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Theorem fta1 21733
Description: The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg ( F ) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
fta1.1  |-  R  =  ( `' F " { 0 } )
Assertion
Ref Expression
fta1  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )

Proof of Theorem fta1
Dummy variables  x  g  f  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2441 . 2  |-  (deg `  F )  =  (deg
`  F )
2 dgrcl 21660 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
32adantr 462 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
(deg `  F )  e.  NN0 )
4 eqeq2 2450 . . . . . . 7  |-  ( x  =  0  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  0 ) )
54imbi1d 317 . . . . . 6  |-  ( x  =  0  ->  (
( (deg `  f
)  =  x  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  ( (deg `  f )  =  0  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
65ralbidv 2733 . . . . 5  |-  ( x  =  0  ->  ( A. f  e.  (
(Poly `  CC )  \  { 0p }
) ( (deg `  f )  =  x  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  0  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
7 eqeq2 2450 . . . . . . 7  |-  ( x  =  d  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  d ) )
87imbi1d 317 . . . . . 6  |-  ( x  =  d  ->  (
( (deg `  f
)  =  x  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  ( (deg `  f )  =  d  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
98ralbidv 2733 . . . . 5  |-  ( x  =  d  ->  ( A. f  e.  (
(Poly `  CC )  \  { 0p }
) ( (deg `  f )  =  x  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  d  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
10 eqeq2 2450 . . . . . . 7  |-  ( x  =  ( d  +  1 )  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  ( d  +  1 ) ) )
1110imbi1d 317 . . . . . 6  |-  ( x  =  ( d  +  1 )  ->  (
( (deg `  f
)  =  x  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  ( (deg `  f )  =  ( d  +  1 )  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
1211ralbidv 2733 . . . . 5  |-  ( x  =  ( d  +  1 )  ->  ( A. f  e.  (
(Poly `  CC )  \  { 0p }
) ( (deg `  f )  =  x  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  ( d  +  1 )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
13 eqeq2 2450 . . . . . . 7  |-  ( x  =  (deg `  F
)  ->  ( (deg `  f )  =  x  <-> 
(deg `  f )  =  (deg `  F )
) )
1413imbi1d 317 . . . . . 6  |-  ( x  =  (deg `  F
)  ->  ( (
(deg `  f )  =  x  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )  <->  ( (deg `  f )  =  (deg
`  F )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
1514ralbidv 2733 . . . . 5  |-  ( x  =  (deg `  F
)  ->  ( A. f  e.  ( (Poly `  CC )  \  {
0p } ) ( (deg `  f
)  =  x  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
16 eldifsni 3998 . . . . . . . . . . 11  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  =/=  0p )
1716adantr 462 . . . . . . . . . 10  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  =/=  0p )
18 simplr 749 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  (deg `  f
)  =  0 )
19 eldifi 3475 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  e.  (Poly `  CC ) )
2019ad2antrr 720 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  e.  (Poly `  CC ) )
21 0dgrb 21673 . . . . . . . . . . . . . . . 16  |-  ( f  e.  (Poly `  CC )  ->  ( (deg `  f )  =  0  <-> 
f  =  ( CC 
X.  { ( f `
 0 ) } ) ) )
2220, 21syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( (deg `  f )  =  0  <-> 
f  =  ( CC 
X.  { ( f `
 0 ) } ) ) )
2318, 22mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { ( f `  0 ) } ) )
2423fveq1d 5690 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  ( ( CC  X.  {
( f `  0
) } ) `  x ) )
2519adantr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  e.  (Poly `  CC ) )
26 plyf 21625 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
27 ffn 5556 . . . . . . . . . . . . . . . . . . . 20  |-  ( f : CC --> CC  ->  f  Fn  CC )
28 fniniseg 5821 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  CC  ->  (
x  e.  ( `' f " { 0 } )  <->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) ) )
2925, 26, 27, 284syl 21 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( x  e.  ( `' f " { 0 } )  <-> 
( x  e.  CC  /\  ( f `  x
)  =  0 ) ) )
3029biimpa 481 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) )
3130simprd 460 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  0 )
3230simpld 456 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  x  e.  CC )
33 fvex 5698 . . . . . . . . . . . . . . . . . . 19  |-  ( f `
 0 )  e. 
