MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fta1lem Unicode version

Theorem fta1lem 20084
Description: Lemma for fta1 20085. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
fta1.1  |-  R  =  ( `' F " { 0 } )
fta1.2  |-  ( ph  ->  D  e.  NN0 )
fta1.3  |-  ( ph  ->  F  e.  ( (Poly `  CC )  \  {
0 p } ) )
fta1.4  |-  ( ph  ->  (deg `  F )  =  ( D  + 
1 ) )
fta1.5  |-  ( ph  ->  A  e.  ( `' F " { 0 } ) )
fta1.6  |-  ( ph  ->  A. g  e.  ( (Poly `  CC )  \  { 0 p }
) ( (deg `  g )  =  D  ->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
Assertion
Ref Expression
fta1lem  |-  ( ph  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
Distinct variable groups:    A, g    D, g    g, F
Allowed substitution hints:    ph( g)    R( g)

Proof of Theorem fta1lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fta1.3 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( (Poly `  CC )  \  {
0 p } ) )
2 eldifsn 3863 . . . . . . . . . 10  |-  ( F  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0 p ) )
31, 2sylib 189 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  (Poly `  CC )  /\  F  =/=  0 p ) )
43simpld 446 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  CC ) )
5 fta1.5 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( `' F " { 0 } ) )
6 plyf 19977 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
74, 6syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F : CC --> CC )
8 ffn 5524 . . . . . . . . . . . 12  |-  ( F : CC --> CC  ->  F  Fn  CC )
97, 8syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  CC )
10 fniniseg 5783 . . . . . . . . . . 11  |-  ( F  Fn  CC  ->  ( A  e.  ( `' F " { 0 } )  <->  ( A  e.  CC  /\  ( F `
 A )  =  0 ) ) )
119, 10syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( `' F " { 0 } )  <->  ( A  e.  CC  /\  ( F `
 A )  =  0 ) ) )
125, 11mpbid 202 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  ( F `  A
)  =  0 ) )
1312simpld 446 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1412simprd 450 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  0 )
15 eqid 2380 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { A } ) )  =  ( X p  o F  -  ( CC  X.  { A } ) )
1615facth 20083 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )
174, 13, 14, 16syl3anc 1184 . . . . . . 7  |-  ( ph  ->  F  =  ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )
1817cnveqd 4981 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )
1918imaeq1d 5135 . . . . 5  |-  ( ph  ->  ( `' F " { 0 } )  =  ( `' ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) )
" { 0 } ) )
20 cnex 8997 . . . . . . 7  |-  CC  e.  _V
2120a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
22 ssid 3303 . . . . . . . . 9  |-  CC  C_  CC
23 ax-1cn 8974 . . . . . . . . 9  |-  1  e.  CC
24 plyid 19988 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  X p  e.  (Poly `  CC ) )
2522, 23, 24mp2an 654 . . . . . . . 8  |-  X p  e.  (Poly `  CC )
26 plyconst 19985 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
2722, 13, 26sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( CC  X.  { A } )  e.  (Poly `  CC ) )
28 plysubcl 20001 . . . . . . . 8  |-  ( ( X p  e.  (Poly `  CC )  /\  ( CC  X.  { A }
)  e.  (Poly `  CC ) )  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
2925, 27, 28sylancr 645 . . . . . . 7  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
30 plyf 19977 . . . . . . 7  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC )  ->  ( X p  o F  -  ( CC  X.  { A }
) ) : CC --> CC )
3129, 30syl 16 . . . . . 6  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { A } ) ) : CC --> CC )
3215plyremlem 20081 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
( X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { A } ) )
" { 0 } )  =  { A } ) )
3313, 32syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { A } ) )
" { 0 } )  =  { A } ) )
3433simp2d 970 . . . . . . . . . 10  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  1 )
35 ax-1ne0 8985 . . . . . . . . . . 11  |-  1  =/=  0
3635a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3734, 36eqnetrd 2561 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =/=  0 )
38 fveq2 5661 . . . . . . . . . . 11  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  (deg `  0 p
) )
39 dgr0 20040 . . . . . . . . . . 11  |-  (deg ` 
0 p )  =  0
4038, 39syl6eq 2428 . . . . . . . . . 10  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  0 )
4140necon3i 2582 . . . . . . . . 9  |-  ( (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )  =/=  0  ->  ( X p  o F  -  ( CC  X.  { A }
) )  =/=  0 p )
4237, 41syl 16 . . . . . . . 8  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { A } ) )  =/=  0 p )
43 quotcl2 20079 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC )  /\  ( X p  o F  -  ( CC  X.  { A }
) )  =/=  0 p )  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC ) )
444, 29, 42, 43syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC ) )
45 plyf 19977 . . . . . . 7  |-  ( ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) )  e.  (Poly `  CC )  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) : CC --> CC )
4644, 45syl 16 . . . . . 6  |-  ( ph  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) : CC --> CC )
47 ofmulrt 20059 . . . . . 6  |-  ( ( CC  e.  _V  /\  ( X p  o F  -  ( CC  X.  { A } ) ) : CC --> CC  /\  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) : CC --> CC )  -> 
( `' ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) " { 0 } )  =  ( ( `' ( X p  o F  -  ( CC  X.  { A } ) ) " { 0 } )  u.  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
4821, 31, 46, 47syl3anc 1184 . . . . 5  |-  ( ph  ->  ( `' ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) " { 0 } )  =  ( ( `' ( X p  o F  -  ( CC  X.  { A } ) ) " { 0 } )  u.  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
