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Theorem fta1 19688
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  =/=  0 p )  -> 
( 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 2283 . 2  |-  (deg `  F )  =  (deg
`  F )
2 dgrcl 19615 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
32adantr 451 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  -> 
(deg `  F )  e.  NN0 )
4 eqeq2 2292 . . . . . . 7  |-  ( x  =  0  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  0 ) )
54imbi1d 308 . . . . . 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 2563 . . . . 5  |-  ( x  =  0  ->  ( A. f  e.  (
(Poly `  CC )  \  { 0 p }
) ( (deg `  f )  =  x  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  0  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
7 eqeq2 2292 . . . . . . 7  |-  ( x  =  d  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  d ) )
87imbi1d 308 . . . . . 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 2563 . . . . 5  |-  ( x  =  d  ->  ( A. f  e.  (
(Poly `  CC )  \  { 0 p }
) ( (deg `  f )  =  x  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  d  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
10 eqeq2 2292 . . . . . . 7  |-  ( x  =  ( d  +  1 )  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  ( d  +  1 ) ) )
1110imbi1d 308 . . . . . 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 2563 . . . . 5  |-  ( x  =  ( d  +  1 )  ->  ( A. f  e.  (
(Poly `  CC )  \  { 0 p }
) ( (deg `  f )  =  x  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  ( d  +  1 )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
13 eqeq2 2292 . . . . . . 7  |-  ( x  =  (deg `  F
)  ->  ( (deg `  f )  =  x  <-> 
(deg `  f )  =  (deg `  F )
) )
1413imbi1d 308 . . . . . 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 2563 . . . . 5  |-  ( x  =  (deg `  F
)  ->  ( A. f  e.  ( (Poly `  CC )  \  {
0 p } ) ( (deg `  f
)  =  x  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
16 eldifsni 3750 . . . . . . . . . . 11  |-  ( f  e.  ( (Poly `  CC )  \  { 0 p } )  -> 
f  =/=  0 p )
1716adantr 451 . . . . . . . . . 10  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  f  =/=  0 p )
18 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  (deg `  f
)  =  0 )
19 eldifi 3298 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  ( (Poly `  CC )  \  { 0 p } )  -> 
f  e.  (Poly `  CC ) )
2019ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  e.  (Poly `  CC ) )
21 0dgrb 19628 . . . . . . . . . . . . . . . 16  |-  ( f  e.  (Poly `  CC )  ->  ( (deg `  f )  =  0  <-> 
f  =  ( CC 
X.  { ( f `
 0 ) } ) ) )
2220, 21syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( (deg `  f )  =  0  <-> 
f  =  ( CC 
X.  { ( f `
 0 ) } ) ) )
2318, 22mpbid 201 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { ( f `  0 ) } ) )
2423fveq1d 5527 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  ( ( CC  X.  {
( f `  0
) } ) `  x ) )
2519adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  f  e.  (Poly `  CC ) )
26 plyf 19580 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
27 ffn 5389 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : CC --> CC  ->  f  Fn  CC )
2825, 26, 273syl 18 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  f  Fn  CC )
29 fniniseg 5646 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  CC  ->  (
x  e.  ( `' f " { 0 } )  <->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) ) )
3028, 29syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  ( x  e.  ( `' f " { 0 } )  <-> 
( x  e.  CC  /\  ( f `  x
)  =  0 ) ) )
3130biimpa 470 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) )
3231simprd 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  0 )
3331simpld 445 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  x  e.  CC )
34 fvex 5539 . . . . . . . . . . . . . . . . . . 19  |-  ( f `
 0 )  e. 
_V
3534fvconst2 5729 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  (
( CC  X.  {
( f `  0
) } ) `  x )  =  ( f `  0 ) )
3633, 35syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( ( CC 
X.  { ( f `
 0 ) } ) `  x )  =  ( f ` 
0 ) )
3724, 32, 363eqtr3rd 2324 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f ` 
0 )  =  0 )
3837sneqd 3653 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  { ( f `
 0 ) }  =  { 0 } )
3938xpeq2d 4713 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( CC  X.  { ( f ` 
0 ) } )  =  ( CC  X.  { 0 } ) )
4023, 39eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { 0 } ) )
41 df-0p 19025 . . . . . . . . . . . . 13  |-  0 p  =  ( CC  X.  { 0 } )
4240, 41syl6eqr 2333 . . . . . . . . . . . 12  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0 p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  0 p )
4342ex 423 . . . . . . . . . . 11  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  ( x  e.  ( `' f " { 0 } )  ->  f  =  0 p ) )
4443necon3ad 2482 . . . . . . . . . 10  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  ( f  =/=  0 p  ->  -.  x  e.  ( `' f " { 0 } ) ) )
4517, 44mpd 14 . . . . . . . . 9  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  -.  x  e.  ( `' f " {
0 } ) )
4645eq0rdv 3489 . . . . . . . 8  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0 p } )  /\  (deg `  f
)  =  0 )  ->  ( `' f
" { 0 } )  =  (/) )
4746ex 423 . . . . . . 7  |-  ( f  e.  ( (Poly `  CC )  \  { 0 p } )  -> 
( (deg `  f
)  =  0  -> 
( `' f " { 0 } )  =  (/) ) )
48 dgrcl 19615 . . . . . . . . 9  |-  ( f  e.  (Poly `  CC )  ->  (deg `  f
)  e.  NN0 )
49 nn0ge0 9991 . . . . . . . . 9  |-  ( (deg
`  f )  e. 
