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Theorem ply1rem 20039
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12997). 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 16287 . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
16 eqid 2404 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
17 eqid 2404 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
185, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17ply1remlem 20038 . . . . . . . . . 10  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  (
( deg1  `
 R ) `  G )  =  1  /\  ( `' ( O `  G )
" { ( 0g
`  R ) } )  =  { N } ) )
1918simp1d 969 . . . . . . . . 9  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
20 eqid 2404 . . . . . . . . . 10  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
2120, 15mon1puc1p 20026 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
223, 19, 21syl2anc 643 . . . . . . . 8  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
23 ply1rem.e . . . . . . . . 9  |-  E  =  (rem1p `  R )
2423, 5, 6, 20, 16r1pdeglt 20034 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( ( deg1  `  R
) `  ( F E G ) )  < 
( ( deg1  `  R ) `  G ) )
253, 4, 22, 24syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
( deg1  `
 R ) `  G ) )
2618simp2d 970 . . . . . . 7  |-  ( ph  ->  ( ( deg1  `  R ) `  G )  =  1 )
2725, 26breqtrd 4196 . . . . . 6  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  1
)
28 1e0p1 10366 . . . . . 6  |-  1  =  ( 0  +  1 )
2927, 28syl6breq 4211 . . . . 5  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
0  +  1 ) )
30 0nn0 10192 . . . . . 6  |-  0  e.  NN0
31 nn0leltp1 10289 . . . . . 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 653 . . . . 5  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  NN0  ->  ( ( ( deg1  `  R ) `  ( F E G ) )  <_  0  <->  ( ( deg1  `  R ) `  ( F E G ) )  <  ( 0  +  1 ) ) )
3329, 32syl5ibrcom 214 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
NN0  ->  ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0 ) )
34 elsni 3798 . . . . . 6  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }  ->  ( ( deg1  `  R
) `  ( F E G ) )  = 
-oo )
35 0xr 9087 . . . . . . 7  |-  0  e.  RR*
36 mnfle 10685 . . . . . . 7  |-  ( 0  e.  RR*  ->  -oo  <_  0 )
3735, 36ax-mp 8 . . . . . 6  |-  -oo  <_  0
3834, 37syl6eqbr 4209 . . . . 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 20033 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F E G )  e.  B
)
413, 4, 22, 40syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( F E G )  e.  B )
4216, 5, 6deg1cl 19959 . . . . . 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 3448 . . . . 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 189 . . . 4  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  e. 
NN0  \/  ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }
) )
4633, 39, 45mpjaod 371 . . 3  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <_  0
)
4716, 5, 6, 10deg1le0 19987 . . . 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 643 . . 3  |-  ( ph  ->  ( ( ( deg1  `  R
) `  ( F E G ) )  <_ 
0  <->  ( F E G )  =  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
4946, 48mpbid 202 . 2  |-  ( ph  ->  ( F E G )  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
50 eqid 2404 . . . . . . . . 9  |-  (quot1p `  R
)  =  (quot1p `  R
)
51 eqid 2404 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
52 eqid 2404 . . . . . . . . 9  |-  ( +g  `  P )  =  ( +g  `  P )
535, 6, 20, 50, 23, 51, 52r1pid 20035 . . . . . . . 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 1184 . . . . . . 7  |-  ( ph  ->  F  =  ( ( ( F (quot1p `  R
) G ) ( .r `  P ) G ) ( +g  `  P ) ( F E G ) ) )
5554fveq2d 5691 . . . . . 6  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ( +g  `  P
) ( F E G ) ) ) )
56 eqid 2404 . . . . . . . . . 10  |-  ( R  ^s  K )  =  ( R  ^s  K )
5712, 5, 56, 7evl1rhm 19902 . . . . . . . . 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 15781 . . . . . . . 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 16597 . . . . . . . . 9  |-  ( R  e.  Ring  ->  P  e. 
Ring )
623, 61syl 16 . . . . . . . 8  |-  ( ph  ->  P  e.  Ring )
6350, 5, 6, 20q1pcl 20031 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
643, 4, 22, 63syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
655, 6, 15mon1pcl 20020 . . . . . . . . 9  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
6619, 65syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  B )
676, 51rngcl 15632 . . . . . . . 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 1184 . . . . . . 7  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  e.  B )
69 eqid 2404 . . . . . . . 8  |-  ( +g  `  ( R  ^s  K ) )  =  ( +g  `  ( R  ^s  K ) )
706, 52, 69ghmlin 14966 . . . . . . 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 1184 . . . . . 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 2404 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
73 fvex 5701 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
747, 73eqeltri 2474 . . . . . . . 8  |-  K  e. 
