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Theorem ply1rem 19551
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12723). If a polynomial  F is divided by the linear factor  x  -  A, the remainder is equal to  F ( A ), the evaluation of the polynomial at  A (interpreted as a constant polynomial). (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.4  |-  ( ph  ->  F  e.  B )
ply1rem.e  |-  E  =  (rem1p `  R )
Assertion
Ref Expression
ply1rem  |-  ( ph  ->  ( F E G )  =  ( A `
 ( ( O `
 F ) `  N ) ) )

Proof of Theorem ply1rem
StepHypRef Expression
1 ply1rem.1 . . . . . . . . 9  |-  ( ph  ->  R  e. NzRing )
2 nzrrng 16015 . . . . . . . . 9  |-  ( R  e. NzRing  ->  R  e.  Ring )
31, 2syl 15 . . . . . . . 8  |-  ( ph  ->  R  e.  Ring )
4 ply1rem.4 . . . . . . . 8  |-  ( ph  ->  F  e.  B )
5 ply1rem.p . . . . . . . . . . 11  |-  P  =  (Poly1 `  R )
6 ply1rem.b . . . . . . . . . . 11  |-  B  =  ( Base `  P
)
7 ply1rem.k . . . . . . . . . . 11  |-  K  =  ( Base `  R
)
8 ply1rem.x . . . . . . . . . . 11  |-  X  =  (var1 `  R )
9 ply1rem.m . . . . . . . . . . 11  |-  .-  =  ( -g `  P )
10 ply1rem.a . . . . . . . . . . 11  |-  A  =  (algSc `  P )
11 ply1rem.g . . . . . . . . . . 11  |-  G  =  ( X  .-  ( A `  N )
)
12 ply1rem.o . . . . . . . . . . 11  |-  O  =  (eval1 `  R )
13 ply1rem.2 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  CRing )
14 ply1rem.3 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  K )
15 eqid 2285 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
16 eqid 2285 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
17 eqid 2285 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
185, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17ply1remlem 19550 . . . . . . . . . 10  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  (
( deg1  `
 R ) `  G )  =  1  /\  ( `' ( O `  G )
" { ( 0g
`  R ) } )  =  { N } ) )
1918simp1d 967 . . . . . . . . 9  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
20 eqid 2285 . . . . . . . . . 10  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
2120, 15mon1puc1p 19538 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
223, 19, 21syl2anc 642 . . . . . . . 8  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
23 ply1rem.e . . . . . . . . 9  |-  E  =  (rem1p `  R )
2423, 5, 6, 20, 16r1pdeglt 19546 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( ( deg1  `  R
) `  ( F E G ) )  < 
( ( deg1  `  R ) `  G ) )
253, 4, 22, 24syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
( deg1  `
 R ) `  G ) )
2618simp2d 968 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  G )  =  1 )
2725, 26breqtrd 4049 . . . . . 6  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  1
)
28 1e0p1 10154 . . . . . 6  |-  1  =  ( 0  +  1 )
2927, 28syl6breq 4064 . . . . 5  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
0  +  1 ) )
30 0nn0 9982 . . . . . 6  |-  0  e.  NN0
31 nn0leltp1 10077 . . . . . 6  |-  ( ( ( ( deg1  `  R ) `  ( F E G ) )  e.  NN0  /\  0  e.  NN0 )  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0  <->  ( ( deg1  `  R
) `  ( F E G ) )  < 
( 0  +  1 ) ) )
3230, 31mpan2 652 . . . . 5  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  NN0  ->  ( ( ( deg1  `  R ) `  ( F E G ) )  <_  0  <->  ( ( deg1  `  R ) `  ( F E G ) )  <  ( 0  +  1 ) ) )
3329, 32syl5ibrcom 213 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
NN0  ->  ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0 ) )
34 elsni 3666 . . . . . 6  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }  ->  ( ( deg1  `  R
) `  ( F E G ) )  = 
-oo )
35 0xr 8880 . . . . . . 7  |-  0  e.  RR*
36 mnfle 10472 . . . . . . 7  |-  ( 0  e.  RR*  ->  -oo  <_  0 )
3735, 36ax-mp 8 . . . . . 6  |-  -oo  <_  0
3834, 37syl6eqbr 4062 . . . . 5  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }  ->  ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0 )
3938a1i 10 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
{  -oo }  ->  (
( deg1  `
 R ) `  ( F E G ) )  <_  0 ) )
4023, 5, 6, 20r1pcl 19545 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F E G )  e.  B
)
413, 4, 22, 40syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( F E G )  e.  B )
4216, 5, 6deg1cl 19471 . . . . . 6  |-  ( ( F E G )  e.  B  ->  (
( deg1  `
 R ) `  ( F E G ) )  e.  ( NN0 
u.  {  -oo } ) )
4341, 42syl 15 . . . . 5  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  e.  ( NN0  u.  {  -oo } ) )
44 elun 3318 . . . . 5  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  ( NN0 
u.  {  -oo } )  <-> 
( ( ( deg1  `  R
) `  ( F E G ) )  e. 
