Step |
Hyp |
Ref |
Expression |
1 |
|
cntzsdrg.b |
|- B = ( Base ` R ) |
2 |
|
cntzsdrg.m |
|- M = ( mulGrp ` R ) |
3 |
|
cntzsdrg.z |
|- Z = ( Cntz ` M ) |
4 |
|
simpl |
|- ( ( R e. DivRing /\ S C_ B ) -> R e. DivRing ) |
5 |
|
drngring |
|- ( R e. DivRing -> R e. Ring ) |
6 |
1 2 3
|
cntzsubr |
|- ( ( R e. Ring /\ S C_ B ) -> ( Z ` S ) e. ( SubRing ` R ) ) |
7 |
5 6
|
sylan |
|- ( ( R e. DivRing /\ S C_ B ) -> ( Z ` S ) e. ( SubRing ` R ) ) |
8 |
|
oveq2 |
|- ( y = ( 0g ` R ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) ) |
9 |
|
oveq1 |
|- ( y = ( 0g ` R ) -> ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) = ( ( 0g ` R ) ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
10 |
8 9
|
eqeq12d |
|- ( y = ( 0g ` R ) -> ( ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) <-> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) = ( ( 0g ` R ) ( .r ` R ) ( ( invr ` R ) ` x ) ) ) ) |
11 |
|
eldifsn |
|- ( y e. ( S \ { ( 0g ` R ) } ) <-> ( y e. S /\ y =/= ( 0g ` R ) ) ) |
12 |
|
eqid |
|- ( Unit ` R ) = ( Unit ` R ) |
13 |
2
|
oveq1i |
|- ( M |`s ( Unit ` R ) ) = ( ( mulGrp ` R ) |`s ( Unit ` R ) ) |
14 |
|
eqid |
|- ( invr ` R ) = ( invr ` R ) |
15 |
12 13 14
|
invrfval |
|- ( invr ` R ) = ( invg ` ( M |`s ( Unit ` R ) ) ) |
16 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
17 |
1 12 16
|
isdrng |
|- ( R e. DivRing <-> ( R e. Ring /\ ( Unit ` R ) = ( B \ { ( 0g ` R ) } ) ) ) |
18 |
17
|
simprbi |
|- ( R e. DivRing -> ( Unit ` R ) = ( B \ { ( 0g ` R ) } ) ) |
19 |
18
|
oveq2d |
|- ( R e. DivRing -> ( M |`s ( Unit ` R ) ) = ( M |`s ( B \ { ( 0g ` R ) } ) ) ) |
20 |
19
|
fveq2d |
|- ( R e. DivRing -> ( invg ` ( M |`s ( Unit ` R ) ) ) = ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ) |
21 |
15 20
|
syl5eq |
|- ( R e. DivRing -> ( invr ` R ) = ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ) |
22 |
21
|
ad2antrr |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( invr ` R ) = ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ) |
23 |
22
|
fveq1d |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( invr ` R ) ` x ) = ( ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` x ) ) |
24 |
2
|
oveq1i |
|- ( M |`s ( B \ { ( 0g ` R ) } ) ) = ( ( mulGrp ` R ) |`s ( B \ { ( 0g ` R ) } ) ) |
25 |
1 16 24
|
drngmgp |
|- ( R e. DivRing -> ( M |`s ( B \ { ( 0g ` R ) } ) ) e. Grp ) |
26 |
25
|
ad2antrr |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( M |`s ( B \ { ( 0g ` R ) } ) ) e. Grp ) |
27 |
|
ssdif |
|- ( S C_ B -> ( S \ { ( 0g ` R ) } ) C_ ( B \ { ( 0g ` R ) } ) ) |
28 |
27
|
ad2antlr |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( S \ { ( 0g ` R ) } ) C_ ( B \ { ( 0g ` R ) } ) ) |
29 |
|
difss |
|- ( B \ { ( 0g ` R ) } ) C_ B |
30 |
|
eqid |
|- ( M |`s ( B \ { ( 0g ` R ) } ) ) = ( M |`s ( B \ { ( 0g ` R ) } ) ) |
31 |
2 1
|
mgpbas |
|- B = ( Base ` M ) |
32 |
30 31
|
ressbas2 |
|- ( ( B \ { ( 0g ` R ) } ) C_ B -> ( B \ { ( 0g ` R ) } ) = ( Base ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ) |
33 |
29 32
|
ax-mp |
|- ( B \ { ( 0g ` R ) } ) = ( Base ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) |
34 |
|
eqid |
|- ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) = ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) |
35 |
33 34
|
cntzsubg |
|- ( ( ( M |`s ( B \ { ( 0g ` R ) } ) ) e. Grp /\ ( S \ { ( 0g ` R ) } ) C_ ( B \ { ( 0g ` R ) } ) ) -> ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) e. ( SubGrp ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ) |
