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Theorem fta1glem2 21597
Description: Lemma for fta1g 21598. (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 2441 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  ^s  K )  =  ( R  ^s  K )
3 fta1glem.k . . . . . . . . . . . . . . . . . . . . 21  |-  K  =  ( Base `  R
)
4 eqid 2441 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
5 fta1g.1 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  R  e. IDomn )
6 fvex 5698 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( Base `  R )  e.  _V
73, 6eqeltri 2511 . . . . . . . . . . . . . . . . . . . . . 22  |-  K  e. 
_V
87a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  K  e.  _V )
9 isidom 17354 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
109simplbi 457 . . . . . . . . . . . . . . . . . . . . . . . . 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 17736 . . . . . . . . . . . . . . . . . . . . . . . 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 16804 . . . . . . . . . . . . . . . . . . . . . . 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 5841 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
212, 3, 4, 5, 8, 20pwselbas 14423 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( O `  F
) : K --> K )
22 ffn 5556 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
2321, 22syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( O `  F
)  Fn  K )
24 fniniseg 5821 . . . . . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( T  e.  K  /\  ( ( O `  F ) `  T
)  =  W ) )
2726simprd 460 . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. IDomn  ->  R  e. Domn )
33 domnnzr 17345 . . . . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  e.  K )
37 fta1g.w . . . . . . . . . . . . . . . . 17  |-  W  =  ( 0g `  R
)
38 eqid 2441 . . . . . . . . . . . . . . . . 17  |-  ( ||r `  P
)  =  ( ||r `  P
)
3913, 16, 3, 28, 29, 30, 31, 12, 35, 11, 36, 19, 37, 38facth1 21595 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
4027, 39mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G ( ||r `
 P ) F )
41 nzrrng 17321 . . . . . . . . . . . . . . . . 17  |-  ( R  e. NzRing  ->  R  e.  Ring )
4235, 41syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e.  Ring )
43 eqid 2441 . . . . . . . . . . . . . . . . . . 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 21593 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4645simp1d 995 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
47 eqid 2441 . . . . . . . . . . . . . . . . . 18  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4847, 43mon1puc1p 21581 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
4942, 46, 48syl2anc 656 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
50 eqid 2441 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
51 eqid 2441 . . . . . . . . . . . . . . . . 17  |-  (quot1p `  R
)  =  (quot1p `  R
)
5213, 38, 16, 47, 50, 51dvdsq1p 21591 . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
5440, 53mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )
5554fveq2d 5692 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) )
5651, 13, 16, 47q1pcl 21586 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
5742, 19, 49, 56syl3anc 1213 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
5813, 16, 43mon1pcl 21575 . . . . . . . . . . . . . . 15  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
5946, 58syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  B )
60 eqid 2441 . . . . . . . . . . . . . . 15  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
6116, 50, 60rhmmul 16805 . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
6318, 57ffvelrnd 5841 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
6418, 59ffvelrnd 5841 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
65 eqid 2441 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
662, 4, 5, 8, 63, 64, 65, 60pwsmulrval 14425 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) )  =  ( ( O `
 ( F (quot1p `  R ) G ) )  oF ( .r `  R ) ( O `  G
) ) )
6755, 62, 663eqtrd 2477 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( F (quot1p `  R ) G ) )  oF ( .r `  R
) ( O `  G ) ) )
6867fveq1d 5690 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  F ) `  x
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) )  oF ( .r `  R
) ( O `  G ) ) `  x ) )
6968adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) )  oF ( .r `  R
) ( O `  G ) ) `  x ) )
702, 3, 4, 5, 8, 63pwselbas 14423 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
71 ffn 5556 . . . . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  ( F
(quot1p `
 R ) G ) )  Fn  K
)
742, 3, 4, 5, 8, 64pwselbas 14423 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  G
) : K --> K )
75 ffn 5556 . . . . . . . . . . . . 13  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
7674, 75syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  G
)  Fn  K )
7776adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  G )  Fn  K )
787a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  K  e.  _V )
79 simpr 458 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
80 fnfvof 6332 . . . . . . . . . . 11  |-  ( ( ( ( O `  ( F (quot1p `  R ) G ) )  Fn  K  /\  ( O `  G
)  Fn  K )  /\  ( K  e. 
