce1 16254  coe1cco1 16255   deg1 cdg1 19440  Monic1pcmn1 19511
This theorem is referenced by:  ply1rem  19549  facth1  19550  fta1glem1  19551  fta1glem2  19552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-nzr 16010  df-rlreg 16024  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-evl1 16261  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442  df-mon1 19516
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