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Theorem ply1rem 19764
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12929). 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 16223 . . . . . . . . 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 2366 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
16 eqid 2366 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
17 eqid 2366 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
185, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17ply1remlem 19763 . . . . . . . . . 10  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  (
( deg1  `
 R ) `  G )  =  1  /\  ( `' ( O `  G )
" { ( 0g
`  R ) } )  =  { N } ) )
1918simp1d 968 . . . . . . . . 9  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
20 eqid 2366 . . . . . . . . . 10  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
2120, 15mon1puc1p 19751 . . . . . . . . 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 19759 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( ( deg1  `  R
) `  ( F E G ) )  < 
( ( deg1  `  R ) `  G ) )
253, 4, 22, 24syl3anc 1183 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
( deg1  `
 R ) `  G ) )
2618simp2d 969 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  G )  =  1 )
2725, 26breqtrd 4149 . . . . . 6  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  1
)
28 1e0p1 10303 . . . . . 6  |-  1  =  ( 0  +  1 )
2927, 28syl6breq 4164 . . . . 5  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
0  +  1 ) )
30 0nn0 10129 . . . . . 6  |-  0  e.  NN0
31 nn0leltp1 10226 . . . . . 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 3753 . . . . . 6  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }  ->  ( ( deg1  `  R
) `  ( F E G ) )  = 
-oo )
35 0xr 9025 . . . . . . 7  |-  0  e.  RR*
36 mnfle 10622 . . . . . . 7  |-  ( 0  e.  RR*  ->  -oo  <_  0 )
3735, 36ax-mp 8 . . . . . 6  |-  -oo  <_  0
3834, 37syl6eqbr 4162 . . . . 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 19758 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F E G )  e.  B
)
413, 4, 22, 40syl3anc 1183 . . . . . 6  |-  ( ph  ->  ( F E G )  e.  B )
4216, 5, 6deg1cl 19684 . . . . . 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 3404 . . . . 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 19712 . . . 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 2366 . . . . . . . . 9  |-  (quot1p `  R
)  =  (quot1p `  R
)
51 eqid 2366 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
52 eqid 2366 . . . . . . . . 9  |-  ( +g  `  P )  =  ( +g  `  P )
535, 6, 20, 50, 23, 51, 52r1pid 19760 . . . . . . . 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 1183 . . . . . . 7  |-  ( ph  ->  F  =  ( ( ( F (quot1p `  R
) G ) ( .r `  P ) G ) ( +g  `  P ) ( F E G ) ) )
5554fveq2d 5636 . . . . . 6  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ( +g  `  P
) ( F E G ) ) ) )
56 eqid 2366 . . . . . . . . . 10  |-  ( R  ^s  K )  =  ( R  ^s  K )
5712, 5, 56, 7evl1rhm 19627 . . . . . . . . 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 15713 . . . . . . . 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 16536 . . . . . . . . 9  |-  ( R  e.  Ring  ->  P  e. 
Ring )
623, 61syl 15 . . . . . . . 8  |-  ( ph  ->  P  e.  Ring )
6350, 5, 6, 20q1pcl 19756 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
643, 4, 22, 63syl3anc 1183 . . . . . . . 8  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
655, 6, 15mon1pcl 19745 . . . . . . . . 9  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
6619, 65syl 15 . . . . . . . 8  |-  ( ph  ->  G  e.  B )
676, 51rngcl 15564 . . . . . . . 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 1183 . . . . . . 7  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  e.  B )
69 eqid 2366 . . . . . . . 8  |-  ( +g  `  ( R  ^s  K ) )  =  ( +g  `  ( R  ^s  K ) )
706, 52, 69ghmlin 14898 . . . . . . 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 1183 . . . . . 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 2366 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
73 fvex 5646 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
747, 73eqeltri 2436 . . . . . . . 8  |-  K  e. 