NN0  \/  ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }
) )
4543, 44sylib 188 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
NN0  \/  ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }
) )
4633, 39, 45mpjaod 370 . . 3  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <_  0
)
4716, 5, 6, 10deg1le0 19499 . . . 4  |-  ( ( R  e.  Ring  /\  ( F E G )  e.  B )  ->  (
( ( deg1  `  R ) `  ( F E G ) )  <_  0  <->  ( F E G )  =  ( A `  ( (coe1 `  ( F E G ) ) ` 
0 ) ) ) )
483, 41, 47syl2anc 642 . . 3  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0  <->  ( F E G )  =  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
4946, 48mpbid 201 . 2  |-  ( ph  ->  ( F E G )  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
50 eqid 2285 . . . . . . . . 9  |-  (quot1p `  R
)  =  (quot1p `  R
)
51 eqid 2285 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
52 eqid 2285 . . . . . . . . 9  |-  ( +g  `  P )  =  ( +g  `  P )
535, 6, 20, 50, 23, 51, 52r1pid 19547 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  F  =  ( ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ( +g  `  P ) ( F E G ) ) )
543, 4, 22, 53syl3anc 1182 . . . . . . 7  |-  ( ph  ->  F  =  ( ( ( F (quot1p `  R
) G ) ( .r `  P ) G ) ( +g  `  P ) ( F E G ) ) )
5554fveq2d 5531 . . . . . 6  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ( +g  `  P
) ( F E G ) ) ) )
56 eqid 2285 . . . . . . . . . 10  |-  ( R  ^s  K )  =  ( R  ^s  K )
5712, 5, 56, 7evl1rhm 19414 . . . . . . . . 9  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
5813, 57syl 15 . . . . . . . 8  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
59 rhmghm 15505 . . . . . . . 8  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O  e.  ( P  GrpHom  ( R  ^s  K )
) )
6058, 59syl 15 . . . . . . 7  |-  ( ph  ->  O  e.  ( P 
GrpHom  ( R  ^s  K ) ) )
615ply1rng 16328 . . . . . . . . 9  |-  ( R  e.  Ring  ->  P  e. 
Ring )
623, 61syl 15 . . . . . . . 8  |-  ( ph  ->  P  e.  Ring )
6350, 5, 6, 20q1pcl 19543 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
643, 4, 22, 63syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
655, 6, 15mon1pcl 19532 . . . . . . . . 9  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
6619, 65syl 15 . . . . . . . 8  |-  ( ph  ->  G  e.  B )
676, 51rngcl 15356 . . . . . . . 8  |-  ( ( P  e.  Ring  /\  ( F (quot1p `  R ) G )  e.  B  /\  G  e.  B )  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  e.  B )
6862, 64, 66, 67syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  e.  B )
69 eqid 2285 . . . . . . . 8  |-  ( +g  `  ( R  ^s  K ) )  =  ( +g  `  ( R  ^s  K ) )
706, 52, 69ghmlin 14690 . . . . . . 7  |-  ( ( O  e.  ( P 
GrpHom  ( R  ^s  K ) )  /\  ( ( F (quot1p `  R ) G ) ( .r `  P ) G )  e.  B  /\  ( F E G )  e.  B )  ->  ( O `  ( (
( F (quot1p `  R
) G ) ( .r `  P ) G ) ( +g  `  P ) ( F E G ) ) )  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) ( +g  `  ( R  ^s  K ) ) ( O `  ( F E G ) ) ) )
7160, 68, 41, 70syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( O `  (
( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ( +g  `  P ) ( F E G ) ) )  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) ( +g  `  ( R  ^s  K ) ) ( O `  ( F E G ) ) ) )
72 eqid 2285 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
73 fvex 5541 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
747, 73eqeltri 2355 . . . . . . . 8  |-  K  e. 
