MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fta1 Structured version   Unicode version

Theorem fta1 23203
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 2428 . 2  |-  (deg `  F )  =  (deg
`  F )
2 dgrcl 23129 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
32adantr 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
(deg `  F )  e.  NN0 )
4 eqeq2 2439 . . . . . . 7  |-  ( x  =  0  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  0 ) )
54imbi1d 318 . . . . . 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 2804 . . . . 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 2439 . . . . . . 7  |-  ( x  =  d  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  d ) )
87imbi1d 318 . . . . . 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 2804 . . . . 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 2439 . . . . . . 7  |-  ( x  =  ( d  +  1 )  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  ( d  +  1 ) ) )
1110imbi1d 318 . . . . . 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 2804 . . . . 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 2439 . . . . . . 7  |-  ( x  =  (deg `  F
)  ->  ( (deg `  f )  =  x  <-> 
(deg `  f )  =  (deg `  F )
) )
1413imbi1d 318 . . . . . 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 2804 . . . . 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 4069 . . . . . . . . . . 11  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  =/=  0p )
1716adantr 466 . . . . . . . . . 10  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  =/=  0p )
18 simplr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  (deg `  f
)  =  0 )
19 eldifi 3530 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  e.  (Poly `  CC ) )
2019ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  e.  (Poly `  CC ) )
21 0dgrb 23142 . . . . . . . . . . . . . . . 16  |-  ( f  e.  (Poly `  CC )  ->  ( (deg `  f )  =  0  <-> 
f  =  ( CC 
X.  { ( f `
 0 ) } ) ) )
2220, 21syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( (deg `  f )  =  0  <-> 
f  =  ( CC 
X.  { ( f `
 0 ) } ) ) )
2318, 22mpbid 213 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { ( f `  0 ) } ) )
2423fveq1d 5827 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  ( ( CC  X.  {
( f `  0
) } ) `  x ) )
2519adantr 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  e.  (Poly `  CC ) )
26 plyf 23094 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
27 ffn 5689 . . . . . . . . . . . . . . . . . . . 20  |-  ( f : CC --> CC  ->  f  Fn  CC )
28 fniniseg 5962 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  Fn  CC  ->  (
x  e.  ( `' f " { 0 } )  <->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) ) )
2925, 26, 27, 284syl 19 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( x  e.  ( `' f " { 0 } )  <-> 
( x  e.  CC  /\  ( f `  x
)  =  0 ) ) )
3029biimpa 486 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) )
3130simprd 464 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  0 )
3230simpld 460 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  x  e.  CC )
33 fvex 5835 . . . . . . . . . . . . . . . . . . 19  |-  ( f `
 0 )  e. 
_V
3433fvconst2 6079 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  (
( CC  X.  {
( f `  0
) } ) `  x )  =  ( f `  0 ) )
3532, 34syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( ( CC 
X.  { ( f `
 0 ) } ) `  x )  =  ( f ` 
0 ) )
3624, 31, 353eqtr3rd 2471 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f ` 
0 )  =  0 )
3736sneqd 3953 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  { ( f `
 0 ) }  =  { 0 } )
3837xpeq2d 4820 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( CC  X.  { ( f ` 
0 ) } )  =  ( CC  X.  { 0 } ) )
3923, 38eqtrd 2462 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { 0 } ) )
40 df-0p 22570 . . . . . . . . . . . . 13  |-  0p  =  ( CC  X.  { 0 } )
4139, 40syl6eqr 2480 . . . . . . . . . . . 12  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  0p )
4241ex 435 . . . . . . . . . . 11  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( x  e.  ( `' f " { 0 } )  ->  f  =  0p ) )
4342necon3ad 2614 . . . . . . . . . 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 3742 . . . . . . . 8  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( `' f
" { 0 } )  =  (/) )
4645ex 435 . . . . . . 7  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( (deg `  f
)  =  0  -> 
( `' f " { 0 } )  =  (/) ) )
47 dgrcl 23129 . . . . . . . . 9  |-  ( f  e.  (Poly `  CC )  ->  (deg `  f
)  e.  NN0 )
48 nn0ge0 10846 . . . . . . . . 9  |-  ( (deg
`  f )  e. 
