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Theorem ply1rem 20047
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 13005). 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 16295 . . . . . . . . 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 2412 . . . . . . . . . . 11  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
16 eqid 2412 . . . . . . . . . . 11  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
17 eqid 2412 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
185, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17ply1remlem 20046 . . . . . . . . . 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 2412 . . . . . . . . . 10  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
2120, 15mon1puc1p 20034 . . . . . . . . 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 20042 . . . . . . . 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 4204 . . . . . 6  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  1
)
28 1e0p1 10374 . . . . . 6  |-  1  =  ( 0  +  1 )
2927, 28syl6breq 4219 . . . . 5  |-  ( ph  ->  ( ( deg1  `  R ) `  ( F E G ) )  <  (
0  +  1 ) )
30 0nn0 10200 . . . . . 6  |-  0  e.  NN0
31 nn0leltp1 10297 . . . . . 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 3806 . . . . . 6  |-  ( ( ( deg1  `  R ) `  ( F E G ) )  e.  {  -oo }  ->  ( ( deg1  `  R
) `  ( F E G ) )  = 
-oo )
35 0xr 9095 . . . . . . 7  |-  0  e.  RR*
36 mnfle 10693 . . . . . . 7  |-  ( 0  e.  RR*  ->  -oo  <_  0 )
3735, 36ax-mp 8 . . . . . 6  |-  -oo  <_  0
3834, 37syl6eqbr 4217 . . . . 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 20041 . . . . . . 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 19967 . . . . . 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 3456 . . . . 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 19995 . . . 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 2412 . . . . . . . . 9  |-  (quot1p `  R
)  =  (quot1p `  R
)
51 eqid 2412 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
52 eqid 2412 . . . . . . . . 9  |-  ( +g  `  P )  =  ( +g  `  P )
535, 6, 20, 50, 23, 51, 52r1pid 20043 . . . . . . . 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 5699 . . . . . 6  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ( +g  `  P
) ( F E G ) ) ) )
56 eqid 2412 . . . . . . . . . 10  |-  ( R  ^s  K )  =  ( R  ^s  K )
5712, 5, 56, 7evl1rhm 19910 . . . . . . . . 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 15789 . . . . . . . 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 16605 . . . . . . . . 9  |-  ( R  e.  Ring  ->  P  e. 
Ring )
623, 61syl 16 . . . . . . . 8  |-  ( ph  ->  P  e.  Ring )
6350, 5, 6, 20q1pcl 20039 . . . . . . . . 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 20028 . . . . . . . . 9  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
6619, 65syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  B )
676, 51rngcl 15640 . . . . . . . 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 2412 . . . . . . . 8  |-  ( +g  `  ( R  ^s  K ) )  =  ( +g  `  ( R  ^s  K ) )
706, 52, 69ghmlin 14974 . . . . . . 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 2412 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
73 fvex 5709 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
747, 73eqeltri 2482 . . . . . . . 8  |-  K  e. 
_V
7574a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
766, 72rhmf 15790 . . . . . . . . 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 5838 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  e.  ( Base `  ( R  ^s  K ) ) )
7977, 41ffvelrnd 5838 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  e.  ( Base `  ( R  ^s  K ) ) )
80 eqid 2412 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
8156, 72, 1, 75, 78, 79, 80, 69pwsplusgval 13675 . . . . . 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 2448 . . . . 5  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )  o F ( +g  `  R ) ( O `  ( F E G ) ) ) )
8382fveq1d 5697 . . . 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 13674 . . . . . . 7  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) ) : K --> K )
85 ffn 5558 . . . . . . 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 13674 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) ) : K --> K )
88 ffn 5558 . . . . . . 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 6284 . . . . . 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 2412 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
936, 51, 92rhmmul 15791 . . . . . . . . . 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 5838 . . . . . . . . . 10  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
9677, 66ffvelrnd 5838 . . . . . . . . . 10  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
97 eqid 2412 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
