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Theorem plyexmo 20222
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 3339 . . . . . . . . . . . . 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 20109 . . . . . . . . . . . . . . . . . . 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 5583 . . . . . . . . . . . . . . . . . 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 20109 . . . . . . . . . . . . . . . . . . 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 5583 . . . . . . . . . . . . . . . . . 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 9063 . . . . . . . . . . . . . . . . 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 3340 . . . . . . . . . . . . . . . 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 6309 . . . . . . . . . . . . . . . 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 5863 . . . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . . . . 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 5737 . . . . . . . . . . . . . . . . . 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 5737 . . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . . 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 9455 . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 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 20133 . . . . . . . . . . . . . . . 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 20109 . . . . . . . . . . . . . . 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 5583 . . . . . . . . . . . . . 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 5843 . . . . . . . . . . . . 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 3346 . . . . . . . . . 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 7321 . . . . . . . . . . 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 2600 . . . . . . . . . . . 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 2435 . . . . . . . . . . . 12  |-  ( `' ( p  o F  -  a ) " { 0 } )  =  ( `' ( p  o F  -  a ) " {
0 } )
5453fta1 20217 . . . . . . . . . . 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 19554 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
6058, 59syl6eq 2483 . . . . . 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 9989 . . . . . . 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 1642 . . 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 2495 . . . . 5  |-  ( p  =  a  ->  (
p  e.  (Poly `  CC )  <->  a  e.  (Poly `  CC ) ) )
68 reseq1 5132 . . . . . 6  |-  ( p  =  a  ->  (
p  |`  D )  =  ( a  |`  D ) )
6968eqeq1d 2443 . . . . 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 2313 . . 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 20111 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
7473sseli 3336 . . . 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 2327 . 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 1549    = wceq 1652    e. wcel 1725   E*wmo 2281    =/= wne 2598   _Vcvv 2948    C_ wss 3312   {csn 3806   class class class wbr 4204    X. cxp 4868   `'ccnv 4869    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Fincfn 7101   CCcc 8980   0cc0 8982    <_ cle 9113    - cmin 9283   #chash 11610   0 pc0p 19553  Polycply 20095  degcdgr 20098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-idp 20100  df-coe 20101  df-dgr 20102  df-quot 20200
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