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Theorem fta1glem2 19650
Description: Lemma for fta1g 19651. (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 2358 . . . . . . . . . . . . . . . . . . . . 21  |-  ( R  ^s  K )  =  ( R  ^s  K )
3 fta1glem.k . . . . . . . . . . . . . . . . . . . . 21  |-  K  =  ( Base `  R
)
4 eqid 2358 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
5 fta1g.1 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  R  e. IDomn )
6 fvex 5619 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( Base `  R )  e.  _V
73, 6eqeltri 2428 . . . . . . . . . . . . . . . . . . . . . 22  |-  K  e. 
_V
87a1i 10 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  K  e.  _V )
9 isidom 16138 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
109simplbi 446 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( R  e. IDomn  ->  R  e.  CRing )
115, 10syl 15 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  R  e.  CRing )
12 fta1g.o . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  O  =  (eval1 `  R )
13 fta1g.p . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  P  =  (Poly1 `  R )
1412, 13, 2, 3evl1rhm 19510 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
1511, 14syl 15 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
16 fta1g.b . . . . . . . . . . . . . . . . . . . . . . . 24  |-  B  =  ( Base `  P
)
1716, 4rhmf 15597 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
1815, 17syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
19 fta1g.2 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  F  e.  B )
20 ffvelrn 5743 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  F  e.  B )  ->  ( O `  F )  e.  ( Base `  ( R  ^s  K ) ) )
2118, 19, 20syl2anc 642 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
222, 3, 4, 5, 8, 21pwselbas 13481 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( O `  F
) : K --> K )
23 ffn 5469 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
2422, 23syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( O `  F
)  Fn  K )
25 fniniseg 5726 . . . . . . . . . . . . . . . . . . 19  |-  ( ( O `  F )  Fn  K  ->  ( T  e.  ( `' ( O `  F )
" { W }
)  <->  ( T  e.  K  /\  ( ( O `  F ) `
 T )  =  W ) ) )
2624, 25syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( T  e.  ( `' ( O `  F ) " { W } )  <->  ( T  e.  K  /\  (
( O `  F
) `  T )  =  W ) ) )
271, 26mpbid 201 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( T  e.  K  /\  ( ( O `  F ) `  T
)  =  W ) )
2827simprd 449 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( O `  F ) `  T
)  =  W )
29 fta1glem.x . . . . . . . . . . . . . . . . 17  |-  X  =  (var1 `  R )
30 fta1glem.m . . . . . . . . . . . . . . . . 17  |-  .-  =  ( -g `  P )
31 fta1glem.a . . . . . . . . . . . . . . . . 17  |-  A  =  (algSc `  P )
32 fta1glem.g . . . . . . . . . . . . . . . . 17  |-  G  =  ( X  .-  ( A `  T )
)
339simprbi 450 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. IDomn  ->  R  e. Domn )
34 domnnzr 16129 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. Domn  ->  R  e. NzRing )
3533, 34syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( R  e. IDomn  ->  R  e. NzRing )
365, 35syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e. NzRing )
3727simpld 445 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  e.  K )
38 fta1g.w . . . . . . . . . . . . . . . . 17  |-  W  =  ( 0g `  R
)
39 eqid 2358 . . . . . . . . . . . . . . . . 17  |-  ( ||r `  P
)  =  ( ||r `  P
)
4013, 16, 3, 29, 30, 31, 32, 12, 36, 11, 37, 19, 38, 39facth1 19648 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
4128, 40mpbird 223 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G ( ||r `
 P ) F )
42 nzrrng 16106 . . . . . . . . . . . . . . . . 17  |-  ( R  e. NzRing  ->  R  e.  Ring )
4336, 42syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e.  Ring )
44 eqid 2358 . . . . . . . . . . . . . . . . . . 19  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
45 fta1g.d . . . . . . . . . . . . . . . . . . 19  |-  D  =  ( deg1  `  R )
4613, 16, 3, 29, 30, 31, 32, 12, 36, 11, 37, 44, 45, 38ply1remlem 19646 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4746simp1d 967 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
48 eqid 2358 . . . . . . . . . . . . . . . . . 18  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4948, 44mon1puc1p 19634 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
5043, 47, 49syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
51 eqid 2358 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  P )  =  ( .r `  P
)
52 eqid 2358 . . . . . . . . . . . . . . . . 17  |-  (quot1p `  R
)  =  (quot1p `  R
)
5313, 39, 16, 48, 51, 52dvdsq1p 19644 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( G (
||r `  P ) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
5443, 19, 50, 53syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
5541, 54mpbid 201 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )
