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Theorem fta1glem2 22861
Description: Lemma for fta1g 22862. (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 2404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  ^s  K )  =  ( R  ^s  K )
3 fta1glem.k . . . . . . . . . . . . . . . . . . . . 21  |-  K  =  ( Base `  R
)
4 eqid 2404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
5 fta1g.1 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  R  e. IDomn )
6 fvex 5861 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( Base `  R )  e.  _V
73, 6eqeltri 2488 . . . . . . . . . . . . . . . . . . . . . 22  |-  K  e. 
_V
87a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  K  e.  _V )
9 isidom 18275 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
109simplbi 460 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( R  e. IDomn  ->  R  e.  CRing )
115, 10syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  R  e.  CRing )
12 fta1g.o . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  O  =  (eval1 `  R )
13 fta1g.p . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  P  =  (Poly1 `  R )
1412, 13, 2, 3evl1rhm 18690 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
1511, 14syl 17 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
16 fta1g.b . . . . . . . . . . . . . . . . . . . . . . . 24  |-  B  =  ( Base `  P
)
1716, 4rhmf 17697 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
1815, 17syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
19 fta1g.2 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  F  e.  B )
2018, 19ffvelrnd 6012 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
212, 3, 4, 5, 8, 20pwselbas 15105 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( O `  F
) : K --> K )
22 ffn 5716 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
2321, 22syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( O `  F
)  Fn  K )
24 fniniseg 5988 . . . . . . . . . . . . . . . . . . 19  |-  ( ( O `  F )  Fn  K  ->  ( T  e.  ( `' ( O `  F )
" { W }
)  <->  ( T  e.  K  /\  ( ( O `  F ) `
 T )  =  W ) ) )
2523, 24syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( T  e.  ( `' ( O `  F ) " { W } )  <->  ( T  e.  K  /\  (
( O `  F
) `  T )  =  W ) ) )
261, 25mpbid 212 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( T  e.  K  /\  ( ( O `  F ) `  T
)  =  W ) )
2726simprd 463 . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. IDomn  ->  R  e. Domn )
33 domnnzr 18266 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. Domn  ->  R  e. NzRing )
3432, 33syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( R  e. IDomn  ->  R  e. NzRing )
355, 34syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e. NzRing )
3626simpld 459 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  e.  K )
37 fta1g.w . . . . . . . . . . . . . . . . 17  |-  W  =  ( 0g `  R
)
38 eqid 2404 . . . . . . . . . . . . . . . . 17  |-  ( ||r `  P
)  =  ( ||r `  P
)
3913, 16, 3, 28, 29, 30, 31, 12, 35, 11, 36, 19, 37, 38facth1 22859 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
4027, 39mpbird 234 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G ( ||r `
 P ) F )
41 nzrring 18231 . . . . . . . . . . . . . . . . 17  |-  ( R  e. NzRing  ->  R  e.  Ring )
4235, 41syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e.  Ring )
43 eqid 2404 . . . . . . . . . . . . . . . . . . 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 22857 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4645simp1d 1011 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
47 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4847, 43mon1puc1p 22845 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
4942, 46, 48syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
50 eqid 2404 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
51 eqid 2404 . . . . . . . . . . . . . . . . 17  |-  (quot1p `  R
)  =  (quot1p `  R
)
5213, 38, 16, 47, 50, 51dvdsq1p 22855 . . . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
5440, 53mpbid 212 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )
5554fveq2d 5855 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) )
5651, 13, 16, 47q1pcl 22850 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
5742, 19, 49, 56syl3anc 1232 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
5813, 16, 43mon1pcl 22839 . . . . . . . . . . . . . . 15  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
5946, 58syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  B )
60 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
6116, 50, 60rhmmul 17698 . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
6318, 57ffvelrnd 6012 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
6418, 59ffvelrnd 6012 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
65 eqid 2404 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
662, 4, 5, 8, 63, 64, 65, 60pwsmulrval 15107 . . . . . . . . . . . . 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 2449 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( F (quot1p `  R ) G ) )  oF ( .r `  R
) ( O `  G ) ) )
6867fveq1d 5853 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  F ) `  x
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) )  oF ( .r `  R
) ( O `  G ) ) `  x ) )
6968adantr 465 . . . . . . . . . 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 15105 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
71 ffn 5716 . . . . . . . . . . . . 13  |-  ( ( O `  ( F (quot1p `  R ) G ) ) : K --> K  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
7270, 71syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
