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Theorem fta1glem2 20089
Description: Lemma for fta1g 20090. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
fta1g.p  |-  P  =  (Poly1 `  R )
fta1g.b  |-  B  =  ( Base `  P
)
fta1g.d  |-  D  =  ( deg1  `  R )
fta1g.o  |-  O  =  (eval1 `  R )
fta1g.w  |-  W  =  ( 0g `  R
)
fta1g.z  |-  .0.  =  ( 0g `  P )
fta1g.1  |-  ( ph  ->  R  e. IDomn )
fta1g.2  |-  ( ph  ->  F  e.  B )
fta1glem.k  |-  K  =  ( Base `  R
)
fta1glem.x  |-  X  =  (var1 `  R )
fta1glem.m  |-  .-  =  ( -g `  P )
fta1glem.a  |-  A  =  (algSc `  P )
fta1glem.g  |-  G  =  ( X  .-  ( A `  T )
)
fta1glem.3  |-  ( ph  ->  N  e.  NN0 )
fta1glem.4  |-  ( ph  ->  ( D `  F
)  =  ( N  +  1 ) )
fta1glem.5  |-  ( ph  ->  T  e.  ( `' ( O `  F
) " { W } ) )
fta1glem.6  |-  ( ph  ->  A. g  e.  B  ( ( D `  g )  =  N  ->  ( # `  ( `' ( O `  g ) " { W } ) )  <_ 
( D `  g
) ) )
Assertion
Ref Expression
fta1glem2  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  <_ 
( D `  F
) )
Distinct variable groups:    B, g    D, g    g, F    g, N    g, O    g, G    P, g    R, g    g, W
Allowed substitution hints:    ph( g)    A( g)    T( g)    K( g)    .- ( g)    X( g)    .0. ( g)

Proof of Theorem fta1glem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fta1glem.5 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  T  e.  ( `' ( O `  F
) " { W } ) )
2 eqid 2436 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  ^s  K )  =  ( R  ^s  K )
3 fta1glem.k . . . . . . . . . . . . . . . . . . . . 21  |-  K  =  ( Base `  R
)
4 eqid 2436 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
5 fta1g.1 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  R  e. IDomn )
6 fvex 5742 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( Base `  R )  e.  _V
73, 6eqeltri 2506 . . . . . . . . . . . . . . . . . . . . . 22  |-  K  e. 
_V
87a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  K  e.  _V )
9 isidom 16364 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
109simplbi 447 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( R  e. IDomn  ->  R  e.  CRing )
115, 10syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  R  e.  CRing )
12 fta1g.o . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  O  =  (eval1 `  R )
13 fta1g.p . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  P  =  (Poly1 `  R )
1412, 13, 2, 3evl1rhm 19949 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
1511, 14syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
16 fta1g.b . . . . . . . . . . . . . . . . . . . . . . . 24  |-  B  =  ( Base `  P
)
1716, 4rhmf 15827 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
1815, 17syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
19 fta1g.2 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  F  e.  B )
2018, 19ffvelrnd 5871 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
212, 3, 4, 5, 8, 20pwselbas 13711 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( O `  F
) : K --> K )
22 ffn 5591 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
2321, 22syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( O `  F
)  Fn  K )
24 fniniseg 5851 . . . . . . . . . . . . . . . . . . 19  |-  ( ( O `  F )  Fn  K  ->  ( T  e.  ( `' ( O `  F )
" { W }
)  <->  ( T  e.  K  /\  ( ( O `  F ) `
 T )  =  W ) ) )
2523, 24syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( T  e.  ( `' ( O `  F ) " { W } )  <->  ( T  e.  K  /\  (
( O `  F
) `  T )  =  W ) ) )
261, 25mpbid 202 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( T  e.  K  /\  ( ( O `  F ) `  T
)  =  W ) )
2726simprd 450 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( O `  F ) `  T
)  =  W )
28 fta1glem.x . . . . . . . . . . . . . . . . 17  |-  X  =  (var1 `  R )
29 fta1glem.m . . . . . . . . . . . . . . . . 17  |-  .-  =  ( -g `  P )
30 fta1glem.a . . . . . . . . . . . . . . . . 17  |-  A  =  (algSc `  P )
31 fta1glem.g . . . . . . . . . . . . . . . . 17  |-  G  =  ( X  .-  ( A `  T )
)
329simprbi 451 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. IDomn  ->  R  e. Domn )
33 domnnzr 16355 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. Domn  ->  R  e. NzRing )
