Step |
Hyp |
Ref |
Expression |
1 |
|
coe1sub.y |
|- Y = ( Poly1 ` R ) |
2 |
|
coe1sub.b |
|- B = ( Base ` Y ) |
3 |
|
coe1sub.p |
|- .- = ( -g ` Y ) |
4 |
|
coe1sub.q |
|- N = ( -g ` R ) |
5 |
|
simpl1 |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> R e. Ring ) |
6 |
1
|
ply1ring |
|- ( R e. Ring -> Y e. Ring ) |
7 |
|
ringgrp |
|- ( Y e. Ring -> Y e. Grp ) |
8 |
6 7
|
syl |
|- ( R e. Ring -> Y e. Grp ) |
9 |
2 3
|
grpsubcl |
|- ( ( Y e. Grp /\ F e. B /\ G e. B ) -> ( F .- G ) e. B ) |
10 |
8 9
|
syl3an1 |
|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( F .- G ) e. B ) |
11 |
10
|
adantr |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( F .- G ) e. B ) |
12 |
|
simpl3 |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> G e. B ) |
13 |
|
simpr |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> X e. NN0 ) |
14 |
|
eqid |
|- ( +g ` Y ) = ( +g ` Y ) |
15 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
16 |
1 2 14 15
|
coe1addfv |
|- ( ( ( R e. Ring /\ ( F .- G ) e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) ` X ) = ( ( ( coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( ( coe1 ` G ) ` X ) ) ) |
17 |
5 11 12 13 16
|
syl31anc |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) ` X ) = ( ( ( coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( ( coe1 ` G ) ` X ) ) ) |
18 |
8
|
3ad2ant1 |
|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> Y e. Grp ) |
19 |
18
|
adantr |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> Y e. Grp ) |
20 |
|
simpl2 |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> F e. B ) |
21 |
2 14 3
|
grpnpcan |
|- ( ( Y e. Grp /\ F e. B /\ G e. B ) -> ( ( F .- G ) ( +g ` Y ) G ) = F ) |
22 |
19 20 12 21
|
syl3anc |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( F .- G ) ( +g ` Y ) G ) = F ) |
23 |
22
|
fveq2d |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) = ( coe1 ` F ) ) |
24 |
23
|
fveq1d |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` ( ( F .- G ) ( +g ` Y ) G ) ) ` X ) = ( ( coe1 ` F ) ` X ) ) |
25 |
17 24
|
eqtr3d |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( ( coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( ( coe1 ` G ) ` X ) ) = ( ( coe1 ` F ) ` X ) ) |
26 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
27 |
26
|
3ad2ant1 |
|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> R e. Grp ) |
28 |
27
|
adantr |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> R e. Grp ) |
29 |
|
eqid |
|- ( coe1 ` F ) = ( coe1 ` F ) |
30 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
31 |
29 2 1 30
|
coe1f |
|- ( F e. B -> ( coe1 ` F ) : NN0 --> ( Base ` R ) ) |
32 |
31
|
3ad2ant2 |
|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( coe1 ` F ) : NN0 --> ( Base ` R ) ) |
33 |
32
|
ffvelrnda |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` F ) ` X ) e. ( Base ` R ) ) |
34 |
|
eqid |
|- ( coe1 ` G ) = ( coe1 ` G ) |
35 |
34 2 1 30
|
coe1f |
|- ( G e. B -> ( coe1 ` G ) : NN0 --> ( Base ` R ) ) |
36 |
35
|
3ad2ant3 |
|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( coe1 ` G ) : NN0 --> ( Base ` R ) ) |
37 |
36
|
ffvelrnda |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` G ) ` X ) e. ( Base ` R ) ) |
38 |
|
eqid |
|- ( coe1 ` ( F .- G ) ) = ( coe1 ` ( F .- G ) ) |
39 |
38 2 1 30
|
coe1f |
|- ( ( F .- G ) e. B -> ( coe1 ` ( F .- G ) ) : NN0 --> ( Base ` R ) ) |
40 |
10 39
|
syl |
|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( coe1 ` ( F .- G ) ) : NN0 --> ( Base ` R ) ) |
41 |
40
|
ffvelrnda |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` ( F .- G ) ) ` X ) e. ( Base ` R ) ) |
42 |
30 15 4
|
grpsubadd |
|- ( ( R e. Grp /\ ( ( ( coe1 ` F ) ` X ) e. ( Base ` R ) /\ ( ( coe1 ` G ) ` X ) e. ( Base ` R ) /\ ( ( coe1 ` ( F .- G ) ) ` X ) e. ( Base ` R ) ) ) -> ( ( ( ( coe1 ` F ) ` X ) N ( ( coe1 ` G ) ` X ) ) = ( ( coe1 ` ( F .- G ) ) ` X ) <-> ( ( ( coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( ( coe1 ` G ) ` X ) ) = ( ( coe1 ` F ) ` X ) ) ) |
43 |
28 33 37 41 42
|
syl13anc |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( ( ( coe1 ` F ) ` X ) N ( ( coe1 ` G ) ` X ) ) = ( ( coe1 ` ( F .- G ) ) ` X ) <-> ( ( ( coe1 ` ( F .- G ) ) ` X ) ( +g ` R ) ( ( coe1 ` G ) ` X ) ) = ( ( coe1 ` F ) ` X ) ) ) |
44 |
25 43
|
mpbird |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( ( coe1 ` F ) ` X ) N ( ( coe1 ` G ) ` X ) ) = ( ( coe1 ` ( F .- G ) ) ` X ) ) |
45 |
44
|
eqcomd |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ X e. NN0 ) -> ( ( coe1 ` ( F .- G ) ) ` X ) = ( ( ( coe1 ` F ) ` X ) N ( ( coe1 ` G ) ` X ) ) ) |