NN0  ->  0  <_  (deg `  f ) )
4919, 47, 483syl 18 . . . . . . . 8  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
0  <_  (deg `  f
) )
50 id 22 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  =  (/) )
51 0fin 7752 . . . . . . . . . . 11  |-  (/)  e.  Fin
5250, 51syl6eqel 2514 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  e.  Fin )
5352biantrurd 510 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
54 fveq2 5825 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  ( # `  (/) ) )
55 hash0 12498 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
5654, 55syl6eq 2478 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  0 )
5756breq1d 4376 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  0  <_  (deg `  f ) ) )
5853, 57bitr3d 258 . . . . . . . 8  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) )  <->  0  <_  (deg
`  f ) ) )
5949, 58syl5ibrcom 225 . . . . . . 7  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( ( `' f
" { 0 } )  =  (/)  ->  (
( `' f " { 0 } )  e.  Fin  /\  ( # `
 ( `' f
" { 0 } ) )  <_  (deg `  f ) ) ) )
6046, 59syld 45 . . . . . 6  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( (deg `  f
)  =  0  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) ) )
6160rgen 2724 . . . . 5  |-  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  0  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
62 fveq2 5825 . . . . . . . . 9  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
6362eqeq1d 2430 . . . . . . . 8  |-  ( f  =  g  ->  (
(deg `  f )  =  d  <->  (deg `  g )  =  d ) )
64 cnveq 4970 . . . . . . . . . . 11  |-  ( f  =  g  ->  `' f  =  `' g
)
6564imaeq1d 5129 . . . . . . . . . 10  |-  ( f  =  g  ->  ( `' f " {
0 } )  =  ( `' g " { 0 } ) )
6665eleq1d 2490 . . . . . . . . 9  |-  ( f  =  g  ->  (
( `' f " { 0 } )  e.  Fin  <->  ( `' g " { 0 } )  e.  Fin )
)
6765fveq2d 5829 . . . . . . . . . 10  |-  ( f  =  g  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' g " { 0 } ) ) )
6867, 62breq12d 4379 . . . . . . . . 9  |-  ( f  =  g  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  ( `' g " { 0 } ) )  <_  (deg `  g ) ) )
6966, 68anbi12d 715 . . . . . . . 8  |-  ( f  =  g  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
7063, 69imbi12d 321 . . . . . . 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 2996 . . . . . 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 731 . . . . . . . . . . . 12  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  0  <_  (deg
`  f ) )
7372, 58syl5ibrcom 225 . . . . . . . . . . 11  |-  ( ( ( d  e.  NN0  /\  f  e.  ( (Poly `  CC )  \  {
0p } ) )  /\  (deg `  f )  =  ( d  +  1 ) )  ->  ( ( `' f " {
0 } )  =  (/)  ->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
7473a1dd 47 . . . . . . . . . 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 3714 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =/=  (/) 
<->  E. x  x  e.  ( `' f " { 0 } ) )
76 eqid 2428 . . . . . . . . . . . . . 14  |-  ( `' f " { 0 } )  =  ( `' f " {
0 } )
77 simplll 766 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . 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 23202 . . . . . . . . . . . . 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 608 . . . . . . . . . . . 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 1772 . . . . . . . . . . 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 220 . . . . . . . . . 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 2687 . . . . . . . . 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 435 . . . . . . . 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 81 . . . . . . 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 2783 . . . . . 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 220 . . . . 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 10981 . . . 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 17 . . 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 23096 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
9493sseli 3403 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
95 eldifsn 4068 . . . . 5  |-  ( F  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0p ) )
96 fveq2 5825 . . . . . . . 8  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
9796eqeq1d 2430 . . . . . . 7  |-  ( f  =  F  ->  (
(deg `  f )  =  (deg `  F )  <->  (deg
`  F )  =  (deg `  F )
) )
98 cnveq 4970 . . . . . . . . . . 11  |-  ( f  =  F  ->  `' f  =  `' F
)
9998imaeq1d 5129 . . . . . . . . . 10  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
100 fta1.1 . . . . . . . . . 10  |-  R  =  ( `' F " { 0 } )
10199, 100syl6eqr 2480 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
102101eleq1d 2490 . . . . . . . 8  |-  ( f  =  F  ->  (
( `' f " { 0 } )  e.  Fin  <->  R  e.  Fin ) )
103101fveq2d 5829 . . . . . . . . 9  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
104103, 96breq12d 4379 . . . . . . . 8  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  R )  <_  (deg `  F )
) )
105102, 104anbi12d 715 . . . . . . 7  |-  ( f  =  F  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( R  e. 
Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) )
10697, 105imbi12d 321 . . . . . 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 3121 . . . . 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 216 . . . 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 473 . . 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 20 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 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2599   A.wral 2714    \ cdif 3376   (/)c0 3704   {csn 3941   class class class wbr 4366    X. cxp 4794   `'ccnv 4795   "cima 4799    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249   Fincfn 7524   CCcc 9488   0cc0 9490   1c1 9491    + caddc 9493    <_ cle 9627   NN0cn0 10820   #chash 12465   0pc0p 22569  Polycply 23080  degcdgr 23083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-fz 11736  df-fzo 11867  df-fl 11978  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-clim 13495  df-rlim 13496  df-sum 13696  df-0p 22570  df-ply 23084  df-idp 23085  df-coe 23086  df-dgr 23087  df-quot 23186
This theorem is referenced by:  vieta1lem2  23206  vieta1  23207  plyexmo  23208  aannenlem1  23226  aalioulem2  23231  basellem4  23952  basellem5  23953  dchrfi  24125
  Copyright terms: Public domain W3C validator