_V
7574a1i 10 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
766, 72rhmf 15506 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
7758, 76syl 15 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
78 ffvelrn 5665 . . . . . . . 8  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  ( ( F (quot1p `  R ) G ) ( .r `  P ) G )  e.  B )  -> 
( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  e.  ( Base `  ( R  ^s  K ) ) )
7977, 68, 78syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  e.  ( Base `  ( R  ^s  K ) ) )
80 ffvelrn 5665 . . . . . . . 8  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  ( F E G )  e.  B )  ->  ( O `  ( F E G ) )  e.  ( Base `  ( R  ^s  K ) ) )
8177, 41, 80syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  e.  ( Base `  ( R  ^s  K ) ) )
82 eqid 2285 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
8356, 72, 1, 75, 79, 81, 82, 69pwsplusgval 13391 . . . . . 6  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) ( +g  `  ( R  ^s  K ) ) ( O `  ( F E G ) ) )  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) )
8455, 71, 833eqtrd 2321 . . . . 5  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) )
8584fveq1d 5529 . . . 4  |-  ( ph  ->  ( ( O `  F ) `  N
)  =  ( ( ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  o F ( +g  `  R
) ( O `  ( F E G ) ) ) `  N
) )
8656, 7, 72, 1, 75, 79pwselbas 13390 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) ) : K --> K )
87 ffn 5391 . . . . . . 7  |-  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) : K --> K  -> 
( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  Fn  K )
8886, 87syl 15 . . . . . 6  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  Fn  K )
8956, 7, 72, 1, 75, 81pwselbas 13390 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) ) : K --> K )
90 ffn 5391 . . . . . . 7  |-  ( ( O `  ( F E G ) ) : K --> K  -> 
( O `  ( F E G ) )  Fn  K )
9189, 90syl 15 . . . . . 6  |-  ( ph  ->  ( O `  ( F E G ) )  Fn  K )
92 fnfvof 6092 . . . . . 6  |-  ( ( ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) )  Fn  K  /\  ( O `  ( F E G ) )  Fn  K )  /\  ( K  e.  _V  /\  N  e.  K ) )  -> 
( ( ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) `  N )  =  ( ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) `  N ) ( +g  `  R
) ( ( O `
 ( F E G ) ) `  N ) ) )
9388, 91, 75, 14, 92syl22anc 1183 . . . . 5  |-  ( ph  ->  ( ( ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) `  N )  =  ( ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) `  N ) ( +g  `  R
) ( ( O `
 ( F E G ) ) `  N ) ) )
94 eqid 2285 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
956, 51, 94rhmmul 15507 . . . . . . . . . 10  |-  ( ( O  e.  ( P RingHom 
( R  ^s  K ) )  /\  ( F (quot1p `  R ) G )  e.  B  /\  G  e.  B )  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
9658, 64, 66, 95syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
97 ffvelrn 5665 . . . . . . . . . . 11  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  ( F (quot1p `  R ) G )  e.  B )  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
9877, 64, 97syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
99 ffvelrn 5665 . . . . . . . . . . 11  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  G  e.  B )  ->  ( O `  G )  e.  ( Base `  ( R  ^s  K ) ) )
10077, 66, 99syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
101 eqid 2285 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
10256, 72, 1, 75, 98, 100, 101, 94pwsmulrval 13392 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) )  =  ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) )
10396, 102eqtrd 2317 . . . . . . . 8  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) )
104103fveq1d 5529 . . . . . . 7  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) `
 N )  =  ( ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) `  N
) )
10556, 7, 72, 1, 75, 98pwselbas 13390 . . . . . . . . 9  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
106 ffn 5391 . . . . . . . . 9  |-  ( ( O `  ( F (quot1p `  R ) G ) ) : K --> K  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
107105, 106syl 15 . . . . . . . 8  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
10856, 7, 72, 1, 75, 100pwselbas 13390 . . . . . . . . 9  |-  ( ph  ->  ( O `  G
) : K --> K )
109 ffn 5391 . . . . . . . . 9  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
110108, 109syl 15 . . . . . . . 8  |-  ( ph  ->  ( O `  G
)  Fn  K )
111 fnfvof 6092 . . . . . . . 8  |-  ( ( ( ( O `  ( F (quot1p `  R ) G ) )  Fn  K  /\  ( O `  G
)  Fn  K )  /\  ( K  e. 
