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Theorem ply1rem 20087
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 13043). 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 16333 . . . . . . . . 9  |-  ( R  e. NzRing  ->  R  e.  Ring )
31, 2syl 16 . . . . . . . 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 2437 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
16 eqid 2437 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
17 eqid 2437 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
185, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17ply1remlem 20086 . . . . . . . . . 10  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  (
( deg1  `
 R ) `  G )  =  1  /\  ( `' ( O `  G )
" { ( 0g
`  R ) } )  =  { N } ) )
1918simp1d 970 . . . . . . . . 9  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
20 eqid 2437 . . . . . . . . . 10  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
2120, 15mon1puc1p 20074 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
223, 19, 21syl2anc 644 . . . . . . . 8  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
23 ply1rem.e . . . . . . . . 9  |-  E  =  (rem1p `  R )
2423, 5, 6, 20, 16r1pdeglt 20082 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( ( deg1  `  R
) `  ( F E G ) )  < 
( ( deg1  `  R ) `  G ) )
253, 4, 22, 24syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
( deg1  `
 R ) `  G ) )
2618simp2d 971 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  G )  =  1 )
2725, 26breqtrd 4237 . . . . . 6  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  1
)
28 1e0p1 10411 . . . . . 6  |-  1  =  ( 0  +  1 )
2927, 28syl6breq 4252 . . . . 5  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
0  +  1 ) )
30 0nn0 10237 . . . . . 6  |-  0  e.  NN0
31 nn0leltp1 10334 . . . . . 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 654 . . . . 5  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  NN0  ->  ( ( ( deg1  `  R ) `  ( F E G ) )  <_  0  <->  ( ( deg1  `  R ) `  ( F E G ) )  <  ( 0  +  1 ) ) )
3329, 32syl5ibrcom 215 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
NN0  ->  ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0 ) )
34 elsni 3839 . . . . . 6  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }  ->  ( ( deg1  `  R
) `  ( F E G ) )  = 
-oo )
35 0xr 9132 . . . . . . 7  |-  0  e.  RR*
36 mnfle 10730 . . . . . . 7  |-  ( 0  e.  RR*  ->  -oo  <_  0 )
3735, 36ax-mp 8 . . . . . 6  |-  -oo  <_  0
3834, 37syl6eqbr 4250 . . . . 5  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }  ->  ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0 )
3938a1i 11 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
{  -oo }  ->  (
( deg1  `
 R ) `  ( F E G ) )  <_  0 ) )
4023, 5, 6, 20r1pcl 20081 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F E G )  e.  B
)
413, 4, 22, 40syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( F E G )  e.  B )
4216, 5, 6deg1cl 20007 . . . . . 6  |-  ( ( F E G )  e.  B  ->  (
( deg1  `
 R ) `  ( F E G ) )  e.  ( NN0 
u.  {  -oo } ) )
4341, 42syl 16 . . . . 5  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  e.  ( NN0  u.  {  -oo } ) )
44 elun 3489 . . . . 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 190 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
NN0  \/  ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }
) )
4633, 39, 45mpjaod 372 . . 3  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <_  0
)
4716, 5, 6, 10deg1le0 20035 . . . 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 644 . . 3  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0  <->  ( F E G )  =  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
4946, 48mpbid 203 . 2  |-  ( ph  ->  ( F E G )  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
50 eqid 2437 . . . . . . . . 9  |-  (quot1p `  R
)  =  (quot1p `  R
)
51 eqid 2437 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
52 eqid 2437 . . . . . . . . 9  |-  ( +g  `  P )  =  ( +g  `  P )
535, 6, 20, 50, 23, 51, 52r1pid 20083 . . . . . . . 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 1185 . . . . . . 7  |-  ( ph  ->  F  =  ( ( ( F (quot1p `  R
) G ) ( .r `  P ) G ) ( +g  `  P ) ( F E G ) ) )
5554fveq2d 5733 . . . . . 6  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ( +g  `  P
) ( F E G ) ) ) )
56 eqid 2437 . . . . . . . . . 10  |-  ( R  ^s  K )  =  ( R  ^s  K )
5712, 5, 56, 7evl1rhm 19950 . . . . . . . . 9  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
5813, 57syl 16 . . . . . . . 8  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
59 rhmghm 15827 . . . . . . . 8  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O  e.  ( P  GrpHom  ( R  ^s  K )
) )
6058, 59syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  ( P 
GrpHom  ( R  ^s  K ) ) )
615ply1rng 16643 . . . . . . . . 9  |-  ( R  e.  Ring  ->  P  e. 