NN0  ->  0  <_  (deg `  f ) )
5019, 48, 493syl 18 . . . . . . . 8  |-  ( f  e.  ( (Poly `  CC )  \  { 0 p } )  -> 
0  <_  (deg `  f
) )
51 id 19 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  =  (/) )
52 0fin 7087 . . . . . . . . . . 11  |-  (/)  e.  Fin
5351, 52syl6eqel 2371 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  e.  Fin )
5453biantrurd 494 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
55 fveq2 5525 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  ( # `  (/) ) )
56 hash0 11355 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
5755, 56syl6eq 2331 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  0 )
5857breq1d 4033 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  0  <_  (deg `  f ) ) )
5954, 58bitr3d 246 . . . . . . . 8  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) )  <->  0  <_  (deg
`  f ) ) )
6050, 59syl5ibrcom 213 . . . . . . 7  |-  ( f  e.  ( (Poly `  CC )  \  { 0 p } )  -> 
( ( `' f
" { 0 } )  =  (/)  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) ) )
6147, 60syld 40 . . . . . 6  |-  ( f  e.  ( (Poly `  CC )  \  { 0 p } )  -> 
( (deg `  f
)  =  0  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) )
6261rgen 2608 . . . . 5  |-  A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  0  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
63 fveq2 5525 . . . . . . . . 9  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
6463eqeq1d 2291 . . . . . . . 8  |-  ( f  =  g  ->  (
(deg `  f )  =  d  <->  (deg `  g )  =  d ) )
65 cnveq 4855 . . . . . . . . . . 11  |-  ( f  =  g  ->  `' f  =  `' g
)
6665imaeq1d 5011 . . . . . . . . . 10  |-  ( f  =  g  ->  ( `' f " {
0 } )  =  ( `' g " { 0 } ) )
6766eleq1d 2349 . . . . . . . . 9  |-  ( f  =  g  ->  (
( `' f " { 0 } )  e.  Fin  <->  ( `' g " { 0 } )  e.  Fin )
)
6866fveq2d 5529 . . . . . . . . . 10  |-  ( f  =  g  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' g " { 0 } ) ) )
6968, 63breq12d 4036 . . . . . . . . 9  |-  ( f  =  g  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  ( `' g " { 0 } ) )  <_  (deg `  g ) ) )
7067, 69anbi12d 691 . . . . . . . 8  |-  ( f  =  g  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
7164, 70imbi12d 311 . . . . . . 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
) ) ) ) )
7271cbvralv 2764 . . . . . 6  |-  ( A. f  e.  ( (Poly `  CC )  \  {
0 p } ) ( (deg `  f
)  =  d  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  <->  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
7350ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  0  <_  (deg
`  f ) )
7473, 59syl5ibrcom 213 . . . . . . . . . . 11  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( ( `' f " {
0 } )  =  (/)  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
7574a1dd 42 . . . . . . . . . 10  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( ( `' f " {
0 } )  =  (/)  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
76 n0 3464 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =/=  (/) 
<->  E. x  x  e.  ( `' f " { 0 } ) )
77 eqid 2283 . . . . . . . . . . . . . 14  |-  ( `' f " { 0 } )  =  ( `' f " {
0 } )
78 simplll 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0 p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  d  e.  NN0 )
79 simpllr 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0 p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  f  e.  ( (Poly `  CC )  \  { 0 p }
) )
80 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0 p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  (deg `  f
)  =  ( d  +  1 ) )
81 simprl 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0 p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  x  e.  ( `' f " {
0 } ) )
82 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0 p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
8377, 78, 79, 80, 81, 82fta1lem 19687 . . . . . . . . . . . . 13  |-  ( ( ( ( d  e. 