_V  /\  N  e.  K ) )  -> 
( ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) `  N
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  N
) ( .r `  R ) ( ( O `  G ) `
 N ) ) )
112107, 110, 75, 14, 111syl22anc 1183 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) `  N
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  N
) ( .r `  R ) ( ( O `  G ) `
 N ) ) )
113 snidg 3667 . . . . . . . . . . . . 13  |-  ( N  e.  K  ->  N  e.  { N } )
11414, 113syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  { N } )
11518simp3d 969 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( O `
 G ) " { ( 0g `  R ) } )  =  { N }
)
116114, 115eleqtrrd 2362 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( `' ( O `  G
) " { ( 0g `  R ) } ) )
117 fniniseg 5648 . . . . . . . . . . . 12  |-  ( ( O `  G )  Fn  K  ->  ( N  e.  ( `' ( O `  G )
" { ( 0g
`  R ) } )  <->  ( N  e.  K  /\  ( ( O `  G ) `
 N )  =  ( 0g `  R
) ) ) )
118110, 117syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  ( `' ( O `  G ) " {
( 0g `  R
) } )  <->  ( N  e.  K  /\  (
( O `  G
) `  N )  =  ( 0g `  R ) ) ) )
119116, 118mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( N  e.  K  /\  ( ( O `  G ) `  N
)  =  ( 0g
`  R ) ) )
120119simprd 449 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  G ) `  N
)  =  ( 0g
`  R ) )
121120oveq2d 5876 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( ( O `
 G ) `  N ) )  =  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( 0g `  R ) ) )
122 ffvelrn 5665 . . . . . . . . . 10  |-  ( ( ( O `  ( F (quot1p `  R ) G ) ) : K --> K  /\  N  e.  K
)  ->  ( ( O `  ( F
(quot1p `
 R ) G ) ) `  N
)  e.  K )
123105, 14, 122syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) `  N
)  e.  K )
1247, 101, 17rngrz 15380 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
( O `  ( F (quot1p `  R ) G ) ) `  N
)  e.  K )  ->  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  N
) ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
1253, 123, 124syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( 0g `  R ) )  =  ( 0g `  R
) )
126121, 125eqtrd 2317 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( ( O `
 G ) `  N ) )  =  ( 0g `  R
) )
127104, 112, 1263eqtrd 2321 . . . . . 6  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) `
 N )  =  ( 0g `  R
) )
128127oveq1d 5875 . . . . 5  |-  ( ph  ->  ( ( ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) `  N ) ( +g  `  R
) ( ( O `
 ( F E G ) ) `  N ) )  =  ( ( 0g `  R ) ( +g  `  R ) ( ( O `  ( F E G ) ) `
 N ) ) )
129 rnggrp 15348 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1303, 129syl 15 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
131 ffvelrn 5665 . . . . . . 7  |-  ( ( ( O `  ( F E G ) ) : K --> K  /\  N  e.  K )  ->  ( ( O `  ( F E G ) ) `  N )  e.  K )
13289, 14, 131syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  e.  K )
1337, 82, 17grplid 14514 . . . . . 6  |-  ( ( R  e.  Grp  /\  ( ( O `  ( F E G ) ) `  N )  e.  K )  -> 
( ( 0g `  R ) ( +g  `  R ) ( ( O `  ( F E G ) ) `
 N ) )  =  ( ( O `
 ( F E G ) ) `  N ) )
134130, 132, 133syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( 0g `  R ) ( +g  `  R ) ( ( O `  ( F E G ) ) `
 N ) )  =  ( ( O `
 ( F E G ) ) `  N ) )
13593, 128, 1343eqtrd 2321 . . . 4  |-  ( ph  ->  ( ( ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) `  N )  =  ( ( O `
 ( F E G ) ) `  N ) )
13649fveq2d 5531 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
137 eqid 2285 . . . . . . . . . . 11  |-  (coe1 `  ( F E G ) )  =  (coe1 `  ( F E G ) )