_V  /\  x  e.  K ) )  -> 
( ( ( O `
 ( F (quot1p `  R ) G ) )  oF ( .r `  R ) ( O `  G
) ) `  x
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8173, 77, 78, 79, 80syl22anc 1214 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  ( F (quot1p `  R ) G ) )  oF ( .r `  R
) ( O `  G ) ) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8269, 81eqtrd 2473 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8382eqeq1d 2449 . . . . . . . 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 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  R  e. Domn )
8670ffvelrnda 5840 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  e.  K )
8774ffvelrnda 5840 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  e.  K )
883, 65, 37domneq0 17347 . . . . . . . . 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 1213 . . . . . . . 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 253 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  F ) `  x
)  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) )
9190pm5.32da 636 . . . . . 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 857 . . . . . 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 261 . . . . 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 5821 . . . . . 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 3494 . . . . . 6  |-  ( x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  <->  ( x  e.  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  \/  x  e.  { T } ) )
97 fniniseg 5821 . . . . . . . 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 997 . . . . . . . . 9  |-  ( ph  ->  ( `' ( O `
 G ) " { W } )  =  { T } )
10099eleq2d 2508 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  x  e.  { T } ) )
101 fniniseg 5821 . . . . . . . . 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 255 . . . . . . 7  |-  ( ph  ->  ( x  e.  { T }  <->  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) )
10498, 103orbi12d 704 . . . . . 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 257 . . . . 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 285 . . . 4  |-  ( ph  ->  ( x  e.  ( `' ( O `  F ) " { W } )  <->  x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
107106eqrdv 2439 . . 3  |-  ( ph  ->  ( `' ( O `
 F ) " { W } )  =  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )
108107fveq2d 5692 . 2  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  =  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
109 fvex 5698 . . . . . . . . 9  |-  ( O `
 ( F (quot1p `  R ) G ) )  e.  _V
110109cnvex 6524 . . . . . . . 8  |-  `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V
111 imaexg 6514 . . . . . . . 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 21596 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
118 fveq2 5688 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( D `  g )  =  ( D `  ( F (quot1p `  R ) G ) ) )
119118eqeq1d 2449 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( D `
 g )  =  N  <->  ( D `  ( F (quot1p `  R ) G ) )  =  N ) )
120 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( O `  g )  =  ( O `  ( F (quot1p `  R ) G ) ) )
121120cnveqd 5011 . . . . . . . . . . . . . 14  |-  ( g  =  ( F (quot1p `  R ) G )  ->  `' ( O `
 g )  =  `' ( O `  ( F (quot1p `  R ) G ) ) )
122121imaeq1d 5165 . . . . . . . . . . . . 13  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( `' ( O `  g )
" { W }
)  =  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )
123122fveq2d 5692 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( # `  ( `' ( O `  g ) " { W } ) )  =  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) ) )
124123, 118breq12d 4302 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( # `  ( `' ( O `
 g ) " { W } ) )  <_  ( D `  g )  <->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) ) )
125119, 124imbi12d 320 . . . . . . . . . 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 3066 . . . . . . . . 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 61 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) )
128127, 117breqtrd 4313 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_  N )
129 hashbnd 12105 . . . . . . 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 1213 . . . . . 6  |-  ( ph  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin )
131 snfi 7386 . . . . . 6  |-  { T }  e.  Fin
132 unfi 7575 . . . . . 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 657 . . . . 5  |-  ( ph  ->  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin )
134 hashcl 12122 . . . . 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 10633 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e.  RR )
137 hashcl 12122 . . . . . 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 10633 . . . 4  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR )
140 peano2re 9538 . . . 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 10616 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
143113, 142syl 16 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
144116, 143eqeltrd 2515 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
145144nn0red 10633 . . 3  |-  ( ph  ->  ( D `  F
)  e.  RR )
146 hashun2 12142 . . . . 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 657 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) ) )
148 hashsng 12132 . . . . . 6  |-  ( T  e.  ( `' ( O `  F )
" { W }
)  ->  ( # `  { T } )  =  1 )
1491, 148syl 16 . . . . 5  |-  ( ph  ->  ( # `  { T } )  =  1 )
150149oveq2d 6106 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) )  =  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
151147, 150breqtrd 4313 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
152113nn0red 10633 . . . . 5  |-  ( ph  ->  N  e.  RR )
153 1red 9397 . . . . 5  |-  ( ph  ->  1  e.  RR )
154139, 152, 153, 128leadd1dd 9949 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( N  +  1 ) )
155154, 116breqtrrd 4315 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( D `  F )
)
156136, 141, 145, 151, 155letrd 9524 . 2  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( D `  F
) )
157108, 156eqbrtrd 4309 1  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  <_ 
( D `  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    u. cun 3323   {csn 3874   class class class wbr 4289   `'ccnv 4835   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   Fincfn 7306   RRcr 9277   1c1 9279    + caddc 9281    <_ cle 9415   NN0cn0 10575   #chash 12099   Basecbs 14170   .rcmulr 14235   0gc0g 14374    ^s cpws 14381   -gcsg 15409   Ringcrg 16635   CRingccrg 16636   ||rcdsr 16720   RingHom crh 16794  NzRingcnzr 17317  Domncdomn 17329  IDomncidom 17330  algSccascl 17361  var1cv1 17608  Poly1cpl1 17609  eval1ce1 17722   deg1 cdg1 21482  Monic1pcmn1 21556  Unic1pcuc1p 21557  quot1pcq1p 21558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-0g 14376  df-gsum 14377  df-prds 14382  df-pws 14384  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-rnghom 16796  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-nzr 17318  df-rlreg 17332  df-domn 17333  df-idom 17334  df-assa 17362  df-asp 17363  df-ascl 17364  df-psr 17401  df-mvr 17402  df-mpl 17403  df-opsr 17405  df-evls 17564  df-evl 17565  df-psr1 17612  df-vr1 17613  df-ply1 17614  df-coe1 17615  df-evl1 17724  df-cnfld 17778  df-mdeg 21483  df-deg1 21484  df-mon1 21561  df-uc1p 21562  df-q1p 21563  df-r1p 21564
This theorem is referenced by:  fta1g  21598
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