36 |
26 28 35
|
syl2anc |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) e. ( SubGrp ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ) |
37 |
|
simpr |
|- ( ( R e. DivRing /\ S C_ B ) -> S C_ B ) |
38 |
|
difss |
|- ( S \ { ( 0g ` R ) } ) C_ S |
39 |
31 3
|
cntz2ss |
|- ( ( S C_ B /\ ( S \ { ( 0g ` R ) } ) C_ S ) -> ( Z ` S ) C_ ( Z ` ( S \ { ( 0g ` R ) } ) ) ) |
40 |
37 38 39
|
sylancl |
|- ( ( R e. DivRing /\ S C_ B ) -> ( Z ` S ) C_ ( Z ` ( S \ { ( 0g ` R ) } ) ) ) |
41 |
40
|
ssdifssd |
|- ( ( R e. DivRing /\ S C_ B ) -> ( ( Z ` S ) \ { ( 0g ` R ) } ) C_ ( Z ` ( S \ { ( 0g ` R ) } ) ) ) |
42 |
41
|
sselda |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> x e. ( Z ` ( S \ { ( 0g ` R ) } ) ) ) |
43 |
31 3
|
cntzssv |
|- ( Z ` S ) C_ B |
44 |
|
ssdif |
|- ( ( Z ` S ) C_ B -> ( ( Z ` S ) \ { ( 0g ` R ) } ) C_ ( B \ { ( 0g ` R ) } ) ) |
45 |
43 44
|
mp1i |
|- ( ( R e. DivRing /\ S C_ B ) -> ( ( Z ` S ) \ { ( 0g ` R ) } ) C_ ( B \ { ( 0g ` R ) } ) ) |
46 |
45
|
sselda |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> x e. ( B \ { ( 0g ` R ) } ) ) |
47 |
42 46
|
elind |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> x e. ( ( Z ` ( S \ { ( 0g ` R ) } ) ) i^i ( B \ { ( 0g ` R ) } ) ) ) |
48 |
1
|
fvexi |
|- B e. _V |
49 |
48
|
difexi |
|- ( B \ { ( 0g ` R ) } ) e. _V |
50 |
30 3 34
|
resscntz |
|- ( ( ( B \ { ( 0g ` R ) } ) e. _V /\ ( S \ { ( 0g ` R ) } ) C_ ( B \ { ( 0g ` R ) } ) ) -> ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) = ( ( Z ` ( S \ { ( 0g ` R ) } ) ) i^i ( B \ { ( 0g ` R ) } ) ) ) |
51 |
49 28 50
|
sylancr |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) = ( ( Z ` ( S \ { ( 0g ` R ) } ) ) i^i ( B \ { ( 0g ` R ) } ) ) ) |
52 |
47 51
|
eleqtrrd |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> x e. ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) ) |
53 |
|
eqid |
|- ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) = ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) |
54 |
53
|
subginvcl |
|- ( ( ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) e. ( SubGrp ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) /\ x e. ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) ) -> ( ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` x ) e. ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) ) |
55 |
36 52 54
|
syl2anc |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( invg ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` x ) e. ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) ) |
56 |
23 55
|
eqeltrd |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( invr ` R ) ` x ) e. ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) ) |
57 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
58 |
2 57
|
mgpplusg |
|- ( .r ` R ) = ( +g ` M ) |
59 |
30 58
|
ressplusg |
|- ( ( B \ { ( 0g ` R ) } ) e. _V -> ( .r ` R ) = ( +g ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ) |
60 |
49 59
|
ax-mp |
|- ( .r ` R ) = ( +g ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) |
61 |
60 34
|
cntzi |
|- ( ( ( ( invr ` R ) ` x ) e. ( ( Cntz ` ( M |`s ( B \ { ( 0g ` R ) } ) ) ) ` ( S \ { ( 0g ` R ) } ) ) /\ y e. ( S \ { ( 0g ` R ) } ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
62 |
56 61
|
sylan |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. ( S \ { ( 0g ` R ) } ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
