Step |
Hyp |
Ref |
Expression |
1 |
|
subdrgint.1 |
|- L = ( R |`s |^| S ) |
2 |
|
subdrgint.2 |
|- ( ph -> R e. DivRing ) |
3 |
|
subdrgint.3 |
|- ( ph -> S C_ ( SubRing ` R ) ) |
4 |
|
subdrgint.4 |
|- ( ph -> S =/= (/) ) |
5 |
|
subdrgint.5 |
|- ( ( ph /\ s e. S ) -> ( R |`s s ) e. DivRing ) |
6 |
|
subrgint |
|- ( ( S C_ ( SubRing ` R ) /\ S =/= (/) ) -> |^| S e. ( SubRing ` R ) ) |
7 |
3 4 6
|
syl2anc |
|- ( ph -> |^| S e. ( SubRing ` R ) ) |
8 |
1
|
subrgring |
|- ( |^| S e. ( SubRing ` R ) -> L e. Ring ) |
9 |
7 8
|
syl |
|- ( ph -> L e. Ring ) |
10 |
1
|
fveq2i |
|- ( mulGrp ` L ) = ( mulGrp ` ( R |`s |^| S ) ) |
11 |
10
|
oveq1i |
|- ( ( mulGrp ` L ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` ( R |`s |^| S ) ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) |
12 |
|
eqid |
|- ( R |`s |^| S ) = ( R |`s |^| S ) |
13 |
|
eqid |
|- ( mulGrp ` R ) = ( mulGrp ` R ) |
14 |
12 13
|
mgpress |
|- ( ( R e. DivRing /\ |^| S e. ( SubRing ` R ) ) -> ( ( mulGrp ` R ) |`s |^| S ) = ( mulGrp ` ( R |`s |^| S ) ) ) |
15 |
2 7 14
|
syl2anc |
|- ( ph -> ( ( mulGrp ` R ) |`s |^| S ) = ( mulGrp ` ( R |`s |^| S ) ) ) |
16 |
15
|
oveq1d |
|- ( ph -> ( ( ( mulGrp ` R ) |`s |^| S ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` ( R |`s |^| S ) ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) |
17 |
|
difssd |
|- ( ph -> ( ( Base ` L ) \ { ( 0g ` L ) } ) C_ ( Base ` L ) ) |
18 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
19 |
18
|
subrgss |
|- ( |^| S e. ( SubRing ` R ) -> |^| S C_ ( Base ` R ) ) |
20 |
1 18
|
ressbas2 |
|- ( |^| S C_ ( Base ` R ) -> |^| S = ( Base ` L ) ) |
21 |
7 19 20
|
3syl |
|- ( ph -> |^| S = ( Base ` L ) ) |
22 |
17 21
|
sseqtrrd |
|- ( ph -> ( ( Base ` L ) \ { ( 0g ` L ) } ) C_ |^| S ) |
23 |
|
ressabs |
|- ( ( |^| S e. ( SubRing ` R ) /\ ( ( Base ` L ) \ { ( 0g ` L ) } ) C_ |^| S ) -> ( ( ( mulGrp ` R ) |`s |^| S ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) |
24 |
7 22 23
|
syl2anc |
|- ( ph -> ( ( ( mulGrp ` R ) |`s |^| S ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) |
25 |
16 24
|
eqtr3d |
|- ( ph -> ( ( mulGrp ` ( R |`s |^| S ) ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) |
26 |
|
intiin |
|- |^| S = |^|_ s e. S s |
27 |
26 21
|
syl5reqr |
|- ( ph -> ( Base ` L ) = |^|_ s e. S s ) |
28 |
27
|
difeq1d |
|- ( ph -> ( ( Base ` L ) \ { ( 0g ` L ) } ) = ( |^|_ s e. S s \ { ( 0g ` L ) } ) ) |
29 |
28
|
oveq2d |
|- ( ph -> ( ( mulGrp ` R ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( |^|_ s e. S s \ { ( 0g ` L ) } ) ) ) |
30 |
|
vex |
|- s e. _V |
31 |
30
|
difexi |