Ring )
623, 61syl 16 . . . . . . . 8  |-  ( ph  ->  P  e.  Ring )
6350, 5, 6, 20q1pcl 20079 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
643, 4, 22, 63syl3anc 1185 . . . . . . . 8  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
655, 6, 15mon1pcl 20068 . . . . . . . . 9  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
6619, 65syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  B )
676, 51rngcl 15678 . . . . . . . 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 1185 . . . . . . 7  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  e.  B )
69 eqid 2437 . . . . . . . 8  |-  ( +g  `  ( R  ^s  K ) )  =  ( +g  `  ( R  ^s  K ) )
706, 52, 69ghmlin 15012 . . . . . . 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 1185 . . . . . 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 2437 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
73 fvex 5743 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
747, 73eqeltri 2507 . . . . . . . 8  |-  K  e. 
_V
7574a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
766, 72rhmf 15828 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
7758, 76syl 16 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
7877, 68ffvelrnd 5872 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  e.  ( Base `  ( R  ^s  K ) ) )
7977, 41ffvelrnd 5872 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  e.  ( Base `  ( R  ^s  K ) ) )
80 eqid 2437 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
8156, 72, 1, 75, 78, 79, 80, 69pwsplusgval 13713 . . . . . 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 ) ) ) )
8255, 71, 813eqtrd 2473 . . . . 5  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) )
8382fveq1d 5731 . . . 4  |-  ( ph  ->  ( ( O `  F ) `  N
)  =  ( ( ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  o F ( +g  `  R
) ( O `  ( F E G ) ) ) `  N
) )
8456, 7, 72, 1, 75, 78pwselbas 13712 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) ) : K --> K )
85 ffn 5592 . . . . . . 7  |-  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) : K --> K  -> 
( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  Fn  K )
8684, 85syl 16 . . . . . 6  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  Fn  K )
8756, 7, 72, 1, 75, 79pwselbas 13712 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) ) : K --> K )
88 ffn 5592 . . . . . . 7  |-  ( ( O `  ( F E G ) ) : K --> K  -> 
( O `  ( F E G ) )  Fn  K )
8987, 88syl 16 . . . . . 6  |-  ( ph  ->  ( O `  ( F E G ) )  Fn  K )
90 fnfvof 6318 . . . . . 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 ) ) )
9186, 89, 75, 14, 90syl22anc 1186 . . . . 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 ) ) )
92 eqid 2437 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
936, 51, 92rhmmul 15829 . . . . . . . . . 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 ) ) )
9458, 64, 66, 93syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
9577, 64ffvelrnd 5872 . . . . . . . . . 10  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
9677, 66ffvelrnd 5872 . . . . . . . . . 10  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
97 eqid 2437 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
9856, 72, 1, 75, 95, 96, 97, 92pwsmulrval 13714 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) )  =  ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) )
9994, 98eqtrd 2469 . . . . . . . 8  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) )
10099fveq1d 5731 . . . . . . 7  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) `
