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Theorem plyexmo 20183
Description: An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
Assertion
Ref Expression
plyexmo  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  E* p ( p  e.  (Poly `  S
)  /\  ( p  |`  D )  =  F ) )
Distinct variable groups:    S, p    F, p    D, p

Proof of Theorem plyexmo
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 732 . . . . . . . . 9  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  -.  D  e.  Fin )
2 simpll 731 . . . . . . . . . . . . . 14  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  D  C_  CC )
32sseld 3307 . . . . . . . . . . . . 13  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( b  e.  D  ->  b  e.  CC ) )
4 simprll 739 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  p  e.  (Poly `  CC ) )
5 plyf 20070 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e.  (Poly `  CC )  ->  p : CC --> CC )
64, 5syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  p : CC --> CC )
7 ffn 5550 . . . . . . . . . . . . . . . . . 18  |-  ( p : CC --> CC  ->  p  Fn  CC )
86, 7syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  p  Fn  CC )
98adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  p  Fn  CC )
10 simprrl 741 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
a  e.  (Poly `  CC ) )
11 plyf 20070 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  (Poly `  CC )  ->  a : CC --> CC )
1210, 11syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
a : CC --> CC )
13 ffn 5550 . . . . . . . . . . . . . . . . . 18  |-  ( a : CC --> CC  ->  a  Fn  CC )
1412, 13syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
a  Fn  CC )
1514adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  a  Fn  CC )
16 cnex 9027 . . . . . . . . . . . . . . . . 17  |-  CC  e.  _V
1716a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  CC  e.  _V )
182sselda 3308 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  b  e.  CC )
19 fnfvof 6276 . . . . . . . . . . . . . . . 16  |-  ( ( ( p  Fn  CC  /\  a  Fn  CC )  /\  ( CC  e.  _V  /\  b  e.  CC ) )  ->  (
( p  o F  -  a ) `  b )  =  ( ( p `  b
)  -  ( a `
 b ) ) )
209, 15, 17, 18, 19syl22anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( (
p  o F  -  a ) `  b
)  =  ( ( p `  b )  -  ( a `  b ) ) )
216adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  p : CC
--> CC )
2221, 18ffvelrnd 5830 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( p `  b )  e.  CC )
23 simprlr 740 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  |`  D )  =  F )
24 simprrr 742 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( a  |`  D )  =  F )
2523, 24eqtr4d 2439 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  |`  D )  =  ( a  |`  D ) )
2625adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( p  |`  D )  =  ( a  |`  D )
)
2726fveq1d 5689 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( (
p  |`  D ) `  b )  =  ( ( a  |`  D ) `
 b ) )
28 fvres 5704 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  D  ->  (
( p  |`  D ) `
 b )  =  ( p `  b
) )
2928adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( (
p  |`  D ) `  b )  =  ( p `  b ) )
30 fvres 5704 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  D  ->  (
( a  |`  D ) `
 b )  =  ( a `  b
) )
3130adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( (
a  |`  D ) `  b )  =  ( a `  b ) )
3227, 29, 313eqtr3d 2444 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( p `  b )  =  ( a `  b ) )
3322, 32subeq0bd 9419 . . . . . . . . . . . . . . 15  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( (
p `  b )  -  ( a `  b ) )  =  0 )
3420, 33eqtrd 2436 . . . . . . . . . . . . . 14  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  ( (
p  o F  -  a ) `  b
)  =  0 )
3534ex 424 . . . . . . . . . . . . 13  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( b  e.  D  ->  ( ( p  o F  -  a ) `
 b )  =  0 ) )
363, 35jcad 520 . . . . . . . . . . . 12  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( b  e.  D  ->  ( b  e.  CC  /\  ( ( p  o F  -  a ) `
 b )  =  0 ) ) )
37 plysubcl 20094 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  (Poly `  CC )  /\  a  e.  (Poly `  CC )
)  ->  ( p  o F  -  a
)  e.  (Poly `  CC ) )
384, 10, 37syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  o F  -  a )  e.  (Poly `  CC )
)
39 plyf 20070 . . . . . . . . . . . . . . 15  |-  ( ( p  o F  -  a )  e.  (Poly `  CC )  ->  (
p  o F  -  a ) : CC --> CC )
4038, 39syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  o F  -  a ) : CC --> CC )
41 ffn 5550 . . . . . . . . . . . . . 14  |-  ( ( p  o F  -  a ) : CC --> CC  ->  ( p  o F  -  a )  Fn  CC )
4240, 41syl 16 . . . . . . . . . . . . 13  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  o F  -  a )  Fn  CC )
43 fniniseg 5810 . . . . . . . . . . . . 13  |-  ( ( p  o F  -  a )  Fn  CC  ->  ( b  e.  ( `' ( p  o F  -  a )
" { 0 } )  <->  ( b  e.  CC  /\  ( ( p  o F  -  a ) `  b
)  =  0 ) ) )
4442, 43syl 16 . . . . . . . . . . . 12  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( b  e.  ( `' ( p  o F  -  a )
" { 0 } )  <->  ( b  e.  CC  /\  ( ( p  o F  -  a ) `  b
)  =  0 ) ) )
4536, 44sylibrd 226 . . . . . . . . . . 11  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( b  e.  D  ->  b  e.  ( `' ( p  o F  -  a ) " { 0 } ) ) )
4645ssrdv 3314 . . . . . . . . . 10  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  D  C_  ( `' ( p  o F  -  a ) " {
0 } ) )
47 ssfi 7288 . . . . . . . . . . 11  |-  ( ( ( `' ( p  o F  -  a
) " { 0 } )  e.  Fin  /\  D  C_  ( `' ( p  o F  -  a ) " { 0 } ) )  ->  D  e.  Fin )
4847expcom 425 . . . . . . . . . 10  |-  ( D 
C_  ( `' ( p  o F  -  a ) " {
0 } )  -> 
( ( `' ( p  o F  -  a ) " {
0 } )  e. 