7372adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  ( F
(quot1p `
 R ) G ) )  Fn  K
)
742, 3, 4, 5, 8, 64pwselbas 15105 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  G
) : K --> K )
75 ffn 5716 . . . . . . . . . . . . 13  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
7674, 75syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  G
)  Fn  K )
7776adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  G )  Fn  K )
787a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  K  e.  _V )
79 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
80 fnfvof 6537 . . . . . . . . . . 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 1233 . . . . . . . . . 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 2445 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8382eqeq1d 2406 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  F ) `  x
)  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) )  =  W ) )
845, 32syl 17 . . . . . . . . . 10  |-  ( ph  ->  R  e. Domn )
8584adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  R  e. Domn )
8670ffvelrnda 6011 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  e.  K )
8774ffvelrnda 6011 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  e.  K )
883, 65, 37domneq0 18268 . . . . . . . . 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 1232 . . . . . . . 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 255 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  F ) `  x
)  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) )
9190pm5.32da 641 . . . . . 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 870 . . . . . 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 263 . . . . 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 5988 . . . . . 6  |-  ( ( O `  F )  Fn  K  ->  (
x  e.  ( `' ( O `  F
) " { W } )  <->  ( x  e.  K  /\  (
( O `  F
) `  x )  =  W ) ) )
9523, 94syl 17 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' ( O `  F ) " { W } )  <->  ( x  e.  K  /\  (
( O `  F
) `  x )  =  W ) ) )
96 elun 3586 . . . . . 6  |-  ( x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  <->  ( x  e.  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  \/  x  e.  { T } ) )
97 fniniseg 5988 . . . . . . . 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 17 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  <->  ( x  e.  K  /\  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W ) ) )
9945simp3d 1013 . . . . . . . . 9  |-  ( ph  ->  ( `' ( O `
 G ) " { W } )  =  { T } )
10099eleq2d 2474 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  x  e.  { T } ) )
101 fniniseg 5988 . . . . . . . . 9  |-  ( ( O `  G )  Fn  K  ->  (
x  e.  ( `' ( O `  G
) " { W } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  W ) ) )
10276, 101syl 17 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  W ) ) )
103100, 102bitr3d 257 . . . . . . 7  |-  ( ph  ->  ( x  e.  { T }  <->  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) )
10498, 103orbi12d 710 . . . . . 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 259 . . . . 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 287 . . . 4  |-  ( ph  ->  ( x  e.  ( `' ( O `  F ) " { W } )  <->  x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
107106eqrdv 2401 . . 3  |-  ( ph  ->  ( `' ( O `
 F ) " { W } )  =  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )
108107fveq2d 5855 . 2  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  =  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
109 fvex 5861 . . . . . . . . 9  |-  ( O `
 ( F (quot1p `  R ) G ) )  e.  _V
110109cnvex 6733 . . . . . . . 8  |-  `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V
111 imaexg 6723 . . . . . . . 8  |-  ( `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  _V )
112110, 111mp1i 13 . . . . . . 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 22860 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
118 fveq2 5851 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( D `  g )  =  ( D `  ( F (quot1p `  R ) G ) ) )
119118eqeq1d 2406 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( D `
 g )  =  N  <->  ( D `  ( F (quot1p `  R ) G ) )  =  N ) )
120 fveq2 5851 . . . . . . . . . . . . . . 15  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( O `  g )  =  ( O `  ( F (quot1p `  R ) G ) ) )
121120cnveqd 5001 . . . . . . . . . . . . . 14  |-  ( g  =  ( F (quot1p `  R ) G )  ->  `' ( O `
 g )  =  `' ( O `  ( F (quot1p `  R ) G ) ) )
122121imaeq1d 5158 . . . . . . . . . . . . 13  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( `' ( O `  g )
" { W }
)  =  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )
123122fveq2d 5855 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( # `  ( `' ( O `  g ) " { W } ) )  =  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) ) )
124123, 118breq12d 4410 . . . . . . . . . . 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 3158 . . . . . . . . 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 62 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) )
128127, 117breqtrd 4421 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_  N )
129 hashbnd 12460 . . . . . . 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 1232 . . . . . 6  |-  ( ph  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin )
131 snfi 7636 . . . . . 6  |-  { T }  e.  Fin
132 unfi 7823 . . . . . 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 662 . . . . 5  |-  ( ph  ->  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin )
134 hashcl 12477 . . . . 5  |-  ( ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e. 