3432, 33syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( R  e. IDomn  ->  R  e. NzRing )
355, 34syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e. NzRing )
3626simpld 446 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  e.  K )
37 fta1g.w . . . . . . . . . . . . . . . . 17  |-  W  =  ( 0g `  R
)
38 eqid 2436 . . . . . . . . . . . . . . . . 17  |-  ( ||r `  P
)  =  ( ||r `  P
)
3913, 16, 3, 28, 29, 30, 31, 12, 35, 11, 36, 19, 37, 38facth1 20087 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
4027, 39mpbird 224 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G ( ||r `
 P ) F )
41 nzrrng 16332 . . . . . . . . . . . . . . . . 17  |-  ( R  e. NzRing  ->  R  e.  Ring )
4235, 41syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e.  Ring )
43 eqid 2436 . . . . . . . . . . . . . . . . . . 19  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
44 fta1g.d . . . . . . . . . . . . . . . . . . 19  |-  D  =  ( deg1  `  R )
4513, 16, 3, 28, 29, 30, 31, 12, 35, 11, 36, 43, 44, 37ply1remlem 20085 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4645simp1d 969 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
47 eqid 2436 . . . . . . . . . . . . . . . . . 18  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4847, 43mon1puc1p 20073 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
4942, 46, 48syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
50 eqid 2436 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
51 eqid 2436 . . . . . . . . . . . . . . . . 17  |-  (quot1p `  R
)  =  (quot1p `  R
)
5213, 38, 16, 47, 50, 51dvdsq1p 20083 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( G (
||r `  P ) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
5342, 19, 49, 52syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
5440, 53mpbid 202 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )
5554fveq2d 5732 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) )
5651, 13, 16, 47q1pcl 20078 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
5742, 19, 49, 56syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
5813, 16, 43mon1pcl 20067 . . . . . . . . . . . . . . 15  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
5946, 58syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  B )
60 eqid 2436 . . . . . . . . . . . . . . 15  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
6116, 50, 60rhmmul 15828 . . . . . . . . . . . . . 14  |-  ( ( 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 ) ) )
6215, 57, 59, 61syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
6318, 57ffvelrnd 5871 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
6418, 59ffvelrnd 5871 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
65 eqid 2436 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
662, 4, 5, 8, 63, 64, 65, 60pwsmulrval 13713 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) )  =  ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) )
6755, 62, 663eqtrd 2472 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) )
6867fveq1d 5730 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  F ) `  x
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) `  x ) )
6968adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) `  x ) )
702, 3, 4, 5, 8, 63pwselbas 13711 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
71 ffn 5591 . . . . . . . . . . . . 13  |-  ( ( O `  ( F (quot1p `  R ) G ) ) : K --> K  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
7270, 71syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
7372adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  ( F
(quot1p `
 R ) G ) )  Fn  K
)
742, 3, 4, 5, 8, 64pwselbas 13711 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  G
) : K --> K )
75 ffn 5591 . . . . . . . . . . . . 13  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
7674, 75syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  G
)  Fn  K )
7776adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  G )  Fn  K )
787a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  K  e.  _V )
79 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
80 fnfvof 6317 . . . . . . . . . . 11  |-  ( ( ( ( O `  ( F (quot1p `  R ) G ) )  Fn  K  /\  ( O `  G
)  Fn  K )  /\  ( K  e. 
