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Theorem plyexmo 20097
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 3290 . . . . . . . . . . . . 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 19984 . . . . . . . . . . . . . . . . . . 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 5531 . . . . . . . . . . . . . . . . . 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 19984 . . . . . . . . . . . . . . . . . . 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 5531 . . . . . . . . . . . . . . . . . 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 9004 . . . . . . . . . . . . . . . . 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 3291 . . . . . . . . . . . . . . . 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 6256 . . . . . . . . . . . . . . . 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 5810 . . . . . . . . . . . . . . . 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 2422 . . . . . . . . . . . . . . . . . . 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 5670 . . . . . . . . . . . . . . . . 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 5685 . . . . . . . . . . . . . . . . . 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 5685 . . . . . . . . . . . . . . . . . 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 2427 . . . . . . . . . . . . . . . 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 9395 . . . . . . . . . . . . . . 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 2419 . . . . . . . . . . . . . 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 20008 . . . . . . . . . . . . . . . 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 19984 . . . . . . . . . . . . . . 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 5531 . . . . . . . . . . . . . 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 5790 . . . . . . . . . . . . 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 3297 . . . . . . . . . 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 7265 . . . . . . . . . . 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 2552 . . . . . . . . . . . 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 2387 . . . . . . . . . . . 12  |-  ( `' ( p  o F  -  a ) " { 0 } )  =  ( `' ( p  o F  -  a ) " {
0 } )
5453fta1 20092 . . . . . . . . . . 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 19429 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
6058, 59syl6eq 2435 . . . . . 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 9929 . . . . . . 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 2447 . . . . 5  |-  ( p  =  a  ->  (
p  e.  (Poly `  CC )  <->  a  e.  (Poly `  CC ) ) )
68 reseq1 5080 . . . . . 6  |-  ( p  =  a  ->  (
p  |`  D )  =  ( a  |`  D ) )
6968eqeq1d 2395 . . . . 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 2271 . . 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 19986 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
7473sseli 3287 . . . 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 2285 . 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 1717   E*wmo 2239    =/= wne 2550   _Vcvv 2899    C_ wss 3263   {csn 3757   class class class wbr 4153    X. cxp 4816   `'ccnv 4817    |` cres 4820   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Fcof 6242   Fincfn 7045   CCcc 8921   0cc0 8923    <_ cle 9054    - cmin 9223   #chash 11545   0 pc0p 19428  Polycply 19970  degcdgr 19973
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-0p 19429  df-ply 19974  df-idp 19975  df-coe 19976  df-dgr 19977  df-quot 20075
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