_V
7574a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
766, 72rhmf 15782 . . . . . . . . 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 5830 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  e.  ( Base `  ( R  ^s  K ) ) )
7977, 41ffvelrnd 5830 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  e.  ( Base `  ( R  ^s  K ) ) )
80 eqid 2404 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
8156, 72, 1, 75, 78, 79, 80, 69pwsplusgval 13667 . . . . . 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 2440 . . . . 5  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) )
8382fveq1d 5689 . . . 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 13666 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) ) : K --> K )
85 ffn 5550 . . . . . . 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 13666 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) ) : K --> K )
88 ffn 5550 . . . . . . 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 6276 . . . . . 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 1185 . . . . 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 2404 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
936, 51, 92rhmmul 15783 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
9577, 64ffvelrnd 5830 . . . . . . . . . 10  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
9677, 66ffvelrnd 5830 . . . . . . . . . 10  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
97 eqid 2404 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
9856, 72, 1, 75, 95, 96, 97, 92pwsmulrval 13668 . . . . . . . . 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 2436 . . . . . . . 8  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) )
10099fveq1d 5689 . . . . . . 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 13666 . . . . . . . . 9  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
102 ffn 5550 . . . . . . . . 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 13666 . . . . . . . . 9  |-  ( ph  ->  ( O `  G
) : K --> K )
105 ffn 5550 . . . . . . . . 9  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
106104, 105syl 16 . . . . . . . 8  |-  ( ph  ->  ( O `  G
)  Fn  K )
107 fnfvof 6276 . . . . . . . 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 1185 . . . . . . 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 3799 . . . . . . . . . . . . 13  |-  ( N  e.  K  ->  N  e.  { N } )
11014, 109syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  { N } )
11118simp3d 971 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' ( O `
 G ) " { ( 0g `  R ) } )  =  { N }
)
112110, 111eleqtrrd 2481 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( `' ( O `  G
) " { ( 0g `  R ) } ) )
113 fniniseg 5810 . . . . . . . . . . . 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 202 . . . . . . . . . 10  |-  ( ph  ->  ( N  e.  K  /\  ( ( O `  G ) `  N
)  =  ( 0g
`  R ) ) )
116115simprd 450 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  G ) `  N
)  =  ( 0g
`  R ) )
117116oveq2d 6056 . . . . . . . 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 5830 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) `  N
)  e.  K )
1197, 97, 17rngrz 15656 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( 0g `  R ) )  =  ( 0g `  R
) )
121117, 120eqtrd 2436 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( ( O `
 G ) `  N ) )  =  ( 0g `  R
) )
122100, 108, 1213eqtrd 2440 . . . . . 6  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) `
 N )  =  ( 0g `  R
) )
123122oveq1d 6055 . . . . 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 15624 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1253, 124syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
12687, 14ffvelrnd 5830 . . . . . 6  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  e.  K )
1277, 80, 17grplid 14790 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( ( 0g `  R ) ( +g  `  R ) ( ( O `  ( F E G ) ) `
 N ) )  =  ( ( O `
 ( F E G ) ) `  N ) )
12991, 123, 1283eqtrd 2440 . . . 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 5691 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
131 eqid 2404 . . . . . . . . . . 11  |-  (coe1 `  ( F E G ) )  =  (coe1 `  ( F E G ) )
132131, 6, 5, 7coe1f 16564 . . . . . . . . . 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 5827 . . . . . . . . 9  |-  ( ( (coe1 `  ( F E G ) ) : NN0 --> K  /\  0  e.  NN0 )  ->  (
(coe1 `  ( F E G ) ) ` 
0 )  e.  K
)
135133, 30, 134sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( (coe1 `  ( F E G ) ) ` 
0 )  e.  K
)
13612, 5, 7, 10evl1sca 19903 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0 ) ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) ` 
0 ) } ) )
138130, 137eqtrd 2436 . . . . . 6  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) )
139138fveq1d 5689 . . . . 5  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( ( K  X.  { ( (coe1 `  ( F E G ) ) `  0
) } ) `  N ) )
140 fvex 5701 . . . . . . 7  |-  ( (coe1 `  ( F E G ) ) `  0
)  e.  _V
141140fvconst2 5906 . . . . . 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 2436 . . . 4  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( (coe1 `  ( F E G ) ) `
 0 ) )
14483, 129, 1433eqtrd 2440 . . 3  |-  ( ph  ->  ( ( O `  F ) `  N
)  =  ( (coe1 `  ( F E G ) ) `  0
) )
145144fveq2d 5691 . 2  |-  ( ph  ->  ( A `  (
( O `  F
) `  N )
)  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
14649, 145eqtr4d 2439 1  |-  ( ph  ->  ( F E G )  =  ( A `
 ( ( O `
 F ) `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278   {csn 3774   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   0cc0 8946   1c1 8947    + caddc 8949    -oocmnf 9074   RR*cxr 9075    < clt 9076    <_ cle 9077   NN0cn0 10177   Basecbs 13424   +g cplusg 13484   .rcmulr 13485    ^s cpws 13625   0gc0g 13678   Grpcgrp 14640   -gcsg 14643    GrpHom cghm 14958   Ringcrg 15615   CRingccrg 15616   RingHom crh 15772  NzRingcnzr 16283  algSccascl 16326  var1cv1 16525  Poly1cpl1 16526  eval1ce1 16528  coe1cco1 16529   deg1 cdg1 19930  Monic1pcmn1 20001  Unic1pcuc1p 20002  quot1pcq1p 20003  rem1pcr1p 20004
This theorem is referenced by:  facth1  20040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-nzr 16284  df-rlreg 16298  df-assa 16327  df-asp 16328  df-ascl 16329  df-psr 16372  df-mvr 16373  df-mpl 16374  df-evls 16375  df-evl 16376  df-opsr 16380  df-psr1 16531  df-vr1 16532  df-ply1 16533  df-evl1 16535  df-coe1 16536  df-cnfld 16659  df-mdeg 19931  df-deg1 19932  df-mon1 20006  df-uc1p 20007  df-q1p 20008  df-r1p 20009
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