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Theorem plyexmo 21738
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 749 . . . . . . . . 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 748 . . . . . . . . . . . . . 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 3352 . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . 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 21625 . . . . . . . . . . . . . . . . . . 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 5556 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . 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 21625 . . . . . . . . . . . . . . . . . . 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 5556 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 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 9359 . . . . . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . 16  |-  ( ( ( p  Fn  CC  /\  a  Fn  CC )  /\  ( CC  e.  _V  /\  b  e.  CC ) )  ->  (
( p  oF  -  a ) `  b )  =  ( ( p `  b
)  -  ( a `
 b ) ) )
209, 15, 17, 18, 19syl22anc 1214 . . . . . . . . . . . . . . 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  oF  -  a ) `  b
)  =  ( ( p `  b )  -  ( a `  b ) ) )
216adantr 462 . . . . . . . . . . . . . . . . 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 5841 . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 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 5690 . . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  D  ->  (
( p  |`  D ) `
 b )  =  ( p `  b
) )
2928adantl 463 . . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  D  ->  (
( a  |`  D ) `
 b )  =  ( a `  b
) )
3130adantl 463 . . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . . 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 9770 . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . 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  oF  -  a ) `  b
)  =  0 )
3534ex 434 . . . . . . . . . . . . 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  oF  -  a ) `
 b )  =  0 ) )
363, 35jcad 530 . . . . . . . . . . . 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  oF  -  a ) `
 b )  =  0 ) ) )
37 plysubcl 21649 . . . . . . . . . . . . . 14  |-  ( ( p  e.  (Poly `  CC )  /\  a  e.  (Poly `  CC )
)  ->  ( p  oF  -  a
)  e.  (Poly `  CC ) )
384, 10, 37syl2anc 656 . . . . . . . . . . . . 13  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  oF  -  a )  e.  (Poly `  CC )
)
39 plyf 21625 . . . . . . . . . . . . 13  |-  ( ( p  oF  -  a )  e.  (Poly `  CC )  ->  (
p  oF  -  a ) : CC --> CC )
40 ffn 5556 . . . . . . . . . . . . 13  |-  ( ( p  oF  -  a ) : CC --> CC  ->  ( p  oF  -  a )  Fn  CC )
41 fniniseg 5821 . . . . . . . . . . . . 13  |-  ( ( p  oF  -  a )  Fn  CC  ->  ( b  e.  ( `' ( p  oF  -  a )
" { 0 } )  <->  ( b  e.  CC  /\  ( ( p  oF  -  a ) `  b
)  =  0 ) ) )
4238, 39, 40, 414syl 21 . . . . . . . . . . . 12  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( b  e.  ( `' ( p  oF  -  a )
" { 0 } )  <->  ( b  e.  CC  /\  ( ( p  oF  -  a ) `  b
)  =  0 ) ) )
4336, 42sylibrd 234 . . . . . . . . . . 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  oF  -  a ) " { 0 } ) ) )
4443ssrdv 3359 . . . . . . . . . 10  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  D  C_  ( `' ( p  oF  -  a ) " {
0 } ) )
45 ssfi 7529 . . . . . . . . . . 11  |-  ( ( ( `' ( p  oF  -  a
) " { 0 } )  e.  Fin  /\  D  C_  ( `' ( p  oF  -  a ) " { 0 } ) )  ->  D  e.  Fin )
4645expcom 435 . . . . . . . . . 10  |-  ( D 
C_  ( `' ( p  oF  -  a ) " {
0 } )  -> 
( ( `' ( p  oF  -  a ) " {
0 } )  e. 
Fin  ->  D  e.  Fin ) )
4744, 46syl 16 . . . . . . . . 9  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( ( `' ( p  oF  -  a ) " {
0 } )  e. 
