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Theorem fta1lem 22570
Description: Lemma for fta1 22571. (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 )  \  {
0p } ) )
fta1.4  |-  ( ph  ->  (deg `  F )  =  ( D  + 
1 ) )
fta1.5  |-  ( ph  ->  A  e.  ( `' F " { 0 } ) )
fta1.6  |-  ( ph  ->  A. g  e.  ( (Poly `  CC )  \  { 0p }
) ( (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 )  \  {
0p } ) )
2 eldifsn 4158 . . . . . . . . . 10  |-  ( F  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0p ) )
31, 2sylib 196 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  (Poly `  CC )  /\  F  =/=  0p ) )
43simpld 459 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  CC ) )
5 fta1.5 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( `' F " { 0 } ) )
6 plyf 22463 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
7 ffn 5737 . . . . . . . . . . 11  |-  ( F : CC --> CC  ->  F  Fn  CC )
8 fniniseg 6009 . . . . . . . . . . 11  |-  ( F  Fn  CC  ->  ( A  e.  ( `' F " { 0 } )  <->  ( A  e.  CC  /\  ( F `
 A )  =  0 ) ) )
94, 6, 7, 84syl 21 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( `' F " { 0 } )  <->  ( A  e.  CC  /\  ( F `
 A )  =  0 ) ) )
105, 9mpbid 210 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  ( F `  A
)  =  0 ) )
1110simpld 459 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1210simprd 463 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  0 )
13 eqid 2467 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { A } ) )  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
1413facth 22569 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) )
154, 11, 12, 14syl3anc 1228 . . . . . . 7  |-  ( ph  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) )
1615cnveqd 5184 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )
1716imaeq1d 5342 . . . . 5  |-  ( ph  ->  ( `' F " { 0 } )  =  ( `' ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) " {
0 } ) )
18 cnex 9585 . . . . . . 7  |-  CC  e.  _V
1918a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
20 ssid 3528 . . . . . . . . 9  |-  CC  C_  CC
21 ax-1cn 9562 . . . . . . . . 9  |-  1  e.  CC
22 plyid 22474 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
2320, 21, 22mp2an 672 . . . . . . . 8  |-  Xp  e.  (Poly `  CC )
24 plyconst 22471 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
2520, 11, 24sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( CC  X.  { A } )  e.  (Poly `  CC ) )
26 plysubcl 22487 . . . . . . . 8  |-  ( ( Xp  e.  (Poly `  CC )  /\  ( CC  X.  { A }
)  e.  (Poly `  CC ) )  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
2723, 25, 26sylancr 663 . . . . . . 7  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { A }
) )  e.  (Poly `  CC ) )
28 plyf 22463 . . . . . . 7  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  CC )  ->  ( Xp  oF  -  ( CC  X.  { A }
) ) : CC --> CC )
2927, 28syl 16 . . . . . 6  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { A }
) ) : CC --> CC )
3013plyremlem 22567 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { A }
) ) " {
0 } )  =  { A } ) )
3111, 30syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { A }
) ) " {
0 } )  =  { A } ) )
3231simp2d 1009 . . . . . . . . . 10  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  1 )
33 ax-1ne0 9573 . . . . . . . . . . 11  |-  1  =/=  0
3433a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3532, 34eqnetrd 2760 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =/=  0 )
36 fveq2 5872 . . . . . . . . . . 11  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  (deg `  0p
) )
37 dgr0 22526 . . . . . . . . . . 11  |-  (deg ` 
0p )  =  0
3836, 37syl6eq 2524 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  0 )
3938necon3i 2707 . . . . . . . . 9  |-  ( (deg
`  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =/=  0  ->  ( Xp  oF  -  ( CC  X.  { A }
) )  =/=  0p )
4035, 39syl 16 . . . . . . . 8  |-  ( ph  ->  ( Xp  oF  -  ( CC 
X.  { A }
) )  =/=  0p )
41 quotcl2 22565 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  CC )  /\  ( Xp  oF  -  ( CC  X.  { A }
) )  =/=  0p )  ->  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  e.  (Poly `  CC )
)
424, 27, 40, 41syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC ) )
43 plyf 22463 . . . . . . 7  |-  ( ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) )  e.  (Poly `  CC )  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) ) : CC --> CC )
4442, 43syl 16 . . . . . 6  |-  ( ph  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) ) : CC --> CC )
45 ofmulrt 22545 . . . . . 6  |-  ( ( CC  e.  _V  /\  ( Xp  oF  -  ( CC 
X.  { A }
) ) : CC --> CC  /\  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) : CC --> CC )  ->  ( `' ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) " {
0 } )  =  ( ( `' ( Xp  oF  -  ( CC  X.  { A } ) )
" { 0 } )  u.  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) ) )
4619, 29, 44, 45syl3anc 1228 . . . . 5  |-  ( ph  ->  ( `' ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )
" { 0 } )  =  ( ( `' ( Xp  oF  -  ( CC  X.  { A }
) ) " {
0 } )  u.  ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } ) ) )
4731simp3d 1010 . . . . . 6  |-  ( ph  ->  ( `' ( Xp  oF  -  ( CC  X.  { A } ) ) " { 0 } )  =  { A }
)
4847uneq1d 3662 . . . . 5  |-  ( ph  ->  ( ( `' ( Xp  oF  -  ( CC  X.  { A } ) )