4933simp3d 971 . . . . . 6  |-  ( ph  ->  ( `' ( X p  o F  -  ( CC  X.  { A } ) ) " { 0 } )  =  { A }
)
5049uneq1d 3436 . . . . 5  |-  ( ph  ->  ( ( `' ( X p  o F  -  ( CC  X.  { A } ) )
" { 0 } )  u.  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  =  ( { A }  u.  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
5119, 48, 503eqtrd 2416 . . . 4  |-  ( ph  ->  ( `' F " { 0 } )  =  ( { A }  u.  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } ) ) )
52 fta1.1 . . . 4  |-  R  =  ( `' F " { 0 } )
53 uncom 3427 . . . 4  |-  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } )  =  ( { A }  u.  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )
5451, 52, 533eqtr4g 2437 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
553simprd 450 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0 p )
5617eqcomd 2385 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )  =  F )
57 0cn 9010 . . . . . . . . . . . 12  |-  0  e.  CC
5857a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
59 mul01 9170 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
6059adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
6121, 31, 58, 58, 60caofid1 6266 . . . . . . . . . 10  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
62 df-0p 19422 . . . . . . . . . . 11  |-  0 p  =  ( CC  X.  { 0 } )
6362oveq2i 6024 . . . . . . . . . 10  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  0 p )  =  ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( CC  X.  { 0 } ) )
6461, 63, 623eqtr4g 2437 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  0 p )  =  0 p )
6555, 56, 643netr4d 2570 . . . . . . . 8  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )  =/=  (
( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  0 p ) )
66 oveq2 6021 . . . . . . . . 9  |-  ( ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  0 p  ->  (
( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) )  =  ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  0 p ) )
6766necon3i 2582 . . . . . . . 8  |-  ( ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) )  =/=  ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  0 p )  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  =/=  0 p )
6865, 67syl 16 . . . . . . 7  |-  ( ph  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  =/=  0 p )
69 eldifsn 3863 . . . . . . 7  |-  ( ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) )  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC )  /\  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  =/=  0 p ) )
7044, 68, 69sylanbrc 646 . . . . . 6  |-  ( ph  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  e.  ( (Poly `  CC )  \  { 0 p } ) )
71 fta1.6 . . . . . 6  |-  ( ph  ->  A. g  e.  ( (Poly `  CC )  \  { 0 p }
) ( (deg `  g )  =  D  ->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
7223a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
73 dgrcl 20012 . . . . . . . . 9  |-  ( ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) )  e.  (Poly `  CC )  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e. 
NN0 )
7444, 73syl 16 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e. 
NN0 )
7574nn0cnd 10201 . . . . . . 7  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e.  CC )
76 fta1.2 . . . . . . . 8  |-  ( ph  ->  D  e.  NN0 )
7776nn0cnd 10201 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
78 addcom 9177 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7923, 77, 78sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
8017fveq2d 5665 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =  (deg `  ( (
X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
81 fta1.4 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =  ( D  + 
1 ) )
82 eqid 2380 . . . . . . . . . . 11  |-  (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  =  (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )
83 eqid 2380 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )
8482, 83dgrmul 20048 . . . . . . . . . 10  |-  ( ( ( ( X p  o F  -  ( CC  X.  { A }
) )  e.  (Poly `  CC )  /\  (
X p  o F  -  ( CC  X.  { A } ) )  =/=  0 p )  /\  ( ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC )  /\  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  =/=  0 p ) )  -> 
(deg `  ( (
X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )  =  ( (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  +  (deg
`  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
8529, 42, 44, 68, 84syl22anc 1185 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( (
X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )  =  ( (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  +  (deg
`  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
8680, 81, 853eqtr3d 2420 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
8734oveq1d 6028 . . . . . . . 8  |-  ( ph  ->  ( (deg `  (
X p  o F  -  ( CC  X.  { A } ) ) )  +  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) ) )  =  ( 1  +  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
8879, 86, 873eqtrrd 2417 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8972, 75, 77, 88addcanad 9196 . . . . . 6  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  =  D )
90 fveq2 5661 . . . . . . . . 9  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) ) )
9190eqeq1d 2388 . . . . . . . 8  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( (deg `  g )  =  D  <-> 
(deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  =  D ) )
92 cnveq 4979 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )
9392imaeq1d 5135 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9493eleq1d 2446 . . . . . . . . 9  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( ( `' g " {
0 } )  e. 
