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Theorem fta1 21786
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 2443 . 2  |-  (deg `  F )  =  (deg
`  F )
2 dgrcl 21713 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
32adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
(deg `  F )  e.  NN0 )
4 eqeq2 2452 . . . . . . 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 2747 . . . . 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 2452 . . . . . . 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 2747 . . . . 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 2452 . . . . . . 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 2747 . . . . 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 2452 . . . . . . 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 2747 . . . . 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 4013 . . . . . . . . . . 11  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  =/=  0p )
1716adantr 465 . . . . . . . . . 10  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  =/=  0p )
18 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  (deg `  f
)  =  0 )
19 eldifi 3490 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  e.  (Poly `  CC ) )
2019ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  e.  (Poly `  CC ) )
21 0dgrb 21726 . . . . . . . . . . . . . . . 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 5705 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  ( ( CC  X.  {
( f `  0
) } ) `  x ) )
2519adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  e.  (Poly `  CC ) )
26 plyf 21678 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
27 ffn 5571 . . . . . . . . . . . . . . . . . . . 20  |-  ( f : CC --> CC  ->  f  Fn  CC )
28 fniniseg 5836 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) )
3130simprd 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  0 )
3230simpld 459 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  x  e.  CC )
33 fvex 5713 . . . . . . . . . . . . . . . . . . 19  |-  ( f `
 0 )  e. 
_V
3433fvconst2 5945 . . . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f ` 
0 )  =  0 )
3736sneqd 3901 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  { ( f `
 0 ) }  =  { 0 } )
3837xpeq2d 4876 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( CC  X.  { ( f ` 
0 ) } )  =  ( CC  X.  { 0 } ) )
3923, 38eqtrd 2475 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { 0 } ) )
40 df-0p 21160 . . . . . . . . . . . . 13  |-  0p  =  ( CC  X.  { 0 } )
4139, 40syl6eqr 2493 . . . . . . . . . . . 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 2656 . . . . . . . . . 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 3684 . . . . . . . 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 21713 . . . . . . . . 9  |-  ( f  e.  (Poly `  CC )  ->  (deg `  f
)  e.  NN0 )
48 nn0ge0 10617 . . . . . . . . 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 7552 . . . . . . . . . . 11  |-  (/)  e.  Fin
5250, 51syl6eqel 2531 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  e.  Fin )
5352biantrurd 508 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
54 fveq2 5703 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  ( # `  (/) ) )
55 hash0 12147 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
5654, 55syl6eq 2491 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  0 )
5756breq1d 4314 . . . . . . . . 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 2793 . . . . 5  |-  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  0  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
62 fveq2 5703 . . . . . . . . 9  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
6362eqeq1d 2451 . . . . . . . 8  |-  ( f  =  g  ->  (
(deg `  f )  =  d  <->  (deg `  g )  =  d ) )
64 cnveq 5025 . . . . . . . . . . 11  |-  ( f  =  g  ->  `' f  =  `' g
)
6564imaeq1d 5180 . . . . . . . . . 10  |-  ( f  =  g  ->  ( `' f " {
0 } )  =  ( `' g " { 0 } ) )
6665eleq1d 2509 . . . . . . . . 9  |-  ( f  =  g  ->  (
( `' f " { 0 } )  e.  Fin  <->  ( `' g " { 0 } )  e.  Fin )
)
6765fveq2d 5707 . . . . . . . . . 10  |-  ( f  =  g  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' g " { 0 } ) ) )
6867, 62breq12d 4317 . . . . . . . . 9  |-  ( f  =  g  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  ( `' g " { 0 } ) )  <_  (deg `  g ) ) )
6966, 68anbi12d 710 . . . . . . . 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 2959 . . . . . 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 726 . . . . . . . . . . . 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 3658 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =/=  (/) 
<->  E. x  x  e.  ( `' f " { 0 } ) )
76 eqid 2443 . . . . . . . . . . . . . 14  |-  ( `' f " { 0 } )  =  ( `' f " {
0 } )
77 simplll 757 . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . 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 21785 . . . . . . . . . . . . 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 605 . . . . . . . . . . . 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 1690 . . . . . . . . . . 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 2700 . . . . . . . . 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 2818 . . . . . 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 10750 . . . 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 21680 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
9493sseli 3364 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
95 eldifsn 4012 . . . . 5  |-  ( F  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0p ) )
96 fveq2 5703 . . . . . . . 8  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
9796eqeq1d 2451 . . . . . . 7  |-  ( f  =  F  ->  (
(deg `  f )  =  (deg `  F )  <->  (deg
`  F )  =  (deg `  F )
) )
98 cnveq 5025 . . . . . . . . . . 11  |-  ( f  =  F  ->  `' f  =  `' F
)
9998imaeq1d 5180 . . . . . . . . . 10  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
100 fta1.1 . . . . . . . . . 10  |-  R  =  ( `' F " { 0 } )
10199, 100syl6eqr 2493 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
102101eleq1d 2509 . . . . . . . 8  |-  ( f  =  F  ->  (
( `' f " { 0 } )  e.  Fin  <->  R  e.  Fin ) )
103101fveq2d 5707 . . . . . . . . 9  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
104103, 96breq12d 4317 . . . . . . . 8  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  R )  <_  (deg `  F )
) )
105102, 104anbi12d 710 . . . . . . 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 3081 . . . . 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 471 . . 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 1369   E.wex 1586    e. wcel 1756    =/= wne 2618   A.wral 2727    \ cdif 3337   (/)c0 3649   {csn 3889   class class class wbr 4304    X. cxp 4850   `'ccnv 4851   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   Fincfn 7322   CCcc 9292   0cc0 9294   1c1 9295    + caddc 9297    <_ cle 9431   NN0cn0 10591   #chash 12115   0pc0p 21159  Polycply 21664  degcdgr 21667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979  df-sum 13176  df-0p 21160  df-ply 21668  df-idp 21669  df-coe 21670  df-dgr 21671  df-quot 21769
This theorem is referenced by:  vieta1lem2  21789  vieta1  21790  plyexmo  21791  aannenlem1  21806  aalioulem2  21811  basellem4  22433  basellem5  22434  dchrfi  22606
  Copyright terms: Public domain W3C validator