Fin  ->  D  e.  Fin ) )
4946, 48syl 16 . . . . . . . . 9  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( ( `' ( p  o F  -  a ) " {
0 } )  e. 
Fin  ->  D  e.  Fin ) )
501, 49mtod 170 . . . . . . . 8  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  -.  ( `' ( p  o F  -  a
) " { 0 } )  e.  Fin )
51 df-ne 2569 . . . . . . . . . . . 12  |-  ( ( p  o F  -  a )  =/=  0 p 
<->  -.  ( p  o F  -  a )  =  0 p )
5251biimpri 198 . . . . . . . . . . 11  |-  ( -.  ( p  o F  -  a )  =  0 p  ->  (
p  o F  -  a )  =/=  0 p )
53 eqid 2404 . . . . . . . . . . . 12  |-  ( `' ( p  o F  -  a ) " { 0 } )  =  ( `' ( p  o F  -  a ) " {
0 } )
5453fta1 20178 . . . . . . . . . . 11  |-  ( ( ( p  o F  -  a )  e.  (Poly `  CC )  /\  ( p  o F  -  a )  =/=  0 p )  -> 
( ( `' ( p  o F  -  a ) " {
0 } )  e. 
Fin  /\  ( # `  ( `' ( p  o F  -  a )
" { 0 } ) )  <_  (deg `  ( p  o F  -  a ) ) ) )
5538, 52, 54syl2an 464 . . . . . . . . . 10  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  -.  ( p  o F  -  a
)  =  0 p )  ->  ( ( `' ( p  o F  -  a )
" { 0 } )  e.  Fin  /\  ( # `  ( `' ( p  o F  -  a ) " { 0 } ) )  <_  (deg `  (
p  o F  -  a ) ) ) )
5655simpld 446 . . . . . . . . 9  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  -.  ( p  o F  -  a
)  =  0 p )  ->  ( `' ( p  o F  -  a ) " { 0 } )  e.  Fin )
5756ex 424 . . . . . . . 8  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( -.  ( p  o F  -  a
)  =  0 p  ->  ( `' ( p  o F  -  a ) " {
0 } )  e. 
Fin ) )
5850, 57mt3d 119 . . . . . . 7  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  o F  -  a )  =  0 p )
59 df-0p 19515 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
6058, 59syl6eq 2452 . . . . . 6  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  o F  -  a )  =  ( CC  X.  {
0 } ) )
6116a1i 11 . . . . . . 7  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  CC  e.  _V )
62 ofsubeq0 9953 . . . . . . 7  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  o F  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
6361, 6, 12, 62syl3anc 1184 . . . . . 6  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( ( p  o F  -  a )  =  ( CC  X.  { 0 } )  <-> 
p  =  a ) )
6460, 63mpbid 202 . . . . 5  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  p  =  a )
6564ex 424 . . . 4  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  ( ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) )  ->  p  =  a ) )
6665alrimivv 1639 . . 3  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  A. p A. a
( ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) )  ->  p  =  a ) )
67 eleq1 2464 . . . . 5  |-  ( p  =  a  ->  (
p  e.  (Poly `  CC )  <->  a  e.  (Poly `  CC ) ) )
68 reseq1 5099 . . . . . 6  |-  ( p  =  a  ->  (
p  |`  D )  =  ( a  |`  D ) )
6968eqeq1d 2412 . . . . 5  |-  ( p  =  a  ->  (
( p  |`  D )  =  F  <->  ( a  |`  D )  =  F ) )
7067, 69anbi12d 692 . . . 4  |-  ( p  =  a  ->  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  <->  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )
7170mo4 2287 . . 3  |-  ( E* p ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  <->  A. p A. a ( ( ( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) )  ->  p  =  a )
)
7266, 71sylibr 204 . 2  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  E* p ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F ) )
73 plyssc 20072 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
7473sseli 3304 . . . 4  |-  ( p  e.  (Poly `  S
)  ->  p  e.  (Poly `  CC ) )
7574anim1i 552 . . 3  |-  ( ( p  e.  (Poly `  S )  /\  (
p  |`  D )  =  F )  ->  (
p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F ) )
7675moimi 2301 . 2  |-  ( E* p ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  ->  E* p ( p  e.  (Poly `  S )  /\  ( p  |`  D )  =  F ) )
7772, 76syl 16 1  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  E* p ( p  e.  (Poly `  S
)  /\  ( p  |`  D )  =  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   E*wmo 2255    =/= wne 2567   _Vcvv 2916    C_ wss 3280   {csn 3774   class class class wbr 4172    X. cxp 4835   `'ccnv 4836    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   Fincfn 7068   CCcc 8944   0cc0 8946    <_ cle 9077    - cmin 9247   #chash 11573   0 pc0p 19514  Polycply 20056  degcdgr 20059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-idp 20061  df-coe 20062  df-dgr 20063  df-quot 20161
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