" { 0 } )  u.  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  =  ( { A }  u.  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
4917, 46, 483eqtrd 2512 . . . 4  |-  ( ph  ->  ( `' F " { 0 } )  =  ( { A }  u.  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) ) )
50 fta1.1 . . . 4  |-  R  =  ( `' F " { 0 } )
51 uncom 3653 . . . 4  |-  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } )  =  ( { A }  u.  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )
5249, 50, 513eqtr4g 2533 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
533simprd 463 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0p )
5415eqcomd 2475 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )  =  F )
55 0cnd 9601 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
56 mul01 9770 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5756adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5819, 29, 55, 55, 57caofid1 6565 . . . . . . . . . 10  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
59 df-0p 21945 . . . . . . . . . . 11  |-  0p  =  ( CC  X.  { 0 } )
6059oveq2i 6306 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  0p )  =  ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )
6158, 60, 593eqtr4g 2533 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  0p )  =  0p )
6253, 54, 613netr4d 2772 . . . . . . . 8  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )  =/=  (
( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  0p ) )
63 oveq2 6303 . . . . . . . . 9  |-  ( ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  0p  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )  =  ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  0p ) )
6463necon3i 2707 . . . . . . . 8  |-  ( ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )  =/=  (
( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  0p )  ->  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  =/=  0p )
6562, 64syl 16 . . . . . . 7  |-  ( ph  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  =/=  0p )
66 eldifsn 4158 . . . . . . 7  |-  ( ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) )  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  e.  (Poly `  CC )  /\  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  =/=  0p ) )
6742, 65, 66sylanbrc 664 . . . . . 6  |-  ( ph  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  e.  ( (Poly `  CC )  \  { 0p } ) )
68 fta1.6 . . . . . 6  |-  ( ph  ->  A. g  e.  ( (Poly `  CC )  \  { 0p }
) ( (deg `  g )  =  D  ->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
6921a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
70 dgrcl 22498 . . . . . . . . 9  |-  ( ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) )  e.  (Poly `  CC )  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  e.  NN0 )
7142, 70syl 16 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  e.  NN0 )
7271nn0cnd 10866 . . . . . . 7  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  e.  CC )
73 fta1.2 . . . . . . . 8  |-  ( ph  ->  D  e.  NN0 )
7473nn0cnd 10866 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
75 addcom 9777 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7621, 74, 75sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7715fveq2d 5876 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =  (deg `  ( (
Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
78 fta1.4 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =  ( D  + 
1 ) )
79 eqid 2467 . . . . . . . . . . 11  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )
80 eqid 2467 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )
8179, 80dgrmul 22534 . . . . . . . . . 10  |-  ( ( ( ( Xp  oF  -  ( CC  X.  { A }
) )  e.  (Poly `  CC )  /\  (
Xp  oF  -  ( CC  X.  { A } ) )  =/=  0p )  /\  ( ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  e.  (Poly `  CC )  /\  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  =/=  0p ) )  ->  (deg `  (
( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )  =  ( (deg `  (
Xp  oF  -  ( CC  X.  { A } ) ) )  +  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) ) ) )
8227, 40, 42, 65, 81syl22anc 1229 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( (
Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) )  =  ( (deg
`  ( Xp  oF  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
8377, 78, 823eqtr3d 2516 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( Xp  oF  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
8432oveq1d 6310 . . . . . . . 8  |-  ( ph  ->  ( (deg `  (
Xp  oF  -  ( CC  X.  { A } ) ) )  +  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) ) )  =  ( 1  +  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
8576, 83, 843eqtrrd 2513 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8669, 72, 74, 85addcanad 9796 . . . . . 6  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D )
87 fveq2 5872 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) ) )
8887eqeq1d 2469 . . . . . . . 8  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( (deg `  g )  =  D  <-> 
(deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D ) )
89 cnveq 5182 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )
9089imaeq1d 5342 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9190eleq1d 2536 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( ( `' g " {
0 } )  e. 
