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Theorem fta1lem 19703
Description: Lemma for fta1 19704. (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 3762 . . . . . . . . . 10  |-  ( F  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0 p ) )
31, 2sylib 188 . . . . . . . . 9  |-  ( ph  ->  ( F  e.  (Poly `  CC )  /\  F  =/=  0 p ) )
43simpld 445 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  CC ) )
5 fta1.5 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( `' F " { 0 } ) )
6 plyf 19596 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
74, 6syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  F : CC --> CC )
8 ffn 5405 . . . . . . . . . . . 12  |-  ( F : CC --> CC  ->  F  Fn  CC )
97, 8syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  CC )
10 fniniseg 5662 . . . . . . . . . . 11  |-  ( F  Fn  CC  ->  ( A  e.  ( `' F " { 0 } )  <->  ( A  e.  CC  /\  ( F `
 A )  =  0 ) ) )
119, 10syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( `' F " { 0 } )  <->  ( A  e.  CC  /\  ( F `
 A )  =  0 ) ) )
125, 11mpbid 201 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  ( F `  A
)  =  0 ) )
1312simpld 445 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1412simprd 449 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  =  0 )
15 eqid 2296 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { A } ) )  =  ( X p  o F  -  ( CC  X.  { A } ) )
1615facth 19702 . . . . . . . 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 1182 . . . . . . 7  |-  ( ph  ->  F  =  ( ( X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )
1817cnveqd 4873 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )
1918imaeq1d 5027 . . . . 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 8834 . . . . . . 7  |-  CC  e.  _V
2120a1i 10 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
22 ssid 3210 . . . . . . . . 9  |-  CC  C_  CC
23 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
24 plyid 19607 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  X p  e.  (Poly `  CC ) )
2522, 23, 24mp2an 653 . . . . . . . 8  |-  X p  e.  (Poly `  CC )
26 plyconst 19604 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
2722, 13, 26sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( CC  X.  { A } )  e.  (Poly `  CC ) )
28 plysubcl 19620 . . . . . . . 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 644 . . . . . . 7  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
30 plyf 19596 . . . . . . 7  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC )  ->  ( X p  o F  -  ( CC  X.  { A }
) ) : CC --> CC )
3129, 30syl 15 . . . . . 6  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { A } ) ) : CC --> CC )
3215plyremlem 19700 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 968 . . . . . . . . . 10  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  1 )
35 ax-1ne0 8822 . . . . . . . . . . 11  |-  1  =/=  0
3635a1i 10 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3734, 36eqnetrd 2477 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =/=  0 )
38 fveq2 5541 . . . . . . . . . . 11  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  (deg `  0 p
) )
39 dgr0 19659 . . . . . . . . . . 11  |-  (deg ` 
0 p )  =  0
4038, 39syl6eq 2344 . . . . . . . . . 10  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  0 )
4140necon3i 2498 . . . . . . . . 9  |-  ( (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )  =/=  0  ->  ( X p  o F  -  ( CC  X.  { A }
) )  =/=  0 p )
4237, 41syl 15 . . . . . . . 8  |-  ( ph  ->  ( X p  o F  -  ( CC  X.  { A } ) )  =/=  0 p )
43 quotcl2 19698 . . . . . . . 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 1182 . . . . . . 7  |-  ( ph  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  e.  (Poly `  CC ) )
45 plyf 19596 . . . . . . 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 15 . . . . . 6  |-  ( ph  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) : CC --> CC )
47 ofmulrt 19678 . . . . . 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 1182 . . . . 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 969 . . . . . 6  |-  ( ph  ->  ( `' ( X p  o F  -  ( CC  X.  { A } ) ) " { 0 } )  =  { A }
)
5049uneq1d 3341 . . . . 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 2332 . . . 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 3332 . . . 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 2353 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
553simprd 449 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0 p )
5617eqcomd 2301 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )  =  F )
57 0cn 8847 . . . . . . . . . . . 12  |-  0  e.  CC
5857a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
59 mul01 9007 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
6059adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
6121, 31, 58, 58, 60caofid1 6123 . . . . . . . . . 10  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
62 df-0p 19041 . . . . . . . . . . 11  |-  0 p  =  ( CC  X.  { 0 } )
6362oveq2i 5885 . . . . . . . . . 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 2353 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  0 p )  =  0 p )
6555, 56, 643netr4d 2486 . . . . . . . 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 5882 . . . . . . . . 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 2498 . . . . . . . 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 15 . . . . . . 7  |-  ( ph  ->  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )  =/=  0 p )
69 eldifsn 3762 . . . . . . 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 645 . . . . . 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
) ) ) )
72 fta1.2 . . . . . . . . . 10  |-  ( ph  ->  D  e.  NN0 )
7372nn0cnd 10036 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
74 addcom 9014 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7523, 73, 74sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7617fveq2d 5545 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =  (deg `  ( (
X p  o F  -  ( CC  X.  { A } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
77 fta1.4 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =  ( D  + 
1 ) )
78 eqid 2296 . . . . . . . . . . 11  |-  (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  =  (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )
79 eqid 2296 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )
8078, 79dgrmul 19667 . . . . . . . . . 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 } ) ) ) ) ) )
8129, 42, 44, 68, 80syl22anc 1183 . . . . . . . . 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 } ) ) ) ) ) )
8276, 77, 813eqtr3d 2336 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
8334oveq1d 5889 . . . . . . . 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 } ) ) ) ) ) )
8475, 82, 833eqtrrd 2333 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8523a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
86 dgrcl 19631 . . . . . . . . . 10  |-  ( ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) )  e.  (Poly `  CC )  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e. 
