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Theorem ply1remlem 22540
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 nzrring 17887 . . . . . . . 8  |-  ( R  e. NzRing  ->  R  e.  Ring )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
5 ply1rem.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
65ply1ring 18267 . . . . . . 7  |-  ( R  e.  Ring  ->  P  e. 
Ring )
74, 6syl 16 . . . . . 6  |-  ( ph  ->  P  e.  Ring )
8 ringgrp 17181 . . . . . 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 18236 . . . . . 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 18304 . . . . . . 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 6017 . . . . 5  |-  ( ph  ->  ( A `  N
)  e.  B )
20 ply1rem.m . . . . . 6  |-  .-  =  ( -g `  P )
2111, 20grpsubcl 16096 . . . . 5  |-  ( ( P  e.  Grp  /\  X  e.  B  /\  ( A `  N )  e.  B )  -> 
( X  .-  ( A `  N )
)  e.  B )
229, 13, 19, 21syl3anc 1229 . . . 4  |-  ( ph  ->  ( X  .-  ( A `  N )
)  e.  B )
231, 22syl5eqel 2535 . . 3  |-  ( ph  ->  G  e.  B )
241fveq2i 5859 . . . . . . 7  |-  ( D `
 G )  =  ( D `  ( X  .-  ( A `  N ) ) )
25 ply1rem.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
2625, 5, 11deg1xrcl 22459 . . . . . . . . . . 11  |-  ( ( A `  N )  e.  B  ->  ( D `  ( A `  N ) )  e. 
RR* )
2719, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  e.  RR* )
28 0xr 9643 . . . . . . . . . . 11  |-  0  e.  RR*
2928a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR* )
30 1re 9598 . . . . . . . . . . 11  |-  1  e.  RR
31 rexr 9642 . . . . . . . . . . 11  |-  ( 1  e.  RR  ->  1  e.  RR* )
3230, 31mp1i 12 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR* )
3325, 5, 15, 14deg1sclle 22490 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  K )  ->  ( D `  ( A `  N ) )  <_ 
0 )
344, 18, 33syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( D `  ( A `  N )
)  <_  0 )
35 0lt1 10082 . . . . . . . . . . 11  |-  0  <  1
3635a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0  <  1 )
3727, 29, 32, 34, 36xrlelttrd 11373 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( A `  N )
)  <  1 )
38 eqid 2443 . . . . . . . . . . . . . 14  |-  (mulGrp `  P )  =  (mulGrp `  P )
3938, 11mgpbas 17125 . . . . . . . . . . . . 13  |-  B  =  ( Base `  (mulGrp `  P ) )
40 eqid 2443 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4139, 40mulg1 16127 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4213, 41syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4342fveq2d 5860 . . . . . . . . . 10  |-  ( ph  ->  ( D `  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( D `
 X ) )
44 1nn0 10818 . . . . . . . . . . 11  |-  1  e.  NN0
4525, 5, 10, 38, 40deg1pw 22498 . . . . . . . . . . 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 2486 . . . . . . . . 9  |-  ( ph  ->  ( D `  X
)  =  1 )
4837, 47breqtrrd 4463 . . . . . . . 8  |-  ( ph  ->  ( D `  ( A `  N )
)  <  ( D `  X ) )
495, 25, 4, 11, 20, 13, 19, 48deg1sub 22486 . . . . . . 7  |-  ( ph  ->  ( D `  ( X  .-  ( A `  N ) ) )  =  ( D `  X ) )
5024, 49syl5eq 2496 . . . . . 6  |-  ( ph  ->  ( D `  G
)  =  ( D `
 X ) )
5150, 47eqtrd 2484 . . . . 5  |-  ( ph  ->  ( D `  G
)  =  1 )
5251, 44syl6eqel 2539 . . . 4  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
53 eqid 2443 . . . . . 6  |-  ( 0g
`  P )  =  ( 0g `  P
)
5425, 5, 53, 11deg1nn0clb 22467 . . . . 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 5860 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( (coe1 `  G ) `  1
) )
581fveq2i 5859 . . . . . 6  |-  (coe1 `  G
)  =  (coe1 `  ( X  .-  ( A `  N ) ) )
5958fveq1i 5857 . . . . 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 18285 . . . . . 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 1232 . . . . 5  |-  ( ph  ->  ( (coe1 `  ( X  .-  ( A `  N ) ) ) `  1
)  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6459, 63syl5eq 2496 . . . 4  |-  ( ph  ->  ( (coe1 `  G ) ` 
1 )  =  ( ( (coe1 `  X ) ` 
1 ) ( -g `  R ) ( (coe1 `  ( A `  N
) ) `  1
) ) )
6542oveq2d 6297 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  ( ( 1r `  R ) ( .s `  P
) X ) )