 N )  =  ( ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) `  N
) )
10156, 7, 72, 1, 75, 95pwselbas 13712 . . . . . . . . 9  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
102 ffn 5592 . . . . . . . . 9  |-  ( ( O `  ( F (quot1p `  R ) G ) ) : K --> K  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
103101, 102syl 16 . . . . . . . 8  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
10456, 7, 72, 1, 75, 96pwselbas 13712 . . . . . . . . 9  |-  ( ph  ->  ( O `  G
) : K --> K )
105 ffn 5592 . . . . . . . . 9  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
106104, 105syl 16 . . . . . . . 8  |-  ( ph  ->  ( O `  G
)  Fn  K )
107 fnfvof 6318 . . . . . . . 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 ) ) )
108103, 106, 75, 14, 107syl22anc 1186 . . . . . . 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 ) ) )
109 snidg 3840 . . . . . . . . . . . . 13  |-  ( N  e.  K  ->  N  e.  { N } )
11014, 109syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  { N } )
11118simp3d 972 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( O `
 G ) " { ( 0g `  R ) } )  =  { N }
)
112110, 111eleqtrrd 2514 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( `' ( O `  G
) " { ( 0g `  R ) } ) )
113 fniniseg 5852 . . . . . . . . . . . 12  |-  ( ( O `  G )  Fn  K  ->  ( N  e.  ( `' ( O `  G )
" { ( 0g
`  R ) } )  <->  ( N  e.  K  /\  ( ( O `  G ) `
 N )  =  ( 0g `  R
) ) ) )
114106, 113syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  ( `' ( O `  G ) " {
( 0g `  R
) } )  <->  ( N  e.  K  /\  (
( O `  G
) `  N )  =  ( 0g `  R ) ) ) )
115112, 114mpbid 203 . . . . . . . . . 10  |-  ( ph  ->  ( N  e.  K  /\  ( ( O `  G ) `  N
)  =  ( 0g
`  R ) ) )
116115simprd 451 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  G ) `  N
)  =  ( 0g
`  R ) )
117116oveq2d 6098 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( ( O `
 G ) `  N ) )  =  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( 0g `  R ) ) )
118101, 14ffvelrnd 5872 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) `  N
)  e.  K )
1197, 97, 17rngrz 15702 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
( O `  ( F (quot1p `  R ) G ) ) `  N
)  e.  K )  ->  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  N
) ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
1203, 118, 119syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( 0g `  R ) )  =  ( 0g `  R
) )
121117, 120eqtrd 2469 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( ( O `
 G ) `  N ) )  =  ( 0g `  R
) )
122100, 108, 1213eqtrd 2473 . . . . . 6  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) `
 N )  =  ( 0g `  R
) )
123122oveq1d 6097 . . . . 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 ) ) )
124 rnggrp 15670 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1253, 124syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
12687, 14ffvelrnd 5872 . . . . . 6  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  e.  K )
1277, 80, 17grplid 14836 . . . . . 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 ) )
128125, 126, 127syl2anc 644 . . . . 5  |-  ( ph  ->  ( ( 0g `  R ) ( +g  `  R ) ( ( O `  ( F E G ) ) `
 N ) )  =  ( ( O `
 ( F E G ) ) `  N ) )
12991, 123, 1283eqtrd 2473 . . . 4  |-  ( ph  ->  ( ( ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) `  N )  =  ( ( O `
 ( F E G ) ) `  N ) )
13049fveq2d 5733 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
131 eqid 2437 . . . . . . . . . . 11  |-  (coe1 `  ( F E G ) )  =  (coe1 `  ( F E G ) )
