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Theorem fta1lem 22681
Description: Lemma for fta1 22682. (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 4140 . . . . . . . . . 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 22573 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
7 ffn 5721 . . . . . . . . . . 11  |-  ( F : CC --> CC  ->  F  Fn  CC )
8 fniniseg 5993 . . . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { A } ) )  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
1413facth 22680 . . . . . . . 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 1229 . . . . . . 7  |-  ( ph  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) )
1615cnveqd 5168 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )
1716imaeq1d 5326 . . . . 5  |-  ( ph  ->  ( `' F " { 0 } )  =  ( `' ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) " {
0 } ) )
18 cnex 9576 . . . . . . 7  |-  CC  e.  _V
1918a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
20 ssid 3508 . . . . . . . . 9  |-  CC  C_  CC
21 ax-1cn 9553 . . . . . . . . 9  |-  1  e.  CC
22 plyid 22584 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
2320, 21, 22mp2an 672 . . . . . . . 8  |-  Xp  e.  (Poly `  CC )
24 plyconst 22581 . . . . . . . . 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 22597 . . . . . . . 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 22573 . . . . . . 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 22678 . . . . . . . . . . . 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 1010 . . . . . . . . . 10  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  1 )
33 ax-1ne0 9564 . . . . . . . . . . 11  |-  1  =/=  0
3433a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3532, 34eqnetrd 2736 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =/=  0 )
36 fveq2 5856 . . . . . . . . . . 11  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  (deg `  0p
) )
37 dgr0 22637 . . . . . . . . . . 11  |-  (deg ` 
0p )  =  0
3836, 37syl6eq 2500 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  0 )
3938necon3i 2683 . . . . . . . . 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 22676 . . . . . . . 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 1229 . . . . . . 7  |-  ( ph  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC ) )
43 plyf 22573 . . . . . . 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 22656 . . . . . 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 1229 . . . . 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 1011 . . . . . 6  |-  ( ph  ->  ( `' ( Xp  oF  -  ( CC  X.  { A } ) ) " { 0 } )  =  { A }
)
4847uneq1d 3642 . . . . 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 2488 . . . 4  |-  ( ph  ->  ( `' F " { 0 } )  =  ( { A }  u.  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) ) )
50 fta1.1 . . . 4  |-  R  =  ( `' F " { 0 } )
51 uncom 3633 . . . 4  |-  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } )  =  ( { A }  u.  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )
5249, 50, 513eqtr4g 2509 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
533simprd 463 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0p )
5415eqcomd 2451 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )  =  F )
55 0cnd 9592 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
56 mul01 9762 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5756adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5819, 29, 55, 55, 57caofid1 6555 . . . . . . . . . 10  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
59 df-0p 22055 . . . . . . . . . . 11  |-  0p  =  ( CC  X.  { 0 } )
6059oveq2i 6292 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  0p )  =  ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )
6158, 60, 593eqtr4g 2509 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  0p )  =  0p )
6253, 54, 613netr4d 2748 . . . . . . . 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 6289 . . . . . . . . 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 2683 . . . . . . . 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 4140 . . . . . . 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 22608 . . . . . . . . 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 10861 . . . . . . 7  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  e.  CC )
73 fta1.2 . . . . . . . 8  |-  ( ph  ->  D  e.  NN0 )
7473nn0cnd 10861 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
75 addcom 9769 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7621, 74, 75sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7715fveq2d 5860 . . . . . . . . 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 2443 . . . . . . . . . . 11  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )
80 eqid 2443 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )
8179, 80dgrmul 22645 . . . . . . . . . 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 1230 . . . . . . . . 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 2492 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( Xp  oF  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
8432oveq1d 6296 . . . . . . . 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 2489 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8669, 72, 74, 85addcanad 9788 . . . . . 6  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D )
87 fveq2 5856 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) ) )
8887eqeq1d 2445 . . . . . . . 8  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( (deg `  g )  =  D  <-> 
(deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D ) )
89 cnveq 5166 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )
9089imaeq1d 5326 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9190eleq1d 2512 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( ( `' g " {
0 } )  e. 
Fin 
<->  ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin ) )
9290fveq2d 5860 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( # `  ( `' g " {
0 } ) )  =  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
9392, 87breq12d 4450 . . . . . . . . 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 3192 . . . . . 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 7598 . . . 4  |-  { A }  e.  Fin
100 unfi 7789 . . . 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 2531 . 2  |-  ( ph  ->  R  e.  Fin )
10352fveq2d 5860 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
104 hashcl 12410 . . . . . 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 10860 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  e.  RR )
107 hashcl 12410 . . . . . . 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 10860 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
110 peano2re 9756 . . . . 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 22608 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1134, 112syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
114113nn0red 10860 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
115 hashun2 12433 . . . . . 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 12420 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
11811, 117syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
119118oveq2d 6297 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  ( # `  { A } ) )  =  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 ) )
120116, 119breqtrd 4461 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (
( # `  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  +  1 ) )
12173nn0red 10860 . . . . . 6  |-  ( ph  ->  D  e.  RR )
122 1red 9614 . . . . . 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 4461 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
125109, 121, 122, 124leadd1dd 10173 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
126125, 78breqtrrd 4463 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
127106, 111, 114, 120, 126letrd 9742 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (deg `  F ) )
128103, 127eqbrtrd 4457 . 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    \ cdif 3458    u. cun 3459    C_ wss 3461   {csn 4014   class class class wbr 4437    X. cxp 4987   `'ccnv 4988   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   Fincfn 7518   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    <_ cle 9632    - cmin 9810   NN0cn0 10802   #chash 12387   0pc0p 22054  Polycply 22559   Xpcidp 22560  degcdgr 22562   quot cquot 22664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-0p 22055  df-ply 22563  df-idp 22564  df-coe 22565  df-dgr 22566  df-quot 22665
This theorem is referenced by:  fta1  22682
  Copyright terms: Public domain W3C validator