138137, 6, 5, 7coe1f 16294 . . . . . . . . . 10  |-  ( ( F E G )  e.  B  ->  (coe1 `  ( F E G ) ) : NN0 --> K )
13941, 138syl 15 . . . . . . . . 9  |-  ( ph  ->  (coe1 `  ( F E G ) ) : NN0 --> K )
140 ffvelrn 5665 . . . . . . . . 9  |-  ( ( (coe1 `  ( F E G ) ) : NN0 --> K  /\  0  e.  NN0 )  ->  (
(coe1 `  ( F E G ) ) ` 
0 )  e.  K
)
141139, 30, 140sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( (coe1 `  ( F E G ) ) ` 
0 )  e.  K
)
14212, 5, 7, 10evl1sca 19415 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  (
(coe1 `  ( F E G ) ) ` 
0 )  e.  K
)  ->  ( O `  ( A `  (
(coe1 `  ( F E G ) ) ` 
0 ) ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) )
14313, 141, 142syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0 ) ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) ` 
0 ) } ) )
144136, 143eqtrd 2317 . . . . . 6  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) )
145144fveq1d 5529 . . . . 5  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( ( K  X.  { ( (coe1 `  ( F E G ) ) `  0
) } ) `  N ) )
146 fvex 5541 . . . . . . 7  |-  ( (coe1 `  ( F E G ) ) `  0
)  e.  _V
147146fvconst2 5731 . . . . . 6  |-  ( N  e.  K  ->  (
( K  X.  {
( (coe1 `  ( F E G ) ) ` 
0 ) } ) `
 N )  =  ( (coe1 `  ( F E G ) ) ` 
0 ) )
14814, 147syl 15 . . . . 5  |-  ( ph  ->  ( ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) `  N )  =  ( (coe1 `  ( F E G ) ) `
 0 ) )
149145, 148eqtrd 2317 . . . 4  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( (coe1 `  ( F E G ) ) `
 0 ) )
15085, 135, 1493eqtrd 2321 . . 3  |-  ( ph  ->  ( ( O `  F ) `  N
)  =  ( (coe1 `  ( F E G ) ) `  0
) )
151150fveq2d 5531 . 2  |-  ( ph  ->  ( A `  (
( O `  F
) `  N )
)  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
15249, 151eqtr4d 2320 1  |-  ( ph  ->  ( F E G )  =  ( A `
 ( ( O `
 F ) `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686   _Vcvv 2790    u. cun 3152   {csn 3642   class class class wbr 4025    X. cxp 4689   `'ccnv 4690   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860    o Fcof 6078   0cc0 8739   1c1 8740    + caddc 8742    -oocmnf 8867   RR*cxr 8868    < clt 8869    <_ cle 8870   NN0cn0 9967   Basecbs 13150   +g cplusg 13210   .rcmulr 13211    ^s cpws 13349   0gc0g 13402   Grpcgrp 14364   -gcsg 14367    GrpHom cghm 14682   Ringcrg 15339   CRingccrg 15340   RingHom crh 15496  NzRingcnzr 16011  algSccascl 16054  var1cv1 16253  Poly1cpl1 16254  eval1ce1 16256  coe1cco1 16257   deg1 cdg1 19442  Monic1pcmn1 19513  Unic1pcuc1p 19514  quot1pcq1p 19515  rem1pcr1p 19516
This theorem is referenced by:  facth1  19552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-ofr 6081  df-1st 6124  df-2nd 6125  df-tpos 6236  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-fz 10785  df-fzo 10873  df-seq 11049  df-hash 11340  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-prds 13350  df-pws 13352  df-0g 13406  df-gsum 13407  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-mhm 14417  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mulg 14494  df-subg 14620  df-ghm 14683  df-cntz 14795  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-cring 15343  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-rnghom 15498  df-subrg 15545  df-lmod 15631  df-lss 15692  df-lsp 15731  df-nzr 16012  df-rlreg 16026  df-assa 16055  df-asp 16056  df-ascl 16057  df-psr 16100  df-mvr 16101  df-mpl 16102  df-evls 16103  df-evl 16104  df-opsr 16108  df-psr1 16259  df-vr1 16260  df-ply1 16261  df-evl1 16263  df-coe1 16264  df-cnfld 16380  df-mdeg 19443  df-deg1 19444  df-mon1 19518  df-uc1p 19519  df-q1p 19520  df-r1p 19521
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