665ply1sca 18272 . . . . . . . . . . . . 13  |-  ( R  e. NzRing  ->  R  =  (Scalar `  P ) )
672, 66syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  P ) )
6867fveq2d 5860 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  R
)  =  ( 1r
`  (Scalar `  P )
) )
6968oveq1d 6296 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) X )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) X ) )
705ply1lmod 18271 . . . . . . . . . . . 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 17518 . . . . . . . . . . 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 2488 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  R ) ( .s
`  P ) ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  X )
7877fveq2d 5860 . . . . . . . 8  |-  ( ph  ->  (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) )  =  (coe1 `  X ) )
7978fveq1d 5858 . . . . . . 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, 80ringidcl 17197 . . . . . . . . 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 18293 . . . . . . . 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 1229 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( ( 1r
`  R ) ( .s `  P ) ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( 1r `  R ) )
8679, 85eqtr3d 2486 . . . . . 6  |-  ( ph  ->  ( (coe1 `  X ) ` 
1 )  =  ( 1r `  R ) )
87 eqid 2443 . . . . . . . . . 10  |-  ( 0g
`  R )  =  ( 0g `  R
)
885, 14, 15, 87coe1scl 18306 . . . . . . . . 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 5858 . . . . . . 7  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( ( x  e.  NN0  |->  if ( x  =  0 ,  N ,  ( 0g `  R ) ) ) `  1
) )
91 ax-1ne0 9564 . . . . . . . . . . 11  |-  1  =/=  0
92 neeq1 2724 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  =/=  0  <->  1  =/=  0 ) )
9391, 92mpbiri 233 . . . . . . . . . 10  |-  ( x  =  1  ->  x  =/=  0 )
94 ifnefalse 3938 . . . . . . . . . 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 5866 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
9895, 96, 97fvmpt 5941 . . . . . . . 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 2500 . . . . . 6  |-  ( ph  ->  ( (coe1 `  ( A `  N ) ) ` 
1 )  =  ( 0g `  R ) )
10186, 100oveq12d 6299 . . . . 5  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( ( 1r `  R ) ( -g `  R ) ( 0g
`  R ) ) )
102 ringgrp 17181 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1034, 102syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
10415, 87, 61grpsubid1 16101 . . . . . 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 2484 . . . 4  |-  ( ph  ->  ( ( (coe1 `  X
) `  1 )
( -g `  R ) ( (coe1 `  ( A `  N ) ) ` 
1 ) )  =  ( 1r `  R
) )
10757, 64, 1063eqtrd 2488 . . 3  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  =  ( 1r
`  R ) )
108 ply1rem.u . . . 4  |-  U  =  (Monic1p `  R )
1095, 11, 53, 25, 108, 80ismon1p 22520 . . 3  |-  ( G  e.  U  <->  ( G  e.  B  /\  G  =/=  ( 0g `  P
)  /\  ( (coe1 `  G ) `  ( D `  G )
)  =  ( 1r
`  R ) ) )
11023, 56, 107, 109syl3anbrc 1181 . 2  |-  ( ph  ->  G  e.  U )
1111fveq2i 5859 . . . . . . . . . 10  |-  ( O `
 G )  =  ( O `  ( X  .-  ( A `  N ) ) )
112111fveq1i 5857 . . . . . . . . 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 18351 . . . . . . . . . . 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 18349 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  (
( A `  N
)  e.  B  /\  ( ( O `  ( A `  N ) ) `  x )  =  N ) )
120113, 5, 15, 11, 115, 116, 117, 119, 20, 61evl1subd 18356 . . . . . . . . . 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 2496 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  =  ( x (
-g `  R ) N ) )
123122eqeq1d 2445 . . . . . . 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 16102 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  K  /\  N  e.  K )  ->  ( ( x (
-g `  R ) N )  =  .0.  <->  x  =  N ) )
126124, 116, 118, 125syl3anc 1229 . . . . . . 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 4028 . . . . . 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 5866 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
13415, 133eqeltri 2527 . . . . . . . 8  |-  K  e. 
