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Theorem fta1lem 19687
Description: Lemma for fta1 19688. (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 3749 . . . . . . . . . 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 19580 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  CC )  ->  F : CC --> CC )
74, 6syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  F : CC --> CC )
8 ffn 5389 . . . . . . . . . . . 12  |-  ( F : CC --> CC  ->  F  Fn  CC )
97, 8syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  CC )
10 fniniseg 5646 . . . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( X p  o F  -  ( CC  X.  { A } ) )  =  ( X p  o F  -  ( CC  X.  { A } ) )
1615facth 19686 . . . . . . . 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 4857 . . . . . 6  |-  ( ph  ->  `' F  =  `' ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )
1918imaeq1d 5011 . . . . 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 8818 . . . . . . 7  |-  CC  e.  _V
2120a1i 10 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
22 ssid 3197 . . . . . . . . 9  |-  CC  C_  CC
23 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
24 plyid 19591 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  X p  e.  (Poly `  CC ) )
2522, 23, 24mp2an 653 . . . . . . . 8  |-  X p  e.  (Poly `  CC )
26 plyconst 19588 . . . . . . . . 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 19604 . . . . . . . 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 19580 . . . . . . 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 19684 . . . . . . . . . . . 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 8806 . . . . . . . . . . 11  |-  1  =/=  0
3635a1i 10 . . . . . . . . . 10  |-  ( ph  ->  1  =/=  0 )
3734, 36eqnetrd 2464 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =/=  0 )
38 fveq2 5525 . . . . . . . . . . 11  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  (deg `  0 p
) )
39 dgr0 19643 . . . . . . . . . . 11  |-  (deg ` 
0 p )  =  0
4038, 39syl6eq 2331 . . . . . . . . . 10  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  =  0 p  -> 
(deg `  ( X p  o F  -  ( CC  X.  { A }
) ) )  =  0 )
4140necon3i 2485 . . . . . . . . 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 19682 . . . . . . . 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 19580 . . . . . . 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 19662 . . . . . 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 3328 . . . . 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 2319 . . . 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 3319 . . . 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 2340 . . 3  |-  ( ph  ->  R  =  ( ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } )  u.  { A } ) )
553simprd 449 . . . . . . . . 9  |-  ( ph  ->  F  =/=  0 p )
5617eqcomd 2288 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )  =  F )
57 0cn 8831 . . . . . . . . . . . 12  |-  0  e.  CC
5857a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  CC )
59 mul01 8991 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
6059adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
6121, 31, 58, 58, 60caofid1 6107 . . . . . . . . . 10  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) )
62 df-0p 19025 . . . . . . . . . . 11  |-  0 p  =  ( CC  X.  { 0 } )
6362oveq2i 5869 . . . . . . . . . 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 2340 . . . . . . . . 9  |-  ( ph  ->  ( ( X p  o F  -  ( CC  X.  { A }
) )  o F  x.  0 p )  =  0 p )
6555, 56, 643netr4d 2473 . . . . . . . 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 5866 . . . . . . . . 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 2485 . . . . . . . 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 3749 . . . . . . 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 10020 . . . . . . . . 9  |-  ( ph  ->  D  e.  CC )
74 addcom 8998 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7523, 73, 74sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 1  +  D
)  =  ( D  +  1 ) )
7617fveq2d 5529 . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  (deg `  ( X p  o F  -  ( CC  X.  { A } ) ) )  =  (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )
79 eqid 2283 . . . . . . . . . . 11  |-  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) )  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )
8078, 79dgrmul 19651 . . . . . . . . . 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 2323 . . . . . . . 8  |-  ( ph  ->  ( D  +  1 )  =  ( (deg
`  ( X p  o F  -  ( CC  X.  { A }
) ) )  +  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) ) ) )
8334oveq1d 5873 . . . . . . . 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 2320 . . . . . . 7  |-  ( ph  ->  ( 1  +  (deg
`  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) ) )  =  ( 1  +  D
) )
8523a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
86 dgrcl 19615 . . . . . . . . . 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 10020 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) )  e.  CC )
8985, 88, 73addcand 9015 . . . . . . 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 5525 . . . . . . . . 9  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  (deg `  g
)  =  (deg `  ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) ) )
9291eqeq1d 2291 . . . . . . . 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 4855 . . . . . . . . . . 11  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  `' g  =  `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) )
9493imaeq1d 5011 . . . . . . . . . 10  |-  ( g  =  ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) )  ->  ( `' g " { 0 } )  =  ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) )
" { 0 } ) )
9594eleq1d 2349 . . . . . . . . 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 5529 . . . . . . . . . 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 4036 . . . . . . . . 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 2880 . . . . . 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 6941 . . . 4  |-  { A }  e.  Fin
104 unfi 7124 . . . 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 2357 . 2  |-  ( ph  ->  R  e.  Fin )
10754fveq2d 5529 . . 3  |-  ( ph  ->  ( # `  R
)  =  ( # `  ( ( `' ( F quot  ( X p  o F  -  ( CC  X.  { A }
) ) ) " { 0 } )  u.  { A }
) ) )
108 hashcl 11350 . . . . . 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 10019 . . . 4  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  e.  RR )
111 hashcl 11350 . . . . . . 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 10019 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  e.  RR )
114 peano2re 8985 . . . . 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 19615 . . . . . 6  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  e.  NN0 )
1174, 116syl 15 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
118117nn0red 10019 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
119 hashun2 11365 . . . . . 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 11356 . . . . . . 7  |-  ( A  e.  CC  ->  ( # `
 { A }
)  =  1 )
12213, 121syl 15 . . . . . 6  |-  ( ph  ->  ( # `  { A } )  =  1 )
123122oveq2d 5874 . . . . 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 4047 . . . 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 10019 . . . . . 6  |-  ( ph  ->  D  e.  RR )
126 1re 8837 . . . . . . 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 4047 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  <_  D )
130113, 125, 127, 129leadd1dd 9386 . . . . 5  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  ( D  +  1 ) )
131130, 77breqtrrd 4049 . . . 4  |-  ( ph  ->  ( ( # `  ( `' ( F quot  (
X p  o F  -  ( CC  X.  { A } ) ) ) " { 0 } ) )  +  1 )  <_  (deg `  F ) )
132110, 115, 118, 124, 131letrd 8973 . . 3  |-  ( ph  ->  ( # `  (
( `' ( F quot  ( X p  o F  -  ( CC  X.  { A } ) ) ) " {
0 } )  u. 
{ A } ) )  <_  (deg `  F
) )
133107, 132eqbrtrd 4043 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   {csn 3640   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037   NN0cn0 9965   #chash 11337   0 pc0p 19024  Polycply 19566   X pcidp 19567  degcdgr 19569   quot cquot 19670
This theorem is referenced by:  fta1  19688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-quot 19671
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