NN0 )
135133, 134syl 17 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e. 
NN0 )
136135nn0red 10896 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e.  RR )
137 hashcl 12477 . . . . . 6  |-  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e. 
NN0 )
138130, 137syl 17 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e. 
NN0 )
139138nn0red 10896 . . . 4  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR )
140 peano2re 9789 . . . 4  |-  ( (
# `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR  ->  ( ( # `
 ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  e.  RR )
141139, 140syl 17 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  e.  RR )
142 peano2nn0 10879 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
143113, 142syl 17 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
144116, 143eqeltrd 2492 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
145144nn0red 10896 . . 3  |-  ( ph  ->  ( D `  F
)  e.  RR )
146 hashun2 12501 . . . . 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 662 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) ) )
148 hashsng 12488 . . . . . 6  |-  ( T  e.  ( `' ( O `  F )
" { W }
)  ->  ( # `  { T } )  =  1 )
1491, 148syl 17 . . . . 5  |-  ( ph  ->  ( # `  { T } )  =  1 )
150149oveq2d 6296 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) )  =  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
151147, 150breqtrd 4421 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
152113nn0red 10896 . . . . 5  |-  ( ph  ->  N  e.  RR )
153 1red 9643 . . . . 5  |-  ( ph  ->  1  e.  RR )
154139, 152, 153, 128leadd1dd 10208 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( N  +  1 ) )
155154, 116breqtrrd 4423 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( D `  F )
)
156136, 141, 145, 151, 155letrd 9775 . 2  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( D `  F
) )
157108, 156eqbrtrd 4417 1  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  <_ 
( D `  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    \/ wo 368    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   _Vcvv 3061    u. cun 3414   {csn 3974   class class class wbr 4397   `'ccnv 4824   "cima 4828    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280    oFcof 6521   Fincfn 7556   RRcr 9523   1c1 9525    + caddc 9527    <_ cle 9661   NN0cn0 10838   #chash 12454   Basecbs 14843   .rcmulr 14912   0gc0g 15056    ^s cpws 15063   -gcsg 16381   Ringcrg 17520   CRingccrg 17521   ||rcdsr 17609   RingHom crh 17683  NzRingcnzr 18227  Domncdomn 18250  IDomncidom 18251  algSccascl 18282  var1cv1 18537  Poly1cpl1 18538  eval1ce1 18673   deg1 cdg1 22746  Monic1pcmn1 22820  Unic1pcuc1p 22821  quot1pcq1p 22822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-ofr 6524  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-tpos 6960  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-sup 7937  df-oi 7971  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-fz 11729  df-fzo 11857  df-seq 12154  df-hash 12455  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-hom 14935  df-cco 14936  df-0g 15058  df-gsum 15059  df-prds 15064  df-pws 15066  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mhm 16292  df-submnd 16293  df-grp 16383  df-minusg 16384  df-sbg 16385  df-mulg 16386  df-subg 16524  df-ghm 16591  df-cntz 16681  df-cmn 17126  df-abl 17127  df-mgp 17464  df-ur 17476  df-srg 17480  df-ring 17522  df-cring 17523  df-oppr 17594  df-dvdsr 17612  df-unit 17613  df-invr 17643  df-rnghom 17686  df-subrg 17749  df-lmod 17836  df-lss 17901  df-lsp 17940  df-nzr 18228  df-rlreg 18253  df-domn 18254  df-idom 18255  df-assa 18283  df-asp 18284  df-ascl 18285  df-psr 18327  df-mvr 18328  df-mpl 18329  df-opsr 18331  df-evls 18493  df-evl 18494  df-psr1 18541  df-vr1 18542  df-ply1 18543  df-coe1 18544  df-evl1 18675  df-cnfld 18743  df-mdeg 22747  df-deg1 22748  df-mon1 22825  df-uc1p 22826  df-q1p 22827  df-r1p 22828
This theorem is referenced by:  fta1g  22862
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