_V
135134a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
136113, 5, 131, 15evl1rhm 18346 . . . . . . . . . 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 17353 . . . . . . . . 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 6017 . . . . . . 7  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
141131, 15, 132, 2, 135, 140pwselbas 14867 . . . . . 6  |-  ( ph  ->  ( O `  G
) : K --> K )
142 ffn 5721 . . . . . 6  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
143141, 142syl 16 . . . . 5  |-  ( ph  ->  ( O `  G
)  Fn  K )
144 fniniseg 5993 . . . . 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 4160 . . . . . 6  |-  ( ph  ->  { N }  C_  K )
147146sseld 3488 . . . . 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 2440 . 2  |-  ( ph  ->  ( `' ( O `
 G ) " {  .0.  } )  =  { N } )
151110, 51, 1503jca 1177 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095   ifcif 3926   {csn 4014   class class class wbr 4437    |-> cmpt 4495   `'ccnv 4988   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495   1c1 9496   RR*cxr 9630    < clt 9631    <_ cle 9632   NN0cn0 10802   Basecbs 14613  Scalarcsca 14681   .scvsca 14682   0gc0g 14818    ^s cpws 14825   Grpcgrp 16031   -gcsg 16033  .gcmg 16034  mulGrpcmgp 17119   1rcur 17131   Ringcrg 17176   CRingccrg 17177   RingHom crh 17339   LModclmod 17490  NzRingcnzr 17883  algSccascl 17938  var1cv1 18193  Poly1cpl1 18194  coe1cco1 18195  eval1ce1 18329   deg1 cdg1 22429  Monic1pcmn1 22503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-fz 11683  df-fzo 11806  df-seq 12089  df-hash 12387  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-starv 14693  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-unif 14701  df-hom 14702  df-cco 14703  df-0g 14820  df-gsum 14821  df-prds 14826  df-pws 14828  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-mhm 15944  df-submnd 15945  df-grp 16035  df-minusg 16036  df-sbg 16037  df-mulg 16038  df-subg 16176  df-ghm 16243  df-cntz 16333  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-srg 17136  df-ring 17178  df-cring 17179  df-oppr 17250  df-dvdsr 17268  df-unit 17269  df-invr 17299  df-rnghom 17342  df-subrg 17405  df-lmod 17492  df-lss 17557  df-lsp 17596  df-nzr 17884  df-rlreg 17909  df-assa 17939  df-asp 17940  df-ascl 17941  df-psr 17983  df-mvr 17984  df-mpl 17985  df-opsr 17987  df-evls 18149  df-evl 18150  df-psr1 18197  df-vr1 18198  df-ply1 18199  df-coe1 18200  df-evl1 18331  df-cnfld 18399  df-mdeg 22430  df-deg1 22431  df-mon1 22508
This theorem is referenced by:  ply1rem  22541  facth1  22542  fta1glem1  22543  fta1glem2  22544
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