5655fveq2d 5609 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  =  ( O `
 ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) ) )
5752, 13, 16, 48q1pcl 19639 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
5843, 19, 50, 57syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
5913, 16, 44mon1pcl 19628 . . . . . . . . . . . . . . 15  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
6047, 59syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  B )
61 eqid 2358 . . . . . . . . . . . . . . 15  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
6216, 51, 61rhmmul 15598 . . . . . . . . . . . . . 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 ) ) )
6315, 58, 60, 62syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  (
( F (quot1p `  R
) G ) ( .r `  P ) G ) )  =  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) ) )
64 ffvelrn 5743 . . . . . . . . . . . . . . 15  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  ( F (quot1p `  R ) G )  e.  B )  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
6518, 58, 64syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  e.  (
Base `  ( R  ^s  K ) ) )
66 ffvelrn 5743 . . . . . . . . . . . . . . 15  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  G  e.  B )  ->  ( O `  G )  e.  ( Base `  ( R  ^s  K ) ) )
6718, 60, 66syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( O `  G
)  e.  ( Base `  ( R  ^s  K ) ) )
68 eqid 2358 . . . . . . . . . . . . . 14  |-  ( .r
`  R )  =  ( .r `  R
)
692, 4, 5, 8, 65, 67, 68, 61pwsmulrval 13483 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( O `  ( F (quot1p `  R ) G ) ) ( .r
`  ( R  ^s  K
) ) ( O `
 G ) )  =  ( ( O `
 ( F (quot1p `  R ) G ) )  o F ( .r `  R ) ( O `  G
) ) )
7056, 63, 693eqtrd 2394 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
)  =  ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) )
7170fveq1d 5607 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  F ) `  x
)  =  ( ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) `  x ) )
7271adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) )  o F ( .r `  R
) ( O `  G ) ) `  x ) )
732, 3, 4, 5, 8, 65pwselbas 13481 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) ) : K --> K )
74 ffn 5469 . . . . . . . . . . . . 13  |-  ( ( O `  ( F (quot1p `  R ) G ) ) : K --> K  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
7573, 74syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  ( F (quot1p `  R ) G ) )  Fn  K
)
7675adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  ( F
(quot1p `
 R ) G ) )  Fn  K
)
772, 3, 4, 5, 8, 67pwselbas 13481 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  G
) : K --> K )
78 ffn 5469 . . . . . . . . . . . . 13  |-  ( ( O `  G ) : K --> K  -> 
( O `  G
)  Fn  K )
7977, 78syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  G
)  Fn  K )
8079adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  ( O `  G )  Fn  K )
817a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  K  e.  _V )
82 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  K )  ->  x  e.  K )
83 fnfvof 6174 . . . . . . . . . . 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 ) ) )
8476, 80, 81, 82, 83syl22anc 1183 . . . . . . . . . 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 ) ) )
8572, 84eqtrd 2390 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  F
) `  x )  =  ( ( ( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) ) )
8685eqeq1d 2366 . . . . . . . 8  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  F ) `  x
)  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
) ( .r `  R ) ( ( O `  G ) `
 x ) )  =  W ) )
875, 33syl 15 . . . . . . . . . 10  |-  ( ph  ->  R  e. Domn )
8887adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  R  e. Domn )
89 ffvelrn 5743 . . . . . . . . . 10  |-  ( ( ( O `  ( F (quot1p `  R ) G ) ) : K --> K  /\  x  e.  K
)  ->  ( ( O `  ( F
(quot1p `
 R ) G ) ) `  x
)  e.  K )
9073, 89sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  e.  K )
91 ffvelrn 5743 . . . . . . . . . 10  |-  ( ( ( O `  G
) : K --> K  /\  x  e.  K )  ->  ( ( O `  G ) `  x
)  e.  K )
9277, 91sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  K )  ->  (
( O `  G
) `  x )  e.  K )
933, 68, 38domneq0 16131 . . . . . . . . 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 ) ) )
9488, 90, 92, 93syl3anc 1182 . . . . . . . 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 ) ) )
9586, 94bitrd 244 . . . . . . 7  |-  ( (
ph  /\  x  e.  K )  ->  (
( ( O `  F ) `  x
)  =  W  <->  ( (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) )
9695pm5.32da 622 . . . . . 6  |-  ( ph  ->  ( ( x  e.  K  /\  ( ( O `  F ) `
 x )  =  W )  <->  ( x  e.  K  /\  (
( ( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W  \/  ( ( O `  G ) `  x
)  =  W ) ) ) )
97 andi 837 . . . . . 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 ) ) )
9896, 97syl6bb 252 . . . . 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 ) ) ) )
99 fniniseg 5726 . . . . . 6  |-  ( ( O `  F )  Fn  K  ->  (