Fin 
<->  ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin ) )
9290fveq2d 5876 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( # `  ( `' g " {
0 } ) )  =  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
9392, 87breq12d 4466 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( ( # `
 ( `' g
" { 0 } ) )  <_  (deg `  g )  <->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
9491, 93anbi12d 710 . . . . . . . 8  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( (
( `' g " { 0 } )  e.  Fin  /\  ( # `
 ( `' g
" { 0 } ) )  <_  (deg `  g ) )  <->  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  e.  Fin  /\  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) ) )
9588, 94imbi12d 320 . . . . . . 7  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( (
(deg `  g )  =  D  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  <->  ( (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) ) )  =  D  -> 
( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } )  e.  Fin  /\  ( # `
 ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  <_  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) ) ) )
9695rspcv 3215 . . . . . 6  |-  ( ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) )  e.  ( (Poly `  CC )  \  { 0p } )  ->  ( A. g  e.  (
(Poly `  CC )  \  { 0p }
) ( (deg `  g )  =  D  ->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D  ->  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin  /\  ( # `
 ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  <_  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) ) ) )
9767, 68, 86, 96syl3c 61 . . . . 5  |-  ( ph  ->  ( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } )  e.  Fin  /\  ( # `
 ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  <_  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
9897simpld 459 . . . 4  |-  ( ph  ->  ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin )
99 snfi 7608 . . . 4  |-  { A }  e.  Fin
100 unfi 7799 . . . 4  |-  ( ( ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin  /\  { A }  e.  Fin )  ->  ( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } )  u.  { A } )  e.  Fin )
10198, 99, 100sylancl 662 . . 3  |-  ( ph  ->  ( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
)  e.  Fin )
10252, 101eqeltrd 2555 . 2  |-  ( ph  ->  R  e.  Fin )
10352fveq2d 5876 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
104 hashcl 12408 . . . . . 6  |-  ( ( ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
)  e.  Fin  ->  (
# `  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )  e. 
NN0 )
105101, 104syl 16 . . . . 5  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  e.  NN0 )
106105nn0red 10865 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  e.  RR )
107 hashcl 12408 . . . . . . 7  |-  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  e.  Fin  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e. 
NN0 )
10898, 107syl 16 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e. 
NN0 )
109108nn0red 10865 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
110 peano2re 9764 . . . . 5  |-  ( (
# `  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) )  e.  RR  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  e.  RR )
111109, 110syl 16 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  e.  RR )
112 dgrcl 22498 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1134, 112syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
114113nn0red 10865 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
115 hashun2 12431 . . . . . 6  |-  ( ( ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin  /\  { A }  e.  Fin )  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (
( # `  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  +  (
# `  { A } ) ) )
11698, 99, 115sylancl 662 . . . . 5  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (
( # `  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  +  (
# `  { A } ) ) )
117 hashsng 12418 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
11811, 117syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
119118oveq2d 6311 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  ( # `  { A } ) )  =  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 ) )
120116, 119breqtrd 4477 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (
( # `  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  +  1 ) )
12173nn0red 10865 . . . . . 6  |-  ( ph  ->  D  e.  RR )
122 1red 9623 . . . . . 6  |-  ( ph  ->  1  e.  RR )
12397simprd 463 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) )
124123, 86breqtrd 4477 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
125109, 121, 122, 124leadd1dd 10178 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
126125, 78breqtrrd 4479 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
127106, 111, 114, 120, 126letrd 9750 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (deg `  F ) )
128103, 127eqbrtrd 4473 . 2  |-  ( ph  ->  ( # `  R
)  <_  (deg `  F
) )
129102, 128jca 532 1  |-  ( ph  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    \ cdif 3478    u. cun 3479    C_ wss 3481   {csn 4033   class class class wbr 4453    X. cxp 5003   `'ccnv 5004   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   Fincfn 7528   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    <_ cle 9641    - cmin 9817   NN0cn0 10807   #chash 12385   0pc0p 21944  Polycply 22449   Xpcidp 22450  degcdgr 22452   quot cquot 22553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-0p 21945  df-ply 22453  df-idp 22454  df-coe 22455  df-dgr 22456  df-quot 22554
This theorem is referenced by:  fta1  22571
  Copyright terms: Public domain W3C validator