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Theorem fta1lem 21748
Description: Lemma for fta1 21749. (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 3995 . . . . . . . . . 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 21641 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
7 ffn 5554 . . . . . . . . . . 11  |-  ( F : CC --> CC  ->  F  Fn  CC )
8 fniniseg 5819 . . . . . . . . . . 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 2438 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { A } ) )  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
1413facth 21747 . . . . . . . 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 1218 . . . . . . 7  |-  ( ph  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) )
1615cnveqd 5010 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )
1716imaeq1d 5163 . . . . 5  |-  ( ph  ->  ( `' F " { 0 } )  =  ( `' ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) " {
0 } ) )
18 cnex 9355 . . . . . . 7  |-  CC  e.  _V
1918a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
20 ssid 3370 . . . . . . . . 9  |-  CC  C_  CC
21 ax-1cn 9332 . . . . . . . . 9  |-  1  e.  CC
22 plyid 21652 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
2320, 21, 22mp2an 672 . . . . . . . 8  |-  Xp  e.  (Poly `  CC )
24 plyconst 21649 . . . . . . . . 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 21665 . . . . . . . 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 21641 . . . . . . 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 21745 . . . . . . . . . . . 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 1001 . . . . . . . . . 10  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  1 )
33 ax-1ne0 9343 . . . . . . . . . . 11  |-  1  =/=  0
3433a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3532, 34eqnetrd 2621 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =/=  0 )
36 fveq2 5686 . . . . . . . . . . 11  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  (deg `  0p
) )
37 dgr0 21704 . . . . . . . . . . 11  |-  (deg ` 
0p )  =  0
3836, 37syl6eq 2486 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  0 )
3938necon3i 2645 . . . . . . . . 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 21743 . . . . . . . 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 1218 . . . . . . 7  |-  ( ph  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC ) )
43 plyf 21641 . . . . . . 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 21723 . . . . . 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 1218 . . . . 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 1002 . . . . . 6  |-  ( ph  ->  ( `' ( Xp  oF  -  ( CC  X.  { A } ) ) " { 0 } )  =  { A }
)
4847uneq1d 3504 . . . . 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 2474 . . . 4  |-  ( ph  ->  ( `' F " { 0 } )  =  ( { A }  u.  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) ) )
50 fta1.1 . . . 4  |-  R  =  ( `' F " { 0 } )
51 uncom 3495 . . . 4  |-  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } )  =  ( { A }  u.  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )
5249, 50, 513eqtr4g 2495 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
533simprd 463 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0p )
5415eqcomd 2443 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )  =  F )
55 0cnd 9371 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
56 mul01 9540 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5756adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5819, 29, 55, 55, 57caofid1 6345 . . . . . . . . . 10  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
59 df-0p 21123 . . . . . . . . . . 11  |-  0p  =  ( CC  X.  { 0 } )
6059oveq2i 6097 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  0p )  =  ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )
6158, 60, 593eqtr4g 2495 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  0p )  =  0p )
6253, 54, 613netr4d 2630 . . . . . . . 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 6094 . . . . . . . . 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 2645 . . . . . . . 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 3995 . . . . . . 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 21676 . . . . . . . . 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 10630 . . . . . . 7  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  e.  CC )
73 fta1.2 . . . . . . . 8  |-  ( ph  ->  D  e.  NN0 )
7473nn0cnd 10630 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
75 addcom 9547 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7621, 74, 75sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7715fveq2d 5690 . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )
80 eqid 2438 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )
8179, 80dgrmul 21712 . . . . . . . . . 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 1219 . . . . . . . . 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 2478 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( Xp  oF  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
8432oveq1d 6101 . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8669, 72, 74, 85addcanad 9566 . . . . . 6  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D )
87 fveq2 5686 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) ) )
8887eqeq1d 2446 . . . . . . . 8  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( (deg `  g )  =  D  <-> 
(deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D ) )
89 cnveq 5008 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )
9089imaeq1d 5163 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9190eleq1d 2504 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( ( `' g " {
0 } )  e. 
Fin 
<->  ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin ) )
9290fveq2d 5690 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( # `  ( `' g " {
0 } ) )  =  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
9392, 87breq12d 4300 . . . . . . . . 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 3064 . . . . . 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 7382 . . . 4  |-  { A }  e.  Fin
100 unfi 7571 . . . 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 2512 . 2  |-  ( ph  ->  R  e.  Fin )
10352fveq2d 5690 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
104 hashcl 12118 . . . . . 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 10629 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  e.  RR )
107 hashcl 12118 . . . . . . 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 10629 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
110 peano2re 9534 . . . . 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 21676 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1134, 112syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
114113nn0red 10629 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
115 hashun2 12138 . . . . . 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 12128 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
11811, 117syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
119118oveq2d 6102 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  ( # `  { A } ) )  =  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 ) )
120116, 119breqtrd 4311 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (
( # `  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  +  1 ) )
12173nn0red 10629 . . . . . 6  |-  ( ph  ->  D  e.  RR )
122 1red 9393 . . . . . 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 4311 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
125109, 121, 122, 124leadd1dd 9945 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
126125, 78breqtrrd 4313 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
127106, 111, 114, 120, 126letrd 9520 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (deg `  F ) )
128103, 127eqbrtrd 4307 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   _Vcvv 2967    \ cdif 3320    u. cun 3321    C_ wss 3323   {csn 3872   class class class wbr 4287    X. cxp 4833   `'ccnv 4834   "cima 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    oFcof 6313   Fincfn 7302   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    <_ cle 9411    - cmin 9587   NN0cn0 10571   #chash 12095   0pc0p 21122  Polycply 21627   Xpcidp 21628  degcdgr 21630   quot cquot 21731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-0p 21123  df-ply 21631  df-idp 21632  df-coe 21633  df-dgr 21634  df-quot 21732
This theorem is referenced by:  fta1  21749
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