x  e.  ( `' ( O `  F
) " { W } )  <->  ( x  e.  K  /\  (
( O `  F
) `  x )  =  W ) ) )
10024, 99syl 15 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' ( O `  F ) " { W } )  <->  ( x  e.  K  /\  (
( O `  F
) `  x )  =  W ) ) )
101 elun 3392 . . . . . 6  |-  ( x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  <->  ( x  e.  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  \/  x  e.  { T } ) )
102 fniniseg 5726 . . . . . . . 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 ) ) )
10375, 102syl 15 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  <->  ( x  e.  K  /\  (
( O `  ( F (quot1p `  R ) G ) ) `  x
)  =  W ) ) )
10446simp3d 969 . . . . . . . . 9  |-  ( ph  ->  ( `' ( O `
 G ) " { W } )  =  { T } )
105104eleq2d 2425 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  x  e.  { T } ) )
106 fniniseg 5726 . . . . . . . . 9  |-  ( ( O `  G )  Fn  K  ->  (
x  e.  ( `' ( O `  G
) " { W } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  W ) ) )
10779, 106syl 15 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( `' ( O `  G ) " { W } )  <->  ( x  e.  K  /\  (
( O `  G
) `  x )  =  W ) ) )
108105, 107bitr3d 246 . . . . . . 7  |-  ( ph  ->  ( x  e.  { T }  <->  ( x  e.  K  /\  ( ( O `  G ) `
 x )  =  W ) ) )
109103, 108orbi12d 690 . . . . . 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 ) ) ) )
110101, 109syl5bb 248 . . . . 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 ) ) ) )
11198, 100, 1103bitr4d 276 . . . 4  |-  ( ph  ->  ( x  e.  ( `' ( O `  F ) " { W } )  <->  x  e.  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
112111eqrdv 2356 . . 3  |-  ( ph  ->  ( `' ( O `
 F ) " { W } )  =  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )
113112fveq2d 5609 . 2  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  =  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) ) )
114 fvex 5619 . . . . . . . . 9  |-  ( O `
 ( F (quot1p `  R ) G ) )  e.  _V
115114cnvex 5288 . . . . . . . 8  |-  `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V
116 imaexg 5105 . . . . . . . 8  |-  ( `' ( O `  ( F (quot1p `  R ) G ) )  e.  _V  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  _V )
117115, 116mp1i 11 . . . . . . 7  |-  ( ph  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  _V )
118 fta1glem.3 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
119 fta1glem.6 . . . . . . . . 9  |-  ( ph  ->  A. g  e.  B  ( ( D `  g )  =  N  ->  ( # `  ( `' ( O `  g ) " { W } ) )  <_ 
( D `  g
) ) )
120 fta1g.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  P )
121 fta1glem.4 . . . . . . . . . 10  |-  ( ph  ->  ( D `  F
)  =  ( N  +  1 ) )
12213, 16, 45, 12, 38, 120, 5, 19, 3, 29, 30, 31, 32, 118, 121, 1fta1glem1 19649 . . . . . . . . 9  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
123 fveq2 5605 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( D `  g )  =  ( D `  ( F (quot1p `  R ) G ) ) )
124123eqeq1d 2366 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( D `
 g )  =  N  <->  ( D `  ( F (quot1p `  R ) G ) )  =  N ) )
125 fveq2 5605 . . . . . . . . . . . . . . 15  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( O `  g )  =  ( O `  ( F (quot1p `  R ) G ) ) )
126125cnveqd 4936 . . . . . . . . . . . . . 14  |-  ( g  =  ( F (quot1p `  R ) G )  ->  `' ( O `
 g )  =  `' ( O `  ( F (quot1p `  R ) G ) ) )
127126imaeq1d 5090 . . . . . . . . . . . . 13  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( `' ( O `  g )
" { W }
)  =  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )
128127fveq2d 5609 . . . . . . . . . . . 12  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( # `  ( `' ( O `  g ) " { W } ) )  =  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) ) )
129128, 123breq12d 4115 . . . . . . . . . . 11  |-  ( g  =  ( F (quot1p `  R ) G )  ->  ( ( # `  ( `' ( O `
 g ) " { W } ) )  <_  ( D `  g )  <->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) ) )
130124, 129imbi12d 311 . . . . . . . . . 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 ) ) ) ) )
131130rspcv 2956 . . . . . . . . 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 ) ) ) ) )
13258, 119, 122, 131syl3c 57 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_ 
( D `  ( F (quot1p `  R ) G ) ) )
133132, 122breqtrd 4126 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  <_  N )
134 hashbnd 11433 . . . . . . 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 )
135117, 118, 133, 134syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin )
136 snfi 7026 . . . . . 6  |-  { T }  e.  Fin
137 unfi 7211 . . . . . 6  |-  ( ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin  /\ 
{ T }  e.  Fin )  ->  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin )
138135, 136, 137sylancl 643 . . . . 5  |-  ( ph  ->  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin )
139 hashcl 11440 . . . . 5  |-  ( ( ( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } )  e.  Fin  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e. 
