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Theorem fta1glem2 22440
Description: Lemma for fta1g 22441. (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 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  ^s  K )  =  ( R  ^s  K )
3 fta1glem.k . . . . . . . . . . . . . . . . . . . . 21  |-  K  =  ( Base `  R
)
4 eqid 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
5 fta1g.1 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  R  e. IDomn )
6 fvex 5866 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( Base `  R )  e.  _V
73, 6eqeltri 2527 . . . . . . . . . . . . . . . . . . . . . 22  |-  K  e. 
_V
87a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  K  e.  _V )
9 isidom 17827 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
109simplbi 460 . . . . . . . . . . . . . . . . . . . . . . . . 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 18242 . . . . . . . . . . . . . . . . . . . . . . . 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 17249 . . . . . . . . . . . . . . . . . . . . . . 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 6017 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
212, 3, 4, 5, 8, 20pwselbas 14763 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( O `  F
) : K --> K )
22 ffn 5721 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
2321, 22syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( O `  F
)  Fn  K )
24 fniniseg 5993 . . . . . . . . . . . . . . . . . . 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 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 17818 . . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  e.  K )
37 fta1g.w . . . . . . . . . . . . . . . . 17  |-  W  =  ( 0g `  R
)
38 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  ( ||r `  P
)  =  ( ||r `  P
)
3913, 16, 3, 28, 29, 30, 31, 12, 35, 11, 36, 19, 37, 38facth1 22438 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
4027, 39mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G ( ||r `
 P ) F )
41 nzrring 17783 . . . . . . . . . . . . . . . . 17  |-  ( R  e. NzRing  ->  R  e.  Ring )
4235, 41syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e.  Ring )
43 eqid 2443 . . . . . . . . . . . . . . . . . . 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 22436 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4645simp1d 1009 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
47 eqid 2443 . . . . . . . . . . . . . . . . . 18  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4847, 43mon1puc1p 22424 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
4942, 46, 48syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
50 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
51 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  (quot1p `  R
)  =  (quot1p `  R
)
5213, 38, 16, 47, 50, 51dvdsq1p 22434 . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . 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 5860 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) )
5651, 13, 16, 47q1pcl 22429 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
5742, 19, 49, 56syl3anc 1229 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
5813, 16, 43mon1pcl 22418 . . . . . . . . . . . . . . 15  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
5946, 58syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  B )
60 eqid 2443 . . . . . . . . . . . . . . 15  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
6116, 50, 60rhmmul 17250 . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
6318, 57ffvelrnd 6017 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
6418, 59ffvelrnd 6017 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
65 eqid 2443 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
662, 4, 5, 8, 63, 64, 65, 60pwsmulrval 14765 . . . . . . . . . . . . 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 2488 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( F (quot1p `  R ) G ) )  oF ( .r `  R
) ( O `  G ) ) )
6867fveq1d 5858 . . . . . . . . . . 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 14763 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
71 ffn 5721 . . . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  ( F
(quot1p `
 R ) G ) )  Fn  K
)
742, 3, 4, 5, 8, 64pwselbas 14763 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  G
) : K --> K )
75 ffn 5721 . . . . . . . . . . . . 13  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
7674, 75syl 16 . . . . . . . . . . . 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 6538 . . . . . . . . . . 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 1230 . . . . . . . . . 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 2484 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8382eqeq1d 2445 . . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  R  e. Domn )
8670ffvelrnda 6016 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  e.  K )
8774ffvelrnda 6016 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  e.  K )
883, 65, 37domneq0 17820 . . . . . . . . 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 1229 . . . . . . . 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 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 867 . . . . . 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 5993 . . . . . 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 3630 . . . . . 6  |-  ( x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  <->  ( x  e.  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  \/  x  e.  { T } ) )