9856, 72, 1, 75, 95, 96, 97, 92pwsmulrval 13676 . . . . . . . . 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 2444 . . . . . . . 8  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) )
10099fveq1d 5697 . . . . . . 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 13674 . . . . . . . . 9  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
102 ffn 5558 . . . . . . . . 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 13674 . . . . . . . . 9  |-  ( ph  ->  ( O `  G
) : K --> K )
105 ffn 5558 . . . . . . . . 9  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
106104, 105syl 16 . . . . . . . 8  |-  ( ph  ->  ( O `  G
)  Fn  K )
107 fnfvof 6284 . . . . . . . 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 3807 . . . . . . . . . . . . 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 2489 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( `' ( O `  G
) " { ( 0g `  R ) } ) )
113 fniniseg 5818 . . . . . . . . . . . 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 6064 . . . . . . . 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 5838 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) `  N
)  e.  K )
1197, 97, 17rngrz 15664 . . . . . . . . 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 2444 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  N ) ( .r `  R
) ( ( O `
 G ) `  N ) )  =  ( 0g `  R
) )
122100, 108, 1213eqtrd 2448 . . . . . 6  |-  ( ph  ->  ( ( O `  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) `
 N )  =  ( 0g `  R
) )
123122oveq1d 6063 . . . . 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 15632 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
1253, 124syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
12687, 14ffvelrnd 5838 . . . . . 6  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  e.  K )
1277, 80, 17grplid 14798 . . . . . 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 2448 . . . 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 5699 . . . . . . 7  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( O `  ( A `  ( (coe1 `  ( F E G ) ) `  0
) ) ) )
131 eqid 2412 . . . . . . . . . . 11  |-  (coe1 `  ( F E G ) )  =  (coe1 `  ( F E G ) )
132131, 6, 5, 7coe1f 16572 . . . . . . . . . 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 5835 . . . . . . . . 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 19911 . . . . . . . 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 2444 . . . . . 6  |-  ( ph  ->  ( O `  ( F E G ) )  =  ( K  X.  { ( (coe1 `  ( F E G ) ) `
 0 ) } ) )
139138fveq1d 5697 . . . . 5  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( ( K  X.  { ( (coe1 `  ( F E G ) ) `  0
) } ) `  N ) )
140 fvex 5709 . . . . . . 7  |-  ( (coe1 `  ( F E G ) ) `  0
)  e.  _V
141140fvconst2 5914 . . . . . 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 2444 . . . 4  |-  ( ph  ->  ( ( O `  ( F E G ) ) `  N )  =  ( (coe1 `  ( F E G ) ) `
 0 ) )
14483, 129, 1433eqtrd 2448 . . 3  |-  ( ph  ->  ( ( O `  F ) `  N
)  =  ( (coe1 `  ( F E G ) ) `  0
) )
145144fveq2d 5699 . 2  |-  ( ph  ->  ( A `  (
( O `  F
) `  N )
)  =  ( A `
 ( (coe1 `  ( F E G ) ) `
 0 ) ) )
14649, 145eqtr4d 2447 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 2924    u. cun 3286   {csn 3782   class class class wbr 4180    X. cxp 4843   `'ccnv 4844   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048    o Fcof 6270   0cc0 8954   1c1 8955    + caddc 8957    -oocmnf 9082   RR*cxr 9083    < clt 9084    <_ cle 9085   NN0cn0 10185   Basecbs 13432   +g cplusg 13492   .rcmulr 13493    ^s cpws 13633   0gc0g 13686   Grpcgrp 14648   -gcsg 14651    GrpHom cghm 14966   Ringcrg 15623   CRingccrg 15624   RingHom crh 15780  NzRingcnzr 16291  algSccascl 16334  var1cv1 16533  Poly1cpl1 16534  eval1ce1 16536  coe1cco1 16537   deg1 cdg1 19938  Monic1pcmn1 20009  Unic1pcuc1p 20010  quot1pcq1p 20011  rem1pcr1p 20012
This theorem is referenced by:  facth1  20048
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-ofr 6273  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-fzo 11099  df-seq 11287  df-hash 11582  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-prds 13634  df-pws 13636  df-0g 13690  df-gsum 13691  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-ghm 14967  df-cntz 15079  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-cring 15627  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-rnghom 15782  df-subrg 15829  df-lmod 15915  df-lss 15972  df-lsp 16011  df-nzr 16292  df-rlreg 16306  df-assa 16335  df-asp 16336  df-ascl 16337  df-psr 16380  df-mvr 16381  df-mpl 16382  df-evls 16383  df-evl 16384  df-opsr 16388  df-psr1 16539  df-vr1 16540  df-ply1 16541  df-evl1 16543  df-coe1 16544  df-cnfld 16667  df-mdeg 19939  df-deg1 19940  df-mon1 20014  df-uc1p 20015  df-q1p 20016  df-r1p 20017
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