Fin 
<->  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  e. 
Fin ) )
9593fveq2d 5665 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( # `  ( `' g " {
0 } ) )  =  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
9695, 90breq12d 4159 . . . . . . . . 9  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( ( # `
 ( `' g
" { 0 } ) )  <_  (deg `  g )  <->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
9794, 96anbi12d 692 . . . . . . . 8  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( (
( `' g " { 0 } )  e.  Fin  /\  ( # `
 ( `' g
" { 0 } ) )  <_  (deg `  g ) )  <->  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  e.  Fin  /\  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) ) )
9891, 97imbi12d 312 . . . . . . 7  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( (
(deg `  g )  =  D  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  <->  ( (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  =  D  -> 
( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  e.  Fin  /\  ( # `
 ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  <_  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) ) ) )
9998rspcv 2984 . . . . . 6  |-  ( ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) )  e.  ( (Poly `  CC )  \  { 0 p } )  ->  ( A. g  e.  (
(Poly `  CC )  \  { 0 p }
) ( (deg `  g )  =  D  ->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  =  D  ->  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  e.  Fin  /\  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) ) ) )
10070, 71, 89, 99syl3c 59 . . . . 5  |-  ( ph  ->  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  e.  Fin  /\  ( # `
 ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  <_  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
101100simpld 446 . . . 4  |-  ( ph  ->  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  e. 
Fin )
102 snfi 7116 . . . 4  |-  { A }  e.  Fin
103 unfi 7303 . . . 4  |-  ( ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  e. 
Fin  /\  { A }  e.  Fin )  ->  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
)  e.  Fin )
104101, 102, 103sylancl 644 . . 3  |-  ( ph  ->  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
)  e.  Fin )
10554, 104eqeltrd 2454 . 2  |-  ( ph  ->  R  e.  Fin )
10654fveq2d 5665 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
107 hashcl 11559 . . . . . 6  |-  ( ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } )  e.  Fin  ->  ( # `
 ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )
" { 0 } )  u.  { A } ) )  e. 
NN0 )
108104, 107syl 16 . . . . 5  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  e.  NN0 )
109108nn0red 10200 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  e.  RR )
110 hashcl 11559 . . . . . . 7  |-  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  e.  Fin  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e. 
NN0 )
111101, 110syl 16 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e. 
NN0 )
112111nn0red 10200 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
113 peano2re 9164 . . . . 5  |-  ( (
# `  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  e.  RR  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  e.  RR )
114112, 113syl 16 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  e.  RR )
115 dgrcl 20012 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1164, 115syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
117116nn0red 10200 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
118 hashun2 11577 . . . . . 6  |-  ( ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  e. 
Fin  /\  { A }  e.  Fin )  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  <_  ( ( # `
 ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  +  ( # `  { A } ) ) )
119101, 102, 118sylancl 644 . . . . 5  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  <_  ( ( # `
 ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  +  ( # `  { A } ) ) )
120 hashsng 11567 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
12113, 120syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
122121oveq2d 6029 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  ( # `  { A } ) )  =  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 ) )
123119, 122breqtrd 4170 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  <_  ( ( # `
 ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  +  1 ) )
12476nn0red 10200 . . . . . 6  |-  ( ph  ->  D  e.  RR )
125 1re 9016 . . . . . . 7  |-  1  e.  RR
126125a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
127100simprd 450 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )
128127, 89breqtrd 4170 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
129112, 124, 126, 128leadd1dd 9565 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
130129, 81breqtrrd 4172 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
131109, 114, 117, 123, 130letrd 9152 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  <_  (deg `  F
) )
132106, 131eqbrtrd 4166 . 2  |-  ( ph  ->  ( # `  R
)  <_  (deg `  F
) )
133105, 132jca 519 1  |-  ( ph  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   _Vcvv 2892    \ cdif 3253    u. cun 3254    C_ wss 3256   {csn 3750   class class class wbr 4146    X. cxp 4809   `'ccnv 4810   "cima 4814    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013    o Fcof 6235   Fincfn 7038   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    <_ cle 9047    - cmin 9216   NN0cn0 10146   #chash 11538   0 pc0p 19421  Polycply 19963   X pcidp 19964  degcdgr 19966   quot cquot 20067
This theorem is referenced by:  fta1  20085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-rlim 12203  df-sum 12400  df-0p 19422  df-ply 19967  df-idp 19968  df-coe 19969  df-dgr 19970  df-quot 20068
  Copyright terms: Public domain W3C validator