_V  /\  x  e.  K ) )  -> 
( ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) `  x
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8173, 77, 78, 79, 80syl22anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8269, 81eqtrd 2468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8382eqeq1d 2444 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  F ) `  x
)  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) )  =  W ) )
845, 32syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e. Domn )
8584adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  R  e. Domn )
8670ffvelrnda 5870 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  e.  K )
8774ffvelrnda 5870 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  e.  K )
883, 65, 37domneq0 16357 . . . . . . . . 9  |-  ( ( R  e. Domn  /\  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  e.  K  /\  ( ( O `  G ) `  x
)  e.  K )  ->  ( ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) )  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) )
8985, 86, 87, 88syl3anc 1184 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( ( O `
 ( F (quot1p `  R ) G ) ) `  x ) ( .r `  R
) ( ( O `
 G ) `  x ) )  =  W  <->  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) )
9083, 89bitrd 245 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  F ) `  x
)  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) )
9190pm5.32da 623 . . . . . 6  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  F ) `
 x )  =  W )  <->  ( x  e.  K  /\  (
( ( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) ) )
92 andi 838 . . . . . 6  |-  ( ( x  e.  K  /\  ( ( ( O `
 ( F (quot1p `  R ) G ) ) `  x )  =  W  \/  (
( O `  G
) `  x )  =  W ) )  <->  ( (
x  e.  K  /\  ( ( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W )  \/  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) )
9391, 92syl6bb 253 . . . . 5  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  F ) `
 x )  =  W )  <->  ( (
x  e.  K  /\  ( ( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W )  \/  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) ) )
94 fniniseg 5851 . . . . . 6  |-  ( ( O `  F )  Fn  K  ->  (
x  e.  ( `' ( O `  F
) " { W } )  <->  ( x  e.  K  /\  (
( O `  F
) `  x )  =  W ) ) )
9523, 94syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' ( O `  F ) " { W } )  <->  ( x  e.  K  /\  (
( O `  F
) `  x )  =  W ) ) )
96 elun 3488 . . . . . 6  |-  ( x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  <->  ( x  e.  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  \/  x  e.  { T } ) )
97 fniniseg 5851 . . . . . . . 8  |-  ( ( O `  ( F (quot1p `  R ) G ) )  Fn  K  ->  ( x  e.  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  <->  ( x  e.  K  /\  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W ) ) )
9872, 97syl 16 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  <->  ( x  e.  K  /\  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W ) ) )
9945simp3d 971 . . . . . . . . 9  |-  ( ph  ->  ( `' ( O `
 G ) " { W } )  =  { T } )
10099eleq2d 2503 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  x  e.  { T } ) )
101 fniniseg 5851 . . . . . . . . 9  |-  ( ( O `  G )  Fn  K  ->  (
x  e.  ( `' ( O `  G
) " { W } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  W ) ) )
10276, 101syl 16 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  W ) ) )
103100, 102bitr3d 247 . . . . . . 7  |-  ( ph  ->  ( x  e.  { T }  <->  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) )
10498, 103orbi12d 691 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  \/  x  e.  { T } )  <-> 
( ( x  e.  K  /\  ( ( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W )  \/  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) ) )
10596, 104syl5bb 249 . . . . 5  |-  ( ph  ->  ( x  e.  ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  <->  ( (
x  e.  K  /\  ( ( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W )  \/  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) ) )
10693, 95, 1053bitr4d 277 . . . 4  |-  ( ph  ->  ( x  e.  ( `' ( O `  F ) " { W } )  <->  x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
107106eqrdv 2434 . . 3  |-  ( ph  ->  ( `' ( O `
 F ) " { W } )  =  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )
108107fveq2d 5732 . 2  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  =  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
109 fvex 5742 . . . . . . . . 9  |-  ( O `
 ( F (quot1p `  R ) G ) )  e.  _V
110109cnvex 5406 . . . . . . . 8  |-  `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V
111 imaexg 5217 . . . . . . . 8  |-  ( `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  _V )
112110, 111mp1i 12 . . . . . . 7  |-  ( ph  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  _V )
113 fta1glem.3 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
114 fta1glem.6 . . . . . . . . 9  |-  ( ph  ->  A. g  e.  B  ( ( D `  g )  =  N  ->  ( # `  ( `' ( O `  g ) " { W } ) )  <_ 
( D `  g
) ) )
115 fta1g.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  P )
116 fta1glem.4 . . . . . . . . . 10  |-  ( ph  ->  ( D `  F
)  =  ( N  +  1 ) )
11713, 16, 44, 12, 37, 115, 5, 19, 3, 28, 29, 30, 31, 113, 116, 1fta1glem1 20088 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
118 fveq2 5728 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( D `  g )  =  ( D `  ( F (quot1p `  R ) G ) ) )
119118eqeq1d 2444 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( D `
 g )  =  N  <->  ( D `  ( F (quot1p `  R ) G ) )  =  N ) )