Fin  ->  D  e.  Fin ) )
481, 47mtod 177 . . . . . . . 8  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  -.  ( `' ( p  oF  -  a
) " { 0 } )  e.  Fin )
49 df-ne 2606 . . . . . . . . . . . 12  |-  ( ( p  oF  -  a )  =/=  0p 
<->  -.  ( p  oF  -  a )  =  0p )
5049biimpri 206 . . . . . . . . . . 11  |-  ( -.  ( p  oF  -  a )  =  0p  ->  (
p  oF  -  a )  =/=  0p )
51 eqid 2441 . . . . . . . . . . . 12  |-  ( `' ( p  oF  -  a ) " { 0 } )  =  ( `' ( p  oF  -  a ) " {
0 } )
5251fta1 21733 . . . . . . . . . . 11  |-  ( ( ( p  oF  -  a )  e.  (Poly `  CC )  /\  ( p  oF  -  a )  =/=  0p )  -> 
( ( `' ( p  oF  -  a ) " {
0 } )  e. 
Fin  /\  ( # `  ( `' ( p  oF  -  a )
" { 0 } ) )  <_  (deg `  ( p  oF  -  a ) ) ) )
5338, 50, 52syl2an 474 . . . . . . . . . 10  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  -.  ( p  oF  -  a
)  =  0p )  ->  ( ( `' ( p  oF  -  a )
" { 0 } )  e.  Fin  /\  ( # `  ( `' ( p  oF  -  a ) " { 0 } ) )  <_  (deg `  (
p  oF  -  a ) ) ) )
5453simpld 456 . . . . . . . . 9  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  -.  ( p  oF  -  a
)  =  0p )  ->  ( `' ( p  oF  -  a ) " { 0 } )  e.  Fin )
5554ex 434 . . . . . . . 8  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( -.  ( p  oF  -  a
)  =  0p  ->  ( `' ( p  oF  -  a ) " {
0 } )  e. 
Fin ) )
5648, 55mt3d 125 . . . . . . 7  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  oF  -  a )  =  0p )
57 df-0p 21107 . . . . . . 7  |-  0p  =  ( CC  X.  { 0 } )
5856, 57syl6eq 2489 . . . . . 6  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  oF  -  a )  =  ( CC  X.  {
0 } ) )
5916a1i 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 )
60 ofsubeq0 10315 . . . . . . 7  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  oF  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
6159, 6, 12, 60syl3anc 1213 . . . . . 6  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( ( p  oF  -  a )  =  ( CC  X.  { 0 } )  <-> 
p  =  a ) )
6258, 61mpbid 210 . . . . 5  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  p  =  a )
6362ex 434 . . . 4  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  ( ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) )  ->  p  =  a ) )
6463alrimivv 1691 . . 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 ) )
65 eleq1 2501 . . . . 5  |-  ( p  =  a  ->  (
p  e.  (Poly `  CC )  <->  a  e.  (Poly `  CC ) ) )
66 reseq1 5100 . . . . . 6  |-  ( p  =  a  ->  (
p  |`  D )  =  ( a  |`  D ) )
6766eqeq1d 2449 . . . . 5  |-  ( p  =  a  ->  (
( p  |`  D )  =  F  <->  ( a  |`  D )  =  F ) )
6865, 67anbi12d 705 . . . 4  |-  ( p  =  a  ->  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  <->  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )
6968mo4 2320 . . 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 )
)
7064, 69sylibr 212 . 2  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  E* p ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F ) )
71 plyssc 21627 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
7271sseli 3349 . . . 4  |-  ( p  e.  (Poly `  S
)  ->  p  e.  (Poly `  CC ) )
7372anim1i 565 . . 3  |-  ( ( p  e.  (Poly `  S )  /\  (
p  |`  D )  =  F )  ->  (
p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F ) )
7473moimi 2323 . 2  |-  ( E* p ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  ->  E* p ( p  e.  (Poly `  S )  /\  ( p  |`  D )  =  F ) )
7570, 74syl 16 1  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  E* p ( p  e.  (Poly `  S
)  /\  ( p  |`  D )  =  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1362    = wceq 1364    e. wcel 1761   E*wmo 2258    =/= wne 2604   _Vcvv 2970    C_ wss 3325   {csn 3874   class class class wbr 4289    X. cxp 4834   `'ccnv 4835    |` cres 4838   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   Fincfn 7306   CCcc 9276   0cc0 9278    <_ cle 9415    - cmin 9591   #chash 12099   0pc0p 21106  Polycply 21611  degcdgr 21614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-idp 21616  df-coe 21617  df-dgr 21618  df-quot 21716
This theorem is referenced by: (None)
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