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Theorem fta1lem 20224
Description: Lemma for fta1 20225. (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 3927 . . . . . . . . . 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 20117 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
74, 6syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F : CC --> CC )
8 ffn 5591 . . . . . . . . . . . 12  |-  ( F : CC --> CC  ->  F  Fn  CC )
97, 8syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  CC )
10 fniniseg 5851 . . . . . . . . . . 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 2436 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { A } ) )  =  ( X p  o F  -  ( CC  X.  { A } ) )
1615facth 20223 . . . . . . . 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 5048 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )
1918imaeq1d 5202 . . . . 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 9071 . . . . . . 7  |-  CC  e.  _V
2120a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
22 ssid 3367 . . . . . . . . 9  |-  CC  C_  CC
23 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
24 plyid 20128 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  X p  e.  (Poly `  CC ) )
2522, 23, 24mp2an 654 . . . . . . . 8  |-  X p  e.  (Poly `  CC )
26 plyconst 20125 . . . . . . . . 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 20141 . . . . . . . 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 20117 . . . . . . 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 20221 . . . . . . . . . . . 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 9059 . . . . . . . . . . 11  |-  1  =/=  0
3635a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3734, 36eqnetrd 2619 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =/=  0 )
38 fveq2 5728 . . . . . . . . . . 11  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  (deg `  0 p
) )
39 dgr0 20180 . . . . . . . . . . 11  |-  (deg ` 
0 p )  =  0
4038, 39syl6eq 2484 . . . . . . . . . 10  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  0 )
4140necon3i 2643 . . . . . . . . 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 20219 . . . . . . . 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 20117 . . . . . . 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 20199 . . . . . 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 3500 . . . . 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 2472 . . . 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 3491 . . . 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 2493 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
553simprd 450 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0 p )
5617eqcomd 2441 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )  =  F )
57 0cn 9084 . . . . . . . . . . . 12  |-  0  e.  CC
5857a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
59 mul01 9245 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
6059adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
6121, 31, 58, 58, 60caofid1 6334 . . . . . . . . . 10  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
62 df-0p 19562 . . . . . . . . . . 11  |-  0 p  =  ( CC  X.  { 0 } )
6362oveq2i 6092 . . . . . . . . . 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 2493 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  0 p )  =  0 p )
6555, 56, 643netr4d 2628 . . . . . . . 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 6089 . . . . . . . . 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 2643 . . . . . . . 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 3927 . . . . . . 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 20152 . . . . . . . . 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 10276 . . . . . . 7  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e.  CC )
76 fta1.2 . . . . . . . 8  |-  ( ph  ->  D  e.  NN0 )
7776nn0cnd 10276 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
78 addcom 9252 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7923, 77, 78sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
8017fveq2d 5732 . . . . . . . . 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 2436 . . . . . . . . . . 11  |-  (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  =  (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )
83 eqid 2436 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )
8482, 83dgrmul 20188 . . . . . . . . . 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 2476 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
8734oveq1d 6096 . . . . . . . 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 2473 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8972, 75, 77, 88addcanad 9271 . . . . . 6  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  =  D )
90 fveq2 5728 . . . . . . . . 9  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) ) )
9190eqeq1d 2444 . . . . . . . 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 5046 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )
9392imaeq1d 5202 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9493eleq1d 2502 . . . . . . . . 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 5732 . . . . . . . . . 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 4225 . . . . . . . . 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 3048 . . . . . 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 7187 . . . 4  |-  { A }  e.  Fin
103 unfi 7374 . . . 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 2510 . 2  |-  ( ph  ->  R  e.  Fin )
10654fveq2d 5732 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
107 hashcl 11639 . . . . . 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 10275 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  e.  RR )
110 hashcl 11639 . . . . . . 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 10275 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
113 peano2re 9239 . . . . 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 20152 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1164, 115syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
117116nn0red 10275 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
118 hashun2 11657 . . . . . 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 11647 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
12113, 120syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
122121oveq2d 6097 . . . . 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 4236 . . . 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 10275 . . . . . 6  |-  ( ph  ->  D  e.  RR )
125 1re 9090 . . . . . . 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 4236 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
129112, 124, 126, 128leadd1dd 9640 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
130129, 81breqtrrd 4238 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
131109, 114, 117, 123, 130letrd 9227 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  <_  (deg `  F
) )
132106, 131eqbrtrd 4232 . 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956    \ cdif 3317    u. cun 3318    C_ wss 3320   {csn 3814   class class class wbr 4212    X. cxp 4876   `'ccnv 4877   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Fincfn 7109   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    <_ cle 9121    - cmin 9291   NN0cn0 10221   #chash 11618   0 pc0p 19561  Polycply 20103   X pcidp 20104  degcdgr 20106   quot cquot 20207
This theorem is referenced by:  fta1  20225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-idp 20108  df-coe 20109  df-dgr 20110  df-quot 20208
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