132131, 6, 5, 7coe1f 16610 . . . . . . . . . 10  |-  ( ( F E G )  e.  B  ->  (coe1 `  ( F E G ) ) : NN0 --> K )
13341, 132syl 16 . . . . . . . . 9  |-  ( ph  ->  (coe1 `  ( F E G ) ) : NN0 --> K )
134 ffvelrn 5869 . . . . . . . . 9  |-  ( ( (coe1 `  ( F E G ) ) : NN0 --> K  /\  0  e.  NN0 )  ->  (
(coe1 `  ( F E G ) ) ` 
0 )  e.  K
)
135133, 30, 134sylancl 645 . . . . . . . 8  |-  ( ph  ->  ( (coe1 `  ( F E G ) ) ` 
0 )  e.  K
)
13612, 5, 7, 10evl1sca 19951 . . . . . . . 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 ) } ) )
13713, 135, 136syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0 ) ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) ` 
0 ) } ) )
138130, 137eqtrd 2469 . . . . . 6  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) )
139138fveq1d 5731 . . . . 5  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( ( K  X.  { ( (coe1 `  ( F E G ) ) `  0
) } ) `  N ) )
140 fvex 5743 . . . . . . 7  |-  ( (coe1 `  ( F E G ) ) `  0
)  e.  _V
141140fvconst2 5948 . . . . . 6  |-  ( N  e.  K  ->  (
( K  X.  {
( (coe1 `  ( F E G ) ) ` 
0 ) } ) `
 N )  =  ( (coe1 `  ( F E G ) ) ` 
0 ) )
14214, 141syl 16 . . . . 5  |-  ( ph  ->  ( ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) `  N )  =  ( (coe1 `  ( F E G ) ) `
 0 ) )
143139, 142eqtrd 2469 . . . 4  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( (coe1 `  ( F E G ) ) `
 0 ) )
14483, 129, 1433eqtrd 2473 . . 3  |-  ( ph  ->  ( ( O `  F ) `  N
)  =  ( (coe1 `  ( F E G ) ) `  0
) )
145144fveq2d 5733 . 2  |-  ( ph  ->  ( A `  (
( O `  F
) `  N )
)  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
14649, 145eqtr4d 2472 1  |-  ( ph  ->  ( F E G )  =  ( A `
 ( ( O `
 F ) `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2957    u. cun 3319   {csn 3815   class class class wbr 4213    X. cxp 4877   `'ccnv 4878   "cima 4882    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082    o Fcof 6304   0cc0 8991   1c1 8992    + caddc 8994    -oocmnf 9119   RR*cxr 9120    < clt 9121    <_ cle 9122   NN0cn0 10222   Basecbs 13470   +g cplusg 13530   .rcmulr 13531    ^s cpws 13671   0gc0g 13724   Grpcgrp 14686   -gcsg 14689    GrpHom cghm 15004   Ringcrg 15661   CRingccrg 15662   RingHom crh 15818  NzRingcnzr 16329  algSccascl 16372  var1cv1 16571  Poly1cpl1 16572  eval1ce1 16574  coe1cco1 16575   deg1 cdg1 19978  Monic1pcmn1 20049  Unic1pcuc1p 20050  quot1pcq1p 20051  rem1pcr1p 20052
This theorem is referenced by:  facth1  20088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-ofr 6307  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-fz 11045  df-fzo 11137  df-seq 11325  df-hash 11620  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-hom 13554  df-cco 13555  df-prds 13672  df-pws 13674  df-0g 13728  df-gsum 13729  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-mhm 14739  df-submnd 14740  df-grp 14813  df-minusg 14814  df-sbg 14815  df-mulg 14816  df-subg 14942  df-ghm 15005  df-cntz 15117  df-cmn 15415  df-abl 15416  df-mgp 15650  df-rng 15664  df-cring 15665  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-rnghom 15820  df-subrg 15867  df-lmod 15953  df-lss 16010  df-lsp 16049  df-nzr 16330  df-rlreg 16344  df-assa 16373  df-asp 16374  df-ascl 16375  df-psr 16418  df-mvr 16419  df-mpl 16420  df-evls 16421  df-evl 16422  df-opsr 16426  df-psr1 16577  df-vr1 16578  df-ply1 16579  df-evl1 16581  df-coe1 16582  df-cnfld 16705  df-mdeg 19979  df-deg1 19980  df-mon1 20054  df-uc1p 20055  df-q1p 20056  df-r1p 20057
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