NN0  /\  f  e.  ( (Poly `  CC )  \  { 0 p }
) )  /\  (deg `  f )  =  ( d  +  1 ) )  /\  ( x  e.  ( `' f
" { 0 } )  /\  A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) ) )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
8483exp32 588 . . . . . . . . . . . 12  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( x  e.  ( `' f " { 0 } )  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8584exlimdv 1664 . . . . . . . . . . 11  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( E. x  x  e.  ( `' f " {
0 } )  -> 
( A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8676, 85syl5bi 208 . . . . . . . . . 10  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( ( `' f " {
0 } )  =/=  (/)  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8775, 86pm2.61dne 2523 . . . . . . . . 9  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( A. g  e.  ( (Poly `  CC )  \  {
0 p } ) ( (deg `  g
)  =  d  -> 
( ( `' g
" { 0 } )  e.  Fin  /\  ( # `  ( `' g " { 0 } ) )  <_ 
(deg `  g )
) )  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) ) )
8887ex 423 . . . . . . . 8  |-  ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0 p } ) )  ->  ( (deg `  f )  =  ( d  +  1 )  ->  ( A. g  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) ) )
8988com23 72 . . . . . . 7  |-  ( ( d  e.  NN0  /\  f  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 )  =  ( d  +  1 )  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) ) ) )
9089ralrimdva 2633 . . . . . 6  |-  ( d  e.  NN0  ->  ( A. g  e.  ( (Poly `  CC )  \  {
0 p } ) ( (deg `  g
)  =  d  -> 
( ( `' g
" { 0 } )  e.  Fin  /\  ( # `  ( `' g " { 0 } ) )  <_ 
(deg `  g )
) )  ->  A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  ( d  +  1 )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
9172, 90syl5bi 208 . . . . 5  |-  ( d  e.  NN0  ->  ( A. f  e.  ( (Poly `  CC )  \  {
0 p } ) ( (deg `  f
)  =  d  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  ( d  +  1 )  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) ) )
926, 9, 12, 15, 62, 91nn0ind 10108 . . . 4  |-  ( (deg
`  F )  e. 
NN0  ->  A. f  e.  ( (Poly `  CC )  \  { 0 p }
) ( (deg `  f )  =  (deg
`  F )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) )
933, 92syl 15 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  ->  A. f  e.  (
(Poly `  CC )  \  { 0 p }
) ( (deg `  f )  =  (deg
`  F )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) )
94 plyssc 19582 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
9594sseli 3176 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
96 eldifsn 3749 . . . . 5  |-  ( F  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0 p ) )
97 fveq2 5525 . . . . . . . 8  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
9897eqeq1d 2291 . . . . . . 7  |-  ( f  =  F  ->  (
(deg `  f )  =  (deg `  F )  <->  (deg
`  F )  =  (deg `  F )
) )
99 cnveq 4855 . . . . . . . . . . 11  |-  ( f  =  F  ->  `' f  =  `' F
)
10099imaeq1d 5011 . . . . . . . . . 10  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
101 fta1.1 . . . . . . . . . 10  |-  R  =  ( `' F " { 0 } )
102100, 101syl6eqr 2333 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
103102eleq1d 2349 . . . . . . . 8  |-  ( f  =  F  ->  (
( `' f " { 0 } )  e.  Fin  <->  R  e.  Fin ) )
104102fveq2d 5529 . . . . . . . . 9  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
105104, 97breq12d 4036 . . . . . . . 8  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  R )  <_  (deg `  F )
) )
106103, 105anbi12d 691 . . . . . . 7  |-  ( f  =  F  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( R  e. 
Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) )
10798, 106imbi12d 311 . . . . . 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 )
) ) ) )
108107rspcv 2880 . . . . 5  |-  ( F  e.  ( (Poly `  CC )  \  { 0 p } )  -> 
( A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  (
(deg `  F )  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) ) )
10996, 108sylbir 204 . . . 4  |-  ( ( F  e.  (Poly `  CC )  /\  F  =/=  0 p )  -> 
( A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  (
(deg `  F )  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) ) )
11095, 109sylan 457 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  -> 
( A. f  e.  ( (Poly `  CC )  \  { 0 p } ) ( (deg
`  f )  =  (deg `  F )  ->  ( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )  ->  (
(deg `  F )  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) ) )
11193, 110mpd 14 . 2  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  -> 
( (deg `  F
)  =  (deg `  F )  ->  ( R  e.  Fin  /\  ( # `
 R )  <_ 
(deg `  F )
) ) )
1121, 111mpi 16 1  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  -> 
( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149   (/)c0 3455   {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   Fincfn 6863   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    <_ cle 8868   NN0cn0 9965   #chash 11337   0 pc0p 19024  Polycply 19566  degcdgr 19569
This theorem is referenced by:  vieta1lem2  19691  vieta1  19692  plyexmo  19693  aannenlem1  19708  aalioulem2  19713  basellem4  20321  basellem5  20322  dchrfi  20494
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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