_V
3433fvconst2 5930 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  (
( CC  X.  {
( f `  0
) } ) `  x )  =  ( f `  0 ) )
3532, 34syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( ( CC 
X.  { ( f `
 0 ) } ) `  x )  =  ( f ` 
0 ) )
3624, 31, 353eqtr3rd 2482 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f ` 
0 )  =  0 )
3736sneqd 3886 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  { ( f `
 0 ) }  =  { 0 } )
3837xpeq2d 4860 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( CC  X.  { ( f ` 
0 ) } )  =  ( CC  X.  { 0 } ) )
3923, 38eqtrd 2473 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { 0 } ) )
40 df-0p 21107 . . . . . . . . . . . . 13  |-  0p  =  ( CC  X.  { 0 } )
4139, 40syl6eqr 2491 . . . . . . . . . . . 12  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  0p )
4241ex 434 . . . . . . . . . . 11  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( x  e.  ( `' f " { 0 } )  ->  f  =  0p ) )
4342necon3ad 2642 . . . . . . . . . 10  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( f  =/=  0p  ->  -.  x  e.  ( `' f " { 0 } ) ) )
4417, 43mpd 15 . . . . . . . . 9  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  -.  x  e.  ( `' f " {
0 } ) )
4544eq0rdv 3669 . . . . . . . 8  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( `' f
" { 0 } )  =  (/) )
4645ex 434 . . . . . . 7  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( (deg `  f
)  =  0  -> 
( `' f " { 0 } )  =  (/) ) )
47 dgrcl 21660 . . . . . . . . 9  |-  ( f  e.  (Poly `  CC )  ->  (deg `  f
)  e.  NN0 )
48 nn0ge0 10601 . . . . . . . . 9  |-  ( (deg
`  f )  e. 
NN0  ->  0  <_  (deg `  f ) )
4919, 47, 483syl 20 . . . . . . . 8  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
0  <_  (deg `  f
) )
50 id 22 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  =  (/) )
51 0fin 7536 . . . . . . . . . . 11  |-  (/)  e.  Fin
5250, 51syl6eqel 2529 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  e.  Fin )
5352biantrurd 505 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
54 fveq2 5688 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  ( # `  (/) ) )
55 hash0 12131 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
5654, 55syl6eq 2489 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  0 )
5756breq1d 4299 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  0  <_  (deg `  f ) ) )
5853, 57bitr3d 255 . . . . . . . 8  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) )  <->  0  <_  (deg
`  f ) ) )
5949, 58syl5ibrcom 222 . . . . . . 7  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( ( `' f
" { 0 } )  =  (/)  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) ) )
6046, 59syld 44 . . . . . 6  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( (deg `  f
)  =  0  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) )
6160rgen 2779 . . . . 5  |-  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  0  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
62 fveq2 5688 . . . . . . . . 9  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
6362eqeq1d 2449 . . . . . . . 8  |-  ( f  =  g  ->  (
(deg `  f )  =  d  <->  (deg `  g )  =  d ) )
64 cnveq 5009 . . . . . . . . . . 11  |-  ( f  =  g  ->  `' f  =  `' g
)
6564imaeq1d 5165 . . . . . . . . . 10  |-  ( f  =  g  ->  ( `' f " {
0 } )  =  ( `' g " { 0 } ) )
6665eleq1d 2507 . . . . . . . . 9  |-  ( f  =  g  ->  (
( `' f " { 0 } )  e.  Fin  <->  ( `' g " { 0 } )  e.  Fin )
)
6765fveq2d 5692 . . . . . . . . . 10  |-  ( f  =  g  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' g " { 0 } ) ) )
6867, 62breq12d 4302 . . . . . . . . 9  |-  ( f  =  g  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  ( `' g " { 0 } ) )  <_  (deg `  g ) ) )
6966, 68anbi12d 705 . . . . . . . 8  |-  ( f  =  g  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
7063, 69imbi12d 320 . . . . . . 7  |-  ( f  =  g  ->  (
( (deg `  f
)  =  d  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  ( (deg `  g )  =  d  ->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )
7170cbvralv 2945 . . . . . 6  |-  ( A. f  e.  ( (Poly `  CC )  \  {
0p } ) ( (deg `  f
)  =  d  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
7249ad2antlr 721 . . . . . . . . . . . 12  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  0  <_  (deg
`  f ) )
7372, 58syl5ibrcom 222 . . . . . . . . . . 11  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( ( `' f " {
0 } )  =  (/)  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
7473a1dd 46 . . . . . . . . . 10  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( ( `' f " {
0 } )  =  (/)  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
75 n0 3643 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =/=  (/) 
<->  E. x  x  e.  ( `' f " { 0 } ) )
76 eqid 2441 . . . . . . . . . . . . . 14  |-  ( `' f " { 0 } )  =  ( `' f " {
0 } )
77 simplll 752 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  d  e.  NN0 )
78 simpllr 753 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  f  e.  ( (Poly `  CC )  \  { 0p }
) )
79 simplr 749 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  (deg `  f
)  =  ( d  +  1 ) )
80 simprl 750 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  x  e.  ( `' f " {
0 } ) )
81 simprr 751 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
