Step |
Hyp |
Ref |
Expression |
1 |
|
ply1scl.p |
|- P = ( Poly1 ` R ) |
2 |
|
ply1scl.a |
|- A = ( algSc ` P ) |
3 |
|
ply1scl0.z |
|- .0. = ( 0g ` R ) |
4 |
|
ply1scl0.y |
|- Y = ( 0g ` P ) |
5 |
|
ply1scln0.k |
|- K = ( Base ` R ) |
6 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
7 |
1 2 5 6
|
ply1sclf1 |
|- ( R e. Ring -> A : K -1-1-> ( Base ` P ) ) |
8 |
7
|
adantr |
|- ( ( R e. Ring /\ X e. K ) -> A : K -1-1-> ( Base ` P ) ) |
9 |
|
simpr |
|- ( ( R e. Ring /\ X e. K ) -> X e. K ) |
10 |
5 3
|
ring0cl |
|- ( R e. Ring -> .0. e. K ) |
11 |
10
|
adantr |
|- ( ( R e. Ring /\ X e. K ) -> .0. e. K ) |
12 |
|
f1fveq |
|- ( ( A : K -1-1-> ( Base ` P ) /\ ( X e. K /\ .0. e. K ) ) -> ( ( A ` X ) = ( A ` .0. ) <-> X = .0. ) ) |
13 |
8 9 11 12
|
syl12anc |
|- ( ( R e. Ring /\ X e. K ) -> ( ( A ` X ) = ( A ` .0. ) <-> X = .0. ) ) |
14 |
13
|
biimpd |
|- ( ( R e. Ring /\ X e. K ) -> ( ( A ` X ) = ( A ` .0. ) -> X = .0. ) ) |
15 |
14
|
necon3d |
|- ( ( R e. Ring /\ X e. K ) -> ( X =/= .0. -> ( A ` X ) =/= ( A ` .0. ) ) ) |
16 |
15
|
3impia |
|- ( ( R e. Ring /\ X e. K /\ X =/= .0. ) -> ( A ` X ) =/= ( A ` .0. ) ) |
17 |
1 2 3 4
|
ply1scl0 |
|- ( R e. Ring -> ( A ` .0. ) = Y ) |
18 |
17
|
3ad2ant1 |
|- ( ( R e. Ring /\ X e. K /\ X =/= .0. ) -> ( A ` .0. ) = Y ) |
19 |
16 18
|
neeqtrd |
|- ( ( R e. Ring /\ X e. K /\ X =/= .0. ) -> ( A ` X ) =/= Y ) |