NN0 )
8744, 86syl 15 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e. 
NN0 )
8887nn0cnd 10036 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e.  CC )
8985, 88, 73addcand 9031 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D )  <->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  =  D ) )
9084, 89mpbid 201 . . . . . 6  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  =  D )
91 fveq2 5541 . . . . . . . . 9  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) ) )
9291eqeq1d 2304 . . . . . . . 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 ) )
93 cnveq 4871 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )
9493imaeq1d 5027 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9594eleq1d 2362 . . . . . . . . 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 ) )
9694fveq2d 5545 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( # `  ( `' g " {
0 } ) )  =  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) ) )
9796, 91breq12d 4052 . . . . . . . . 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 } ) ) ) ) ) )
9895, 97anbi12d 691 . . . . . . . 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 } ) ) ) ) ) ) )
9992, 98imbi12d 311 . . . . . . 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 } ) ) ) ) ) ) ) )
10099rspcv 2893 . . . . . 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 } ) ) ) ) ) ) ) )
10170, 71, 90, 100syl3c 57 . . . . 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 } ) ) ) ) ) )
102101simpld 445 . . . 4  |-  ( ph  ->  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  e. 
Fin )
103 snfi 6957 . . . 4  |-  { A }  e.  Fin
104 unfi 7140 . . . 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 )
105102, 103, 104sylancl 643 . . 3  |-  ( ph  ->  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
)  e.  Fin )
10654, 105eqeltrd 2370 . 2  |-  ( ph  ->  R  e.  Fin )
10754fveq2d 5545 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
108 hashcl 11366 . . . . . 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 )
109105, 108syl 15 . . . . 5  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  e.  NN0 )
110109nn0red 10035 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  e.  RR )
111 hashcl 11366 . . . . . . 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 )
112102, 111syl 15 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e. 
NN0 )
113112nn0red 10035 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
114 peano2re 9001 . . . . 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 )
115113, 114syl 15 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  e.  RR )
116 dgrcl 19631 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1174, 116syl 15 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
118117nn0red 10035 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
119 hashun2 11381 . . . . . 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 } ) ) )
120102, 103, 119sylancl 643 . . . . 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 } ) ) )
121 hashsng 11372 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
12213, 121syl 15 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
123122oveq2d 5890 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  ( # `  { A } ) )  =  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 ) )
124120, 123breqtrd 4063 . . . 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 ) )
12572nn0red 10035 . . . . . 6  |-  ( ph  ->  D  e.  RR )
126 1re 8853 . . . . . . 7  |-  1  e.  RR
127126a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  RR )
128101simprd 449 . . . . . . 7  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_ 
(deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) )
129128, 90breqtrd 4063 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
130113, 125, 127, 129leadd1dd 9402 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
131130, 77breqtrrd 4065 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
132110, 115, 118, 124, 131letrd 8989 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  <_  (deg `  F
) )
133107, 132eqbrtrd 4059 . 2  |-  ( ph  ->  ( # `  R
)  <_  (deg `  F
) )
134106, 133jca 518 1  |-  ( ph  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653   class class class wbr 4039    X. cxp 4703   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   NN0cn0 9981   #chash 11353   0 pc0p 19040  Polycply 19582   X pcidp 19583  degcdgr 19585   quot cquot 19686
This theorem is referenced by:  fta1  19704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-idp 19587  df-coe 19588  df-dgr 19589  df-quot 19687
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