NN0 )
140138, 139syl 15 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e. 
NN0 )
141140nn0red 10108 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  e.  RR )
142 hashcl 11440 . . . . . 6  |-  ( ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } )  e.  Fin  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e. 
NN0 )
143135, 142syl 15 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e. 
NN0 )
144143nn0red 10108 . . . 4  |-  ( ph  ->  ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR )
145 peano2re 9072 . . . 4  |-  ( (
# `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  e.  RR  ->  ( ( # `
 ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  e.  RR )
146144, 145syl 15 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  e.  RR )
147 peano2nn0 10093 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
148118, 147syl 15 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
149121, 148eqeltrd 2432 . . . 4  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
150149nn0red 10108 . . 3  |-  ( ph  ->  ( D `  F
)  e.  RR )
151 hashun2 11455 . . . . 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 } ) ) )
152135, 136, 151sylancl 643 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) ) )
153 hashsng 11446 . . . . . 6  |-  ( T  e.  ( `' ( O `  F )
" { W }
)  ->  ( # `  { T } )  =  1 )
1541, 153syl 15 . . . . 5  |-  ( ph  ->  ( # `  { T } )  =  1 )
155154oveq2d 5958 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  ( # `  { T } ) )  =  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
156152, 155breqtrd 4126 . . 3  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 ) )
157118nn0red 10108 . . . . 5  |-  ( ph  ->  N  e.  RR )
158 1re 8924 . . . . . 6  |-  1  e.  RR
159158a1i 10 . . . . 5  |-  ( ph  ->  1  e.  RR )
160144, 157, 159, 133leadd1dd 9473 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( N  +  1 ) )
161160, 121breqtrrd 4128 . . 3  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( F (quot1p `  R ) G ) ) " { W } ) )  +  1 )  <_  ( D `  F )
)
162141, 146, 150, 156, 161letrd 9060 . 2  |-  ( ph  ->  ( # `  (
( `' ( O `
 ( F (quot1p `  R ) G ) ) " { W } )  u.  { T } ) )  <_ 
( D `  F
) )
163113, 162eqbrtrd 4122 1  |-  ( ph  ->  ( # `  ( `' ( O `  F ) " { W } ) )  <_ 
( D `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    u. cun 3226   {csn 3716   class class class wbr 4102   `'ccnv 4767   "cima 4771    Fn wfn 5329   -->wf 5330   ` cfv 5334  (class class class)co 5942    o Fcof 6160   Fincfn 6948   RRcr 8823   1c1 8825    + caddc 8827    <_ cle 8955   NN0cn0 10054   #chash 11427   Basecbs 13239   .rcmulr 13300    ^s cpws 13440   0gc0g 13493   -gcsg 14458   Ringcrg 15430   CRingccrg 15431   ||rcdsr 15513   RingHom crh 15587  NzRingcnzr 16102  Domncdomn 16114  IDomncidom 16115  algSccascl 16145  var1cv1 16344  Poly1cpl1 16345  eval1ce1 16347   deg1 cdg1 19538  Monic1pcmn1 19609  Unic1pcuc1p 19610  quot1pcq1p 19611
This theorem is referenced by:  fta1g  19651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-ofr 6163  df-1st 6206  df-2nd 6207  df-tpos 6318  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-fz 10872  df-fzo 10960  df-seq 11136  df-hash 11428  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-hom 13323  df-cco 13324  df-prds 13441  df-pws 13443  df-0g 13497  df-gsum 13498  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-mhm 14508  df-submnd 14509  df-grp 14582  df-minusg 14583  df-sbg 14584  df-mulg 14585  df-subg 14711  df-ghm 14774  df-cntz 14886  df-cmn 15184  df-abl 15185  df-mgp 15419  df-rng 15433  df-cring 15434  df-ur 15435  df-oppr 15498  df-dvdsr 15516  df-unit 15517  df-invr 15547  df-rnghom 15589  df-subrg 15636  df-lmod 15722  df-lss 15783  df-lsp 15822  df-nzr 16103  df-rlreg 16117  df-domn 16118  df-idom 16119  df-assa 16146  df-asp 16147  df-ascl 16148  df-psr 16191  df-mvr 16192  df-mpl 16193  df-evls 16194  df-evl 16195  df-opsr 16199  df-psr1 16350  df-vr1 16351  df-ply1 16352  df-evl1 16354  df-coe1 16355  df-cnfld 16477  df-mdeg 19539  df-deg1 19540  df-mon1 19614  df-uc1p 19615  df-q1p 19616  df-r1p 19617
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