_V
7574a1i 10 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
766, 72rhmf 15714 . . . . . . . . 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 5770 . . . . . . . 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 5770 . . . . . . . 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 2366 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
8356, 72, 1, 75, 79, 81, 82, 69pwsplusgval 13599 . . . . . 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 2402 . . . . 5  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) )
8584fveq1d 5634 . . . 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 13598 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) ) : K --> K )
87 ffn 5495 . . . . . . 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 13598 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) ) : K --> K )
90 ffn 5495 . . . . . . 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 6217 . . . . . 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 1184 . . . . 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 2366 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
956, 51, 94rhmmul 15715 . . . . . . . . . 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 1183 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
97 ffvelrn 5770 . . . . . . . . . . 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 5770 . . . . . . . . . . 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 2366 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
10256, 72, 1, 75, 98, 100, 101, 94pwsmulrval 13600 . . . . . . . . 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 2398 . . . . . . . 8  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) )
104103fveq1d 5634 . . . . . . 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 13598 . . . . . . . . 9  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
106 ffn 5495 . . . . . . . . 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 13598 . . . . . . . . 9  |-  ( ph  ->  ( O `  G
) : K --> K )
109 ffn 5495 . . . . . . . . 9  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
110108, 109syl 15 . . . . . . . 8  |-  ( ph  ->  ( O `  G
)  Fn  K )
111 fnfvof 6217 . . . . . . . 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 1184 . . . . . . 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 3754 . . . . . . . . . . . . 13  |-  ( N  e.  K  ->  N  e.  { N } )
11414, 113syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  { N } )
11518simp3d 970 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( O `
 G ) " { ( 0g `  R ) } )  =  { N }
)
116114, 115eleqtrrd 2443 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( `' ( O `  G
) " { ( 0g `  R ) } ) )
117 fniniseg 5753 . . . . . . . . . . . 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 5997 . . . . . . . 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 5770 . . . . . . . . . 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 15588 . . . . . . . . 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 2398 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( ( O `
 G ) `  N ) )  =  ( 0g `  R
) )
127104, 112, 1263eqtrd 2402 . . . . . 6  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) `
 N )  =  ( 0g `  R
) )
128127oveq1d 5996 . . . . 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 15556 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1303, 129syl 15 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
131 ffvelrn 5770 . . . . . . 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 14722 . . . . . 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 2402 . . . 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 5636 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
137 eqid 2366 . . . . . . . . . . 11  |-  (coe1 `  ( F E G ) )  =  (coe1 `  ( F E G ) )
138137, 6, 5, 7coe1f 16502 . . . . . . . . . 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 5770 . . . . . . . . 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 19628 . . . . . . . 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 2398 . . . . . 6  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) )
145144fveq1d 5634 . . . . 5  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( ( K  X.  { ( (coe1 `  ( F E G ) ) `  0
) } ) `  N ) )
146 fvex 5646 . . . . . . 7  |-  ( (coe1 `  ( F E G ) ) `  0
)  e.  _V
147146fvconst2 5847 . . . . . 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 2398 . . . 4  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( (coe1 `  ( F E G ) ) `
 0 ) )
15085, 135, 1493eqtrd 2402 . . 3  |-  ( ph  ->  ( ( O `  F ) `  N
)  =  ( (coe1 `  ( F E G ) ) `  0
) )
151150fveq2d 5636 . 2  |-  ( ph  ->  ( A `  (
( O `  F
) `  N )
)  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
15249, 151eqtr4d 2401 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 1647    e. wcel 1715   _Vcvv 2873    u. cun 3236   {csn 3729   class class class wbr 4125    X. cxp 4790   `'ccnv 4791   "cima 4795    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981    o Fcof 6203   0cc0 8884   1c1 8885    + caddc 8887    -oocmnf 9012   RR*cxr 9013    < clt 9014    <_ cle 9015   NN0cn0 10114   Basecbs 13356   +g cplusg 13416   .rcmulr 13417    ^s cpws 13557   0gc0g 13610   Grpcgrp 14572   -gcsg 14575    GrpHom cghm 14890   Ringcrg 15547   CRingccrg 15548   RingHom crh 15704  NzRingcnzr 16219  algSccascl 16262  var1cv1 16461  Poly1cpl1 16462  eval1ce1 16464  coe1cco1 16465   deg1 cdg1 19655  Monic1pcmn1 19726  Unic1pcuc1p 19727  quot1pcq1p 19728  rem1pcr1p 19729
This theorem is referenced by:  facth1  19765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-ofr 6206  df-1st 6249  df-2nd 6250  df-tpos 6376  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-fzo 11026  df-seq 11211  df-hash 11506  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-prds 13558  df-pws 13560  df-0g 13614  df-gsum 13615  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-mhm 14625  df-submnd 14626  df-grp 14699  df-minusg 14700  df-sbg 14701  df-mulg 14702  df-subg 14828  df-ghm 14891  df-cntz 15003  df-cmn 15301  df-abl 15302  df-mgp 15536  df-rng 15550  df-cring 15551  df-ur 15552  df-oppr 15615  df-dvdsr 15633  df-unit 15634  df-invr 15664  df-rnghom 15706  df-subrg 15753  df-lmod 15839  df-lss 15900  df-lsp 15939  df-nzr 16220  df-rlreg 16234  df-assa 16263  df-asp 16264  df-ascl 16265  df-psr 16308  df-mvr 16309  df-mpl 16310  df-evls 16311  df-evl 16312  df-opsr 16316  df-psr1 16467  df-vr1 16468  df-ply1 16469  df-evl1 16471  df-coe1 16472  df-cnfld 16594  df-mdeg 19656  df-deg1 19657  df-mon1 19731  df-uc1p 19732  df-q1p 19733  df-r1p 19734
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