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Theorem fta1lem 21909
Description: Lemma for fta1 21910. (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 4111 . . . . . . . . . 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 21802 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
7 ffn 5670 . . . . . . . . . . 11  |-  ( F : CC --> CC  ->  F  Fn  CC )
8 fniniseg 5936 . . . . . . . . . . 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 2454 . . . . . . . . 9  |-  ( Xp  oF  -  ( CC  X.  { A } ) )  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
1413facth 21908 . . . . . . . 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 1219 . . . . . . 7  |-  ( ph  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) )
1615cnveqd 5126 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )
1716imaeq1d 5279 . . . . 5  |-  ( ph  ->  ( `' F " { 0 } )  =  ( `' ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) " {
0 } ) )
18 cnex 9477 . . . . . . 7  |-  CC  e.  _V
1918a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
20 ssid 3486 . . . . . . . . 9  |-  CC  C_  CC
21 ax-1cn 9454 . . . . . . . . 9  |-  1  e.  CC
22 plyid 21813 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
2320, 21, 22mp2an 672 . . . . . . . 8  |-  Xp  e.  (Poly `  CC )
24 plyconst 21810 . . . . . . . . 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 21826 . . . . . . . 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 21802 . . . . . . 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 21906 . . . . . . . . . . . 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 9465 . . . . . . . . . . 11  |-  1  =/=  0
3433a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3532, 34eqnetrd 2745 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =/=  0 )
36 fveq2 5802 . . . . . . . . . . 11  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  (deg `  0p
) )
37 dgr0 21865 . . . . . . . . . . 11  |-  (deg ` 
0p )  =  0
3836, 37syl6eq 2511 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  =  0p  -> 
(deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )  =  0 )
3938necon3i 2692 . . . . . . . . 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 21904 . . . . . . . 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 1219 . . . . . . 7  |-  ( ph  ->  ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC ) )
43 plyf 21802 . . . . . . 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 21884 . . . . . 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 1219 . . . . 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 3620 . . . . 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 2499 . . . 4  |-  ( ph  ->  ( `' F " { 0 } )  =  ( { A }  u.  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } ) ) )
50 fta1.1 . . . 4  |-  R  =  ( `' F " { 0 } )
51 uncom 3611 . . . 4  |-  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } )  =  ( { A }  u.  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )
5249, 50, 513eqtr4g 2520 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
533simprd 463 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0p )
5415eqcomd 2462 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )  =  F )
55 0cnd 9493 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
56 mul01 9662 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5756adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5819, 29, 55, 55, 57caofid1 6463 . . . . . . . . . 10  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
59 df-0p 21284 . . . . . . . . . . 11  |-  0p  =  ( CC  X.  { 0 } )
6059oveq2i 6214 . . . . . . . . . 10  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  oF  x.  0p )  =  ( ( Xp  oF  -  ( CC 
X.  { A }
) )  oF  x.  ( CC  X.  { 0 } ) )
6158, 60, 593eqtr4g 2520 . . . . . . . . 9  |-  ( ph  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) )  oF  x.  0p )  =  0p )
6253, 54, 613netr4d 2757 . . . . . . . 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 6211 . . . . . . . . 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 2692 . . . . . . . 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 4111 . . . . . . 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 21837 . . . . . . . . 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 10752 . . . . . . 7  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  e.  CC )
73 fta1.2 . . . . . . . 8  |-  ( ph  ->  D  e.  NN0 )
7473nn0cnd 10752 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
75 addcom 9669 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7621, 74, 75sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7715fveq2d 5806 . . . . . . . . 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 2454 . . . . . . . . . . 11  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { A }
) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { A }
) ) )
80 eqid 2454 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )
8179, 80dgrmul 21873 . . . . . . . . . 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 1220 . . . . . . . . 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 2503 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( Xp  oF  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) ) ) )
8432oveq1d 6218 . . . . . . . 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 2500 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8669, 72, 74, 85addcanad 9688 . . . . . 6  |-  ( ph  ->  (deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D )
87 fveq2 5802 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) ) )
8887eqeq1d 2456 . . . . . . . 8  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( (deg `  g )  =  D  <-> 
(deg `  ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) )  =  D ) )
89 cnveq 5124 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) )
9089imaeq1d 5279 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9190eleq1d 2523 . . . . . . . . 9  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( ( `' g " {
0 } )  e. 
Fin 
<->  ( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  e.  Fin ) )
9290fveq2d 5806 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) )  ->  ( # `  ( `' g " {
0 } ) )  =  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
9392, 87breq12d 4416 . . . . . . . . 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 3175 . . . . . 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 7503 . . . 4  |-  { A }  e.  Fin
100 unfi 7693 . . . 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 2542 . 2  |-  ( ph  ->  R  e.  Fin )
10352fveq2d 5806 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
104 hashcl 12246 . . . . . 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 10751 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  e.  RR )
107 hashcl 12246 . . . . . . 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 10751 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
110 peano2re 9656 . . . . 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 21837 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1134, 112syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
114113nn0red 10751 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
115 hashun2 12267 . . . . . 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 12256 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
11811, 117syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
119118oveq2d 6219 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  ( # `  { A } ) )  =  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 ) )
120116, 119breqtrd 4427 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (
( # `  ( `' ( F quot  ( Xp  oF  -  ( CC  X.  { A } ) ) )
" { 0 } ) )  +  1 ) )
12173nn0red 10751 . . . . . 6  |-  ( ph  ->  D  e.  RR )
122 1red 9515 . . . . . 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 4427 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
125109, 121, 122, 124leadd1dd 10067 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
126125, 78breqtrrd 4429 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
Xp  oF  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
127106, 111, 114, 120, 126letrd 9642 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( Xp  oF  -  ( CC 
X.  { A }
) ) ) " { 0 } )  u.  { A }
) )  <_  (deg `  F ) )
128103, 127eqbrtrd 4423 . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078    \ cdif 3436    u. cun 3437    C_ wss 3439   {csn 3988   class class class wbr 4403    X. cxp 4949   `'ccnv 4950   "cima 4954    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   Fincfn 7423   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    <_ cle 9533    - cmin 9709   NN0cn0 10693   #chash 12223   0pc0p 21283  Polycply 21788   Xpcidp 21789  degcdgr 21791   quot cquot 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fzo 11669  df-fl 11762  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-rlim 13088  df-sum 13285  df-0p 21284  df-ply 21792  df-idp 21793  df-coe 21794  df-dgr 21795  df-quot 21893
This theorem is referenced by:  fta1  21910
  Copyright terms: Public domain W3C validator