120 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( O `  g )  =  ( O `  ( F (quot1p `  R ) G ) ) )
121120cnveqd 5048 . . . . . . . . . . . . . 14  |-  ( g  =  ( F (quot1p `  R ) G )  ->  `' ( O `
 g )  =  `' ( O `  ( F (quot1p `  R ) G ) ) )
122121imaeq1d 5202 . . . . . . . . . . . . 13  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( `' ( O `  g )
" { W }
)  =  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )
123122fveq2d 5732 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( # `  ( `' ( O `  g ) " { W } ) )  =  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) ) )
124123, 118breq12d 4225 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( # `  ( `' ( O `
 g ) " { W } ) )  <_  ( D `  g )  <->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) ) )
125119, 124imbi12d 312 . . . . . . . . . 10  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( ( D `  g )  =  N  ->  ( # `
 ( `' ( O `  g )
" { W }
) )  <_  ( D `  g )
)  <->  ( ( D `
 ( F (quot1p `  R ) G ) )  =  N  -> 
( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) ) ) )
126125rspcv 3048 . . . . . . . . 9  |-  ( ( F (quot1p `  R ) G )  e.  B  -> 
( A. g  e.  B  ( ( D `
 g )  =  N  ->  ( # `  ( `' ( O `  g ) " { W } ) )  <_ 
( D `  g
) )  ->  (
( D `  ( F (quot1p `  R ) G ) )  =  N  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) ) ) )
12757, 114, 117, 126syl3c 59 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) )
128127, 117breqtrd 4236 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_  N )
129 hashbnd 11624 . . . . . . 7  |-  ( ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  _V  /\  N  e.  NN0  /\  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_  N )  ->  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin )
130112, 113, 128, 129syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin )
131 snfi 7187 . . . . . 6  |-  { T }  e.  Fin
132 unfi 7374 . . . . . 6  |-  ( ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin  /\ 
{ T }  e.  Fin )  ->  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin )
133130, 131, 132sylancl 644 . . . . 5  |-  ( ph  ->  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin )
134 hashcl 11639 . . . . 5  |-  ( ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e. 
NN0 )
135133, 134syl 16 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e. 
NN0 )
136135nn0red 10275 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e.  RR )
137 hashcl 11639 . . . . . 6  |-  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e. 
NN0 )
138130, 137syl 16 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e. 
NN0 )
139138nn0red 10275 . . . 4  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR )
140 peano2re 9239 . . . 4  |-  ( (
# `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR  ->  ( ( # `
 ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  e.  RR )
141139, 140syl 16 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  e.  RR )
142 peano2nn0 10260 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
143113, 142syl 16 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
144116, 143eqeltrd 2510 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
145144nn0red 10275 . . 3  |-  ( ph  ->  ( D `  F
)  e.  RR )
146 hashun2 11657 . . . . 5  |-  ( ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin  /\ 
{ T }  e.  Fin )  ->  ( # `  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) ) )
147130, 131, 146sylancl 644 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) ) )
148 hashsng 11647 . . . . . 6  |-  ( T  e.  ( `' ( O `  F )
" { W }
)  ->  ( # `  { T } )  =  1 )
1491, 148syl 16 . . . . 5  |-  ( ph  ->  ( # `  { T } )  =  1 )
150149oveq2d 6097 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) )  =  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
151147, 150breqtrd 4236 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
152113nn0red 10275 . . . . 5  |-  ( ph  ->  N  e.  RR )
153 1re 9090 . . . . . 6  |-  1  e.  RR
154153a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR )
155139, 152, 154, 128leadd1dd 9640 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( N  +  1 ) )
156155, 116breqtrrd 4238 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( D `  F )
)
157136, 141, 145, 151, 156letrd 9227 . 2  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( D `  F
) )
158108, 157eqbrtrd 4232 1  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  <_ 
( D `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    u. cun 3318   {csn 3814   class class class wbr 4212   `'ccnv 4877   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Fincfn 7109   RRcr 8989   1c1 8991    + caddc 8993    <_ cle 9121   NN0cn0 10221   #chash 11618   Basecbs 13469   .rcmulr 13530    ^s cpws 13670   0gc0g 13723   -gcsg 14688   Ringcrg 15660   CRingccrg 15661   ||rcdsr 15743   RingHom crh 15817  NzRingcnzr 16328  Domncdomn 16340  IDomncidom 16341  algSccascl 16371  var1cv1 16570  Poly1cpl1 16571  eval1ce1 16573   deg1 cdg1 19977  Monic1pcmn1 20048  Unic1pcuc1p 20049  quot1pcq1p 20050
This theorem is referenced by:  fta1g  20090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-rnghom 15819  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-nzr 16329  df-rlreg 16343  df-domn 16344  df-idom 16345  df-assa 16372  df-asp 16373  df-ascl 16374  df-psr 16417  df-mvr 16418  df-mpl 16419  df-evls 16420  df-evl 16421  df-opsr 16425  df-psr1 16576  df-vr1 16577  df-ply1 16578  df-evl1 16580  df-coe1 16581  df-cnfld 16704  df-mdeg 19978  df-deg1 19979  df-mon1 20053  df-uc1p 20054  df-q1p 20055  df-r1p 20056
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