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Theorem plyexmo 19693
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 731 . . . . . . . . 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 730 . . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . . . . . 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 19580 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e.  (Poly `  CC )  ->  p : CC --> CC )
64, 5syl 15 . . . . . . . . . . . . . . . . . 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 5389 . . . . . . . . . . . . . . . . . 18  |-  ( p : CC --> CC  ->  p  Fn  CC )
86, 7syl 15 . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . . . . 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 19580 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  (Poly `  CC )  ->  a : CC --> CC )
1210, 11syl 15 . . . . . . . . . . . . . . . . . 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 5389 . . . . . . . . . . . . . . . . . 18  |-  ( a : CC --> CC  ->  a  Fn  CC )
1412, 13syl 15 . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . 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 8818 . . . . . . . . . . . . . . . . 17  |-  CC  e.  _V
1716a1i 10 . . . . . . . . . . . . . . . 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 3180 . . . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . 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 ) ) )
21 simprlr 739 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( p  |`  D )  =  F )
22 simprrr 741 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  -> 
( a  |`  D )  =  F )
2321, 22eqtr4d 2318 . . . . . . . . . . . . . . . . . . 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 ) )
2423adantr 451 . . . . . . . . . . . . . . . . . 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 )
)
2524fveq1d 5527 . . . . . . . . . . . . . . . . 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 ) )
26 fvres 5542 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  D  ->  (
( p  |`  D ) `
 b )  =  ( p `  b
) )
2726adantl 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  |`  D ) `  b )  =  ( p `  b ) )
28 fvres 5542 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  D  ->  (
( a  |`  D ) `
 b )  =  ( a `  b
) )
2928adantl 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
)  ->  ( (
a  |`  D ) `  b )  =  ( a `  b ) )
3025, 27, 293eqtr3d 2323 . . . . . . . . . . . . . . . 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 ) )
316adantr 451 . . . . . . . . . . . . . . . . . 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 : CC
--> CC )
3231, 18ffvelrnd 5666 . . . . . . . . . . . . . . . . 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 `  b )  e.  CC )
3312adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  C_  CC  /\  -.  D  e. 
Fin )  /\  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  /\  (
a  e.  (Poly `  CC )  /\  (
a  |`  D )  =  F ) ) )  /\  b  e.  D
)  ->  a : CC
--> CC )
3433, 18ffvelrnd 5666 . . . . . . . . . . . . . . . . 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 `  b )  e.  CC )
35 subeq0 9073 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p `  b
)  e.  CC  /\  ( a `  b
)  e.  CC )  ->  ( ( ( p `  b )  -  ( a `  b ) )  =  0  <->  ( p `  b )  =  ( a `  b ) ) )
3632, 34, 35syl2anc 642 . . . . . . . . . . . . . . . 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 ) )  =  0  <->  ( p `  b )  =  ( a `  b ) ) )
3730, 36mpbird 223 . . . . . . . . . . . . . . 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 )
3820, 37eqtrd 2315 . . . . . . . . . . . . . 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 )
3938ex 423 . . . . . . . . . . . . 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 ) )
403, 39jcad 519 . . . . . . . . . . . 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 ) ) )
41 plysubcl 19604 . . . . . . . . . . . . . . . 16  |-  ( ( p  e.  (Poly `  CC )  /\  a  e.  (Poly `  CC )
)  ->  ( p  o F  -  a
)  e.  (Poly `  CC ) )
424, 10, 41syl2anc 642 . . . . . . . . . . . . . . 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 )
)
43 plyf 19580 . . . . . . . . . . . . . . 15  |-  ( ( p  o F  -  a )  e.  (Poly `  CC )  ->  (
p  o F  -  a ) : CC --> CC )
4442, 43syl 15 . . . . . . . . . . . . . 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 )
45 ffn 5389 . . . . . . . . . . . . . 14  |-  ( ( p  o F  -  a ) : CC --> CC  ->  ( p  o F  -  a )  Fn  CC )
4644, 45syl 15 . . . . . . . . . . . . 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 )
47 fniniseg 5646 . . . . . . . . . . . . 13  |-  ( ( p  o F  -  a )  Fn  CC  ->  ( b  e.  ( `' ( p  o F  -  a )
" { 0 } )  <->  ( b  e.  CC  /\  ( ( p  o F  -  a ) `  b
)  =  0 ) ) )
4846, 47syl 15 . . . . . . . . . . . 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 ) ) )
4940, 48sylibrd 225 . . . . . . . . . . 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 } ) ) )
5049ssrdv 3185 . . . . . . . . . 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 } ) )
51 ssfi 7083 . . . . . . . . . . 11  |-  ( ( ( `' ( p  o F  -  a
) " { 0 } )  e.  Fin  /\  D  C_  ( `' ( p  o F  -  a ) " { 0 } ) )  ->  D  e.  Fin )
5251expcom 424 . . . . . . . . . 10  |-  ( D 
C_  ( `' ( p  o F  -  a ) " {
0 } )  -> 
( ( `' ( p  o F  -  a ) " {
0 } )  e. 