63 |
11 62
|
sylan2br |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ ( y e. S /\ y =/= ( 0g ` R ) ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
64 |
63
|
anassrs |
|- ( ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. S ) /\ y =/= ( 0g ` R ) ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
65 |
5
|
ad3antrrr |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. S ) -> R e. Ring ) |
66 |
4
|
adantr |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> R e. DivRing ) |
67 |
|
eldifi |
|- ( x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) -> x e. ( Z ` S ) ) |
68 |
67
|
adantl |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> x e. ( Z ` S ) ) |
69 |
43 68
|
sseldi |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> x e. B ) |
70 |
|
eldifsni |
|- ( x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) -> x =/= ( 0g ` R ) ) |
71 |
70
|
adantl |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> x =/= ( 0g ` R ) ) |
72 |
1 16 14
|
drnginvrcl |
|- ( ( R e. DivRing /\ x e. B /\ x =/= ( 0g ` R ) ) -> ( ( invr ` R ) ` x ) e. B ) |
73 |
66 69 71 72
|
syl3anc |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( invr ` R ) ` x ) e. B ) |
74 |
73
|
adantr |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. S ) -> ( ( invr ` R ) ` x ) e. B ) |
75 |
1 57 16
|
ringrz |
|- ( ( R e. Ring /\ ( ( invr ` R ) ` x ) e. B ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) |
76 |
65 74 75
|
syl2anc |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. S ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) |
77 |
1 57 16
|
ringlz |
|- ( ( R e. Ring /\ ( ( invr ` R ) ` x ) e. B ) -> ( ( 0g ` R ) ( .r ` R ) ( ( invr ` R ) ` x ) ) = ( 0g ` R ) ) |
78 |
65 74 77
|
syl2anc |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. S ) -> ( ( 0g ` R ) ( .r ` R ) ( ( invr ` R ) ` x ) ) = ( 0g ` R ) ) |
79 |
76 78
|
eqtr4d |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. S ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) ( 0g ` R ) ) = ( ( 0g ` R ) ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
80 |
10 64 79
|
pm2.61ne |
|- ( ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) /\ y e. S ) -> ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
81 |
80
|
ralrimiva |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> A. y e. S ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) |
82 |
|
simplr |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> S C_ B ) |
83 |
31 58 3
|
cntzel |
|- ( ( S C_ B /\ ( ( invr ` R ) ` x ) e. B ) -> ( ( ( invr ` R ) ` x ) e. ( Z ` S ) <-> A. y e. S ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) ) |
84 |
82 73 83
|
syl2anc |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( ( invr ` R ) ` x ) e. ( Z ` S ) <-> A. y e. S ( ( ( invr ` R ) ` x ) ( .r ` R ) y ) = ( y ( .r ` R ) ( ( invr ` R ) ` x ) ) ) ) |
85 |
81 84
|
mpbird |
|- ( ( ( R e. DivRing /\ S C_ B ) /\ x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ) -> ( ( invr ` R ) ` x ) e. ( Z ` S ) ) |
86 |
85
|
ralrimiva |
|- ( ( R e. DivRing /\ S C_ B ) -> A. x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. ( Z ` S ) ) |
87 |
14 16
|
issdrg2 |
|- ( ( Z ` S ) e. ( SubDRing ` R ) <-> ( R e. DivRing /\ ( Z ` S ) e. ( SubRing ` R ) /\ A. x e. ( ( Z ` S ) \ { ( 0g ` R ) } ) ( ( invr ` R ) ` x ) e. ( Z ` S ) ) ) |
88 |
4 7 86 87
|
syl3anbrc |
|- ( ( R e. DivRing /\ S C_ B ) -> ( Z ` S ) e. ( SubDRing ` R ) ) |