8276, 77, 78, 79, 80, 81fta1lem 21732 . . . . . . . . . . . . 13  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
8382exp32 602 . . . . . . . . . . . 12  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( x  e.  ( `' f " { 0 } )  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8483exlimdv 1695 . . . . . . . . . . 11  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( E. x  x  e.  ( `' f " {
0 } )  -> 
( A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8575, 84syl5bi 217 . . . . . . . . . 10  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( ( `' f " {
0 } )  =/=  (/)  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8674, 85pm2.61dne 2686 . . . . . . . . 9  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( A. g  e.  ( (Poly `  CC )  \  {
0p } ) ( (deg `  g
)  =  d  -> 
( ( `' g
" { 0 } )  e.  Fin  /\  ( # `  ( `' g " { 0 } ) )  <_ 
(deg `  g )
) )  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) ) )
8786ex 434 . . . . . . . 8  |-  ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  ->  ( (deg `  f )  =  ( d  +  1 )  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8887com23 78 . . . . . . 7  |-  ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  ->  ( A. g  e.  ( (Poly `  CC )  \  {
0p } ) ( (deg `  g
)  =  d  -> 
( ( `' g
" { 0 } )  e.  Fin  /\  ( # `  ( `' g " { 0 } ) )  <_ 
(deg `  g )
) )  ->  (
(deg `  f )  =  ( d  +  1 )  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) ) ) )
8988ralrimdva 2804 . . . . . 6  |-  ( d  e.  NN0  ->  ( A. g  e.  ( (Poly `  CC )  \  {
0p } ) ( (deg `  g
)  =  d  -> 
( ( `' g
" { 0 } )  e.  Fin  /\  ( # `  ( `' g " { 0 } ) )  <_ 
(deg `  g )
) )  ->  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  ( d  +  1 )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
9071, 89syl5bi 217 . . . . 5  |-  ( d  e.  NN0  ->  ( A. f  e.  ( (Poly `  CC )  \  {
0p } ) ( (deg `  f
)  =  d  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  ( d  +  1 )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
916, 9, 12, 15, 61, 90nn0ind 10734 . . . 4  |-  ( (deg
`  F )  e. 
NN0  ->  A. f  e.  ( (Poly `  CC )  \  { 0p }
) ( (deg `  f )  =  (deg
`  F )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) )
923, 91syl 16 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  ->  A. f  e.  (
(Poly `  CC )  \  { 0p }
) ( (deg `  f )  =  (deg
`  F )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) )
93 plyssc 21627 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
9493sseli 3349 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
95 eldifsn 3997 . . . . 5  |-  ( F  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0p ) )
96 fveq2 5688 . . . . . . . 8  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
9796eqeq1d 2449 . . . . . . 7  |-  ( f  =  F  ->  (
(deg `  f )  =  (deg `  F )  <->  (deg
`  F )  =  (deg `  F )
) )
98 cnveq 5009 . . . . . . . . . . 11  |-  ( f  =  F  ->  `' f  =  `' F
)
9998imaeq1d 5165 . . . . . . . . . 10  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
100 fta1.1 . . . . . . . . . 10  |-  R  =  ( `' F " { 0 } )
10199, 100syl6eqr 2491 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
102101eleq1d 2507 . . . . . . . 8  |-  ( f  =  F  ->  (
( `' f " { 0 } )  e.  Fin  <->  R  e.  Fin ) )
103101fveq2d 5692 . . . . . . . . 9  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
104103, 96breq12d 4302 . . . . . . . 8  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  R )  <_  (deg `  F )
) )
105102, 104anbi12d 705 . . . . . . 7  |-  ( f  =  F  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( R  e. 
Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) )
10697, 105imbi12d 320 . . . . . 6  |-  ( f  =  F  ->  (
( (deg `  f
)  =  (deg `  F )  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) )  <-> 
( (deg `  F
)  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `
 R )  <_ 
(deg `  F )
) ) ) )
107106rspcv 3066 . . . . 5  |-  ( F  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  (
(deg `  F )  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) ) )
10895, 107sylbir 213 . . . 4  |-  ( ( F  e.  (Poly `  CC )  /\  F  =/=  0p )  -> 
( A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  (
(deg `  F )  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) ) )
10994, 108sylan 468 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
( A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  (
(deg `  F )  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) ) )
11092, 109mpd 15 . 2  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
( (deg `  F
)  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `
 R )  <_ 
(deg `  F )
) ) )
1111, 110mpi 17 1  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713    \ cdif 3322   (/)c0 3634   {csn 3874   class class class wbr 4289    X. cxp 4834   `'ccnv 4835   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Fincfn 7306   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    <_ cle 9415   NN0cn0 10575   #chash 12099   0pc0p 21106  Polycply 21611  degcdgr 21614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-idp 21616  df-coe 21617  df-dgr 21618  df-quot 21716
This theorem is referenced by:  vieta1lem2  21736  vieta1  21737  plyexmo  21738  aannenlem1  21753  aalioulem2  21758  basellem4  22380  basellem5  22381  dchrfi  22553
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