Fin  ->  D  e.  Fin ) )
5350, 52syl 15 . . . . . . . . 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 ) )
541, 53mtod 168 . . . . . . . 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 )
55 df-ne 2448 . . . . . . . . . . . 12  |-  ( ( p  o F  -  a )  =/=  0 p 
<->  -.  ( p  o F  -  a )  =  0 p )
5655biimpri 197 . . . . . . . . . . 11  |-  ( -.  ( p  o F  -  a )  =  0 p  ->  (
p  o F  -  a )  =/=  0 p )
57 eqid 2283 . . . . . . . . . . . 12  |-  ( `' ( p  o F  -  a ) " { 0 } )  =  ( `' ( p  o F  -  a ) " {
0 } )
5857fta1 19688 . . . . . . . . . . 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 ) ) ) )
5942, 56, 58syl2an 463 . . . . . . . . . 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 ) ) ) )
6059simpld 445 . . . . . . . . 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 )
6160ex 423 . . . . . . . 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 ) )
6254, 61mt3d 117 . . . . . . 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 )
63 df-0p 19025 . . . . . . 7  |-  0 p  =  ( CC  X.  { 0 } )
6462, 63syl6eq 2331 . . . . . 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 } ) )
6516a1i 10 . . . . . . 7  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  CC  e.  _V )
66 ofsubeq0 9743 . . . . . . 7  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  o F  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
6765, 6, 12, 66syl3anc 1182 . . . . . 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 ) )
6864, 67mpbid 201 . . . . 5  |-  ( ( ( D  C_  CC  /\ 
-.  D  e.  Fin )  /\  ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )  ->  p  =  a )
6968ex 423 . . . 4  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  ( ( ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  /\  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) )  ->  p  =  a ) )
7069alrimivv 1618 . . 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 ) )
71 eleq1 2343 . . . . 5  |-  ( p  =  a  ->  (
p  e.  (Poly `  CC )  <->  a  e.  (Poly `  CC ) ) )
72 reseq1 4949 . . . . . 6  |-  ( p  =  a  ->  (
p  |`  D )  =  ( a  |`  D ) )
7372eqeq1d 2291 . . . . 5  |-  ( p  =  a  ->  (
( p  |`  D )  =  F  <->  ( a  |`  D )  =  F ) )
7471, 73anbi12d 691 . . . 4  |-  ( p  =  a  ->  (
( p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F )  <->  ( a  e.  (Poly `  CC )  /\  ( a  |`  D )  =  F ) ) )
7574mo4 2176 . . 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 )
)
7670, 75sylibr 203 . 2  |-  ( ( D  C_  CC  /\  -.  D  e.  Fin )  ->  E* p ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F ) )
77 plyssc 19582 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
7877sseli 3176 . . . 4  |-  ( p  e.  (Poly `  S
)  ->  p  e.  (Poly `  CC ) )
7978anim1i 551 . . 3  |-  ( ( p  e.  (Poly `  S )  /\  (
p  |`  D )  =  F )  ->  (
p  e.  (Poly `  CC )  /\  (
p  |`  D )  =  F ) )
8079moimi 2190 . 2  |-  ( E* p ( p  e.  (Poly `  CC )  /\  ( p  |`  D )  =  F )  ->  E* p ( p  e.  (Poly `  S )  /\  ( p  |`  D )  =  F ) )
8176, 80syl 15 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 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144    =/= wne 2446   _Vcvv 2788    C_ wss 3152   {csn 3640   class class class wbr 4023    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Fincfn 6863   CCcc 8735   0cc0 8737    <_ cle 8868    - cmin 9037   #chash 11337   0 pc0p 19024  Polycply 19566  degcdgr 19569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-quot 19671
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