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Theorem fta1 22870
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 2454 . 2  |-  (deg `  F )  =  (deg
`  F )
2 dgrcl 22796 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
32adantr 463 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
(deg `  F )  e.  NN0 )
4 eqeq2 2469 . . . . . . 7  |-  ( x  =  0  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  0 ) )
54imbi1d 315 . . . . . 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 2893 . . . . 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 2469 . . . . . . 7  |-  ( x  =  d  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  d ) )
87imbi1d 315 . . . . . 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 2893 . . . . 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 2469 . . . . . . 7  |-  ( x  =  ( d  +  1 )  ->  (
(deg `  f )  =  x  <->  (deg `  f )  =  ( d  +  1 ) ) )
1110imbi1d 315 . . . . . 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 2893 . . . . 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 2469 . . . . . . 7  |-  ( x  =  (deg `  F
)  ->  ( (deg `  f )  =  x  <-> 
(deg `  f )  =  (deg `  F )
) )
1413imbi1d 315 . . . . . 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 2893 . . . . 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 4142 . . . . . . . . . . 11  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  =/=  0p )
1716adantr 463 . . . . . . . . . 10  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  =/=  0p )
18 simplr 753 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  (deg `  f
)  =  0 )
19 eldifi 3612 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
f  e.  (Poly `  CC ) )
2019ad2antrr 723 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  e.  (Poly `  CC ) )
21 0dgrb 22809 . . . . . . . . . . . . . . . 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 5850 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  ( ( CC  X.  {
( f `  0
) } ) `  x ) )
2519adantr 463 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  f  e.  (Poly `  CC ) )
26 plyf 22761 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
27 ffn 5713 . . . . . . . . . . . . . . . . . . . 20  |-  ( f : CC --> CC  ->  f  Fn  CC )
28 fniniseg 5984 . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( x  e.  CC  /\  ( f `
 x )  =  0 ) )
3130simprd 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f `  x )  =  0 )
3230simpld 457 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  x  e.  CC )
33 fvex 5858 . . . . . . . . . . . . . . . . . . 19  |-  ( f `
 0 )  e. 
_V
3433fvconst2 6103 . . . . . . . . . . . . . . . . . 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 2504 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( f ` 
0 )  =  0 )
3736sneqd 4028 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  { ( f `
 0 ) }  =  { 0 } )
3837xpeq2d 5012 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  ( CC  X.  { ( f ` 
0 ) } )  =  ( CC  X.  { 0 } ) )
3923, 38eqtrd 2495 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  ( CC  X.  { 0 } ) )
40 df-0p 22243 . . . . . . . . . . . . 13  |-  0p  =  ( CC  X.  { 0 } )
4139, 40syl6eqr 2513 . . . . . . . . . . . 12  |-  ( ( ( f  e.  ( (Poly `  CC )  \  { 0p }
)  /\  (deg `  f
)  =  0 )  /\  x  e.  ( `' f " {
0 } ) )  ->  f  =  0p )
4241ex 432 . . . . . . . . . . 11  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( x  e.  ( `' f " { 0 } )  ->  f  =  0p ) )
4342necon3ad 2664 . . . . . . . . . 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 3819 . . . . . . . 8  |-  ( ( f  e.  ( (Poly `  CC )  \  {
0p } )  /\  (deg `  f
)  =  0 )  ->  ( `' f
" { 0 } )  =  (/) )
4645ex 432 . . . . . . 7  |-  ( f  e.  ( (Poly `  CC )  \  { 0p } )  -> 
( (deg `  f
)  =  0  -> 
( `' f " { 0 } )  =  (/) ) )
47 dgrcl 22796 . . . . . . . . 9  |-  ( f  e.  (Poly `  CC )  ->  (deg `  f
)  e.  NN0 )
48 nn0ge0 10817 . . . . . . . . 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 7740 . . . . . . . . . . 11  |-  (/)  e.  Fin
5250, 51syl6eqel 2550 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( `' f
" { 0 } )  e.  Fin )
5352biantrurd 506 . . . . . . . . 9  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( ( # `  ( `' f " { 0 } ) )  <_  (deg `  f
)  <->  ( ( `' f " { 0 } )  e.  Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) ) )
54 fveq2 5848 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  ( # `  (/) ) )
55 hash0 12420 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
5654, 55syl6eq 2511 . . . . . . . . . 10  |-  ( ( `' f " {
0 } )  =  (/)  ->  ( # `  ( `' f " {
0 } ) )  =  0 )