97 fniniseg 5993 . . . . . . . 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 1011 . . . . . . . . 9  |-  ( ph  ->  ( `' ( O `
 G ) " { W } )  =  { T } )
10099eleq2d 2513 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  x  e.  { T } ) )
101 fniniseg 5993 . . . . . . . . 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 709 . . . . . 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 2440 . . 3  |-  ( ph  ->  ( `' ( O `
 F ) " { W } )  =  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )
108107fveq2d 5860 . 2  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  =  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
109 fvex 5866 . . . . . . . . 9  |-  ( O `
 ( F (quot1p `  R ) G ) )  e.  _V
110109cnvex 6732 . . . . . . . 8  |-  `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V
111 imaexg 6722 . . . . . . . 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 22439 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
118 fveq2 5856 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( D `  g )  =  ( D `  ( F (quot1p `  R ) G ) ) )
119118eqeq1d 2445 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( D `
 g )  =  N  <->  ( D `  ( F (quot1p `  R ) G ) )  =  N ) )
120 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( O `  g )  =  ( O `  ( F (quot1p `  R ) G ) ) )
121120cnveqd 5168 . . . . . . . . . . . . . 14  |-  ( g  =  ( F (quot1p `  R ) G )  ->  `' ( O `
 g )  =  `' ( O `  ( F (quot1p `  R ) G ) ) )
122121imaeq1d 5326 . . . . . . . . . . . . 13  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( `' ( O `  g )
" { W }
)  =  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )
123122fveq2d 5860 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( # `  ( `' ( O `  g ) " { W } ) )  =  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) ) )
124123, 118breq12d 4450 . . . . . . . . . . 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 3192 . . . . . . . . 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 4461 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_  N )
129 hashbnd 12390 . . . . . . 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 1229 . . . . . 6  |-  ( ph  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin )
131 snfi 7598 . . . . . 6  |-  { T }  e.  Fin
132 unfi 7789 . . . . . 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 12407 . . . . 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 10859 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e.  RR )
137 hashcl 12407 . . . . . 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 10859 . . . 4  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR )
140 peano2re 9756 . . . 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 10842 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
143113, 142syl 16 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
144116, 143eqeltrd 2531 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
145144nn0red 10859 . . 3  |-  ( ph  ->  ( D `  F
)  e.  RR )
146 hashun2 12430 . . . . 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 12417 . . . . . 6  |-  ( T  e.  ( `' ( O `  F )
" { W }
)  ->  ( # `  { T } )  =  1 )
1491, 148syl 16 . . . . 5  |-  ( ph  ->  ( # `  { T } )  =  1 )
150149oveq2d 6297 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) )  =  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
151147, 150breqtrd 4461 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
152113nn0red 10859 . . . . 5  |-  ( ph  ->  N  e.  RR )
153 1red 9614 . . . . 5  |-  ( ph  ->  1  e.  RR )
154139, 152, 153, 128leadd1dd 10172 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( N  +  1 ) )
155154, 116breqtrrd 4463 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( D `  F )
)
156136, 141, 145, 151, 155letrd 9742 . 2  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( D `  F
) )
157108, 156eqbrtrd 4457 1  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  <_ 
( D `  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    u. cun 3459   {csn 4014   class class class wbr 4437   `'ccnv 4988   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   Fincfn 7518   RRcr 9494   1c1 9496    + caddc 9498    <_ cle 9632   NN0cn0 10801   #chash 12384   Basecbs 14509   .rcmulr 14575   0gc0g 14714    ^s cpws 14721   -gcsg 15929   Ringcrg 17072   CRingccrg 17073   ||rcdsr 17161   RingHom crh 17235  NzRingcnzr 17779  Domncdomn 17802  IDomncidom 17803  algSccascl 17834  var1cv1 18089  Poly1cpl1 18090  eval1ce1 18225   deg1 cdg1 22325  Monic1pcmn1 22399  Unic1pcuc1p 22400  quot1pcq1p 22401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-fzo 11804  df-seq 12087  df-hash 12385  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-0g 14716  df-gsum 14717  df-prds 14722  df-pws 14724  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-ghm 16139  df-cntz 16229  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-srg 17032  df-ring 17074  df-cring 17075  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-rnghom 17238  df-subrg 17301  df-lmod 17388  df-lss 17453  df-lsp 17492  df-nzr 17780  df-rlreg 17805  df-domn 17806  df-idom 17807  df-assa 17835  df-asp 17836  df-ascl 17837  df-psr 17879  df-mvr 17880  df-mpl 17881  df-opsr 17883  df-evls 18045  df-evl 18046  df-psr1 18093  df-vr1 18094  df-ply1 18095  df-coe1 18096  df-evl1 18227  df-cnfld 18295  df-mdeg 22326  df-deg1 22327  df-mon1 22404  df-uc1p 22405  df-q1p 22406  df-r1p 22407
This theorem is referenced by:  fta1g  22441
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