5756breq1d 4449 . . . . . . . . 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 2814 . . . . 5  |-  A. f  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  f )  =  0  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
62 fveq2 5848 . . . . . . . . 9  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
6362eqeq1d 2456 . . . . . . . 8  |-  ( f  =  g  ->  (
(deg `  f )  =  d  <->  (deg `  g )  =  d ) )
64 cnveq 5165 . . . . . . . . . . 11  |-  ( f  =  g  ->  `' f  =  `' g
)
6564imaeq1d 5324 . . . . . . . . . 10  |-  ( f  =  g  ->  ( `' f " {
0 } )  =  ( `' g " { 0 } ) )
6665eleq1d 2523 . . . . . . . . 9  |-  ( f  =  g  ->  (
( `' f " { 0 } )  e.  Fin  <->  ( `' g " { 0 } )  e.  Fin )
)
6765fveq2d 5852 . . . . . . . . . 10  |-  ( f  =  g  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' g " { 0 } ) ) )
6867, 62breq12d 4452 . . . . . . . . 9  |-  ( f  =  g  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  ( `' g " { 0 } ) )  <_  (deg `  g ) ) )
6966, 68anbi12d 708 . . . . . . . 8  |-  ( f  =  g  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( ( `' g " { 0 } )  e.  Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
7063, 69imbi12d 318 . . . . . . 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 3081 . . . . . 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 724 . . . . . . . . . . . 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 3793 . . . . . . . . . . 11  |-  ( ( `' f " {
0 } )  =/=  (/) 
<->  E. x  x  e.  ( `' f " { 0 } ) )
76 eqid 2454 . . . . . . . . . . . . . 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 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
) ) ) ) )  ->  (deg `  f
)  =  ( d  +  1 ) )
80 simprl 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
) ) ) ) )  ->  x  e.  ( `' f " {
0 } ) )
81 simprr 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
) ) ) ) )  ->  A. g  e.  ( (Poly `  CC )  \  { 0p } ) ( (deg
`  g )  =  d  ->  ( ( `' g " {
0 } )  e. 
Fin  /\  ( # `  ( `' g " {
0 } ) )  <_  (deg `  g
) ) ) )
8276, 77, 78, 79, 80, 81fta1lem 22869 . . . . . . . . . . . . 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 603 . . . . . . . . . . . 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 1729 . . . . . . . . . . 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 2771 . . . . . . . . 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 432 . . . . . . . 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 2872 . . . . . 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 10955 . . . 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 22763 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
9493sseli 3485 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
95 eldifsn 4141 . . . . 5  |-  ( F  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( F  e.  (Poly `  CC )  /\  F  =/=  0p ) )
96 fveq2 5848 . . . . . . . 8  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
9796eqeq1d 2456 . . . . . . 7  |-  ( f  =  F  ->  (
(deg `  f )  =  (deg `  F )  <->  (deg
`  F )  =  (deg `  F )
) )
98 cnveq 5165 . . . . . . . . . . 11  |-  ( f  =  F  ->  `' f  =  `' F
)
9998imaeq1d 5324 . . . . . . . . . 10  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
100 fta1.1 . . . . . . . . . 10  |-  R  =  ( `' F " { 0 } )
10199, 100syl6eqr 2513 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
102101eleq1d 2523 . . . . . . . 8  |-  ( f  =  F  ->  (
( `' f " { 0 } )  e.  Fin  <->  R  e.  Fin ) )
103101fveq2d 5852 . . . . . . . . 9  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
104103, 96breq12d 4452 . . . . . . . 8  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )  <->  (
# `  R )  <_  (deg `  F )
) )
105102, 104anbi12d 708 . . . . . . 7  |-  ( f  =  F  ->  (
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
)  <->  ( R  e. 
Fin  /\  ( # `  R
)  <_  (deg `  F
) ) ) )
10697, 105imbi12d 318 . . . . . 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 3203 . . . . 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 469 . . 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 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804    \ cdif 3458   (/)c0 3783   {csn 4016   class class class wbr 4439    X. cxp 4986   `'ccnv 4987   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618   NN0cn0 10791   #chash 12387   0pc0p 22242  Polycply 22747  degcdgr 22750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-0p 22243  df-ply 22751  df-idp 22752  df-coe 22753  df-dgr 22754  df-quot 22853
This theorem is referenced by:  vieta1lem2  22873  vieta1  22874  plyexmo  22875  aannenlem1  22890  aalioulem2  22895  basellem4  23555  basellem5  23556  dchrfi  23728
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