|- ( s \ { ( 0g ` L ) } ) e. _V |
32 |
31
|
dfiin3 |
|- |^|_ s e. S ( s \ { ( 0g ` L ) } ) = |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) |
33 |
|
iindif1 |
|- ( S =/= (/) -> |^|_ s e. S ( s \ { ( 0g ` L ) } ) = ( |^|_ s e. S s \ { ( 0g ` L ) } ) ) |
34 |
4 33
|
syl |
|- ( ph -> |^|_ s e. S ( s \ { ( 0g ` L ) } ) = ( |^|_ s e. S s \ { ( 0g ` L ) } ) ) |
35 |
32 34
|
syl5eqr |
|- ( ph -> |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) = ( |^|_ s e. S s \ { ( 0g ` L ) } ) ) |
36 |
35
|
oveq2d |
|- ( ph -> ( ( mulGrp ` R ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) = ( ( mulGrp ` R ) |`s ( |^|_ s e. S s \ { ( 0g ` L ) } ) ) ) |
37 |
|
difss |
|- ( ( Base ` R ) \ { ( 0g ` R ) } ) C_ ( Base ` R ) |
38 |
|
eqid |
|- ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) = ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |
39 |
13 18
|
mgpbas |
|- ( Base ` R ) = ( Base ` ( mulGrp ` R ) ) |
40 |
38 39
|
ressbas2 |
|- ( ( ( Base ` R ) \ { ( 0g ` R ) } ) C_ ( Base ` R ) -> ( ( Base ` R ) \ { ( 0g ` R ) } ) = ( Base ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
41 |
37 40
|
ax-mp |
|- ( ( Base ` R ) \ { ( 0g ` R ) } ) = ( Base ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) |
42 |
41
|
fvexi |
|- ( ( Base ` R ) \ { ( 0g ` R ) } ) e. _V |
43 |
|
iinssiun |
|- ( S =/= (/) -> |^|_ s e. S ( s \ { ( 0g ` L ) } ) C_ U_ s e. S ( s \ { ( 0g ` L ) } ) ) |
44 |
4 43
|
syl |
|- ( ph -> |^|_ s e. S ( s \ { ( 0g ` L ) } ) C_ U_ s e. S ( s \ { ( 0g ` L ) } ) ) |
45 |
|
subrgsubg |
|- ( s e. ( SubRing ` R ) -> s e. ( SubGrp ` R ) ) |
46 |
45
|
ssriv |
|- ( SubRing ` R ) C_ ( SubGrp ` R ) |
47 |
3 46
|
sstrdi |
|- ( ph -> S C_ ( SubGrp ` R ) ) |
48 |
|
subgint |
|- ( ( S C_ ( SubGrp ` R ) /\ S =/= (/) ) -> |^| S e. ( SubGrp ` R ) ) |
49 |
47 4 48
|
syl2anc |
|- ( ph -> |^| S e. ( SubGrp ` R ) ) |
50 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
51 |
1 50
|
subg0 |
|- ( |^| S e. ( SubGrp ` R ) -> ( 0g ` R ) = ( 0g ` L ) ) |
52 |
49 51
|
syl |
|- ( ph -> ( 0g ` R ) = ( 0g ` L ) ) |
53 |
52
|
adantr |
|- ( ( ph /\ s e. S ) -> ( 0g ` R ) = ( 0g ` L ) ) |
54 |
53
|
sneqd |
|- ( ( ph /\ s e. S ) -> { ( 0g ` R ) } = { ( 0g ` L ) } ) |
55 |
54
|
difeq2d |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` R ) } ) = ( s \ { ( 0g ` L ) } ) ) |
56 |
3
|
sselda |
|- ( ( ph /\ s e. S ) -> s e. ( SubRing ` R ) ) |
57 |
18
|
subrgss |
|- ( s e. ( SubRing ` R ) -> s C_ ( Base ` R ) ) |
58 |
56 57
|
syl |
|- ( ( ph /\ s e. S ) -> s C_ ( Base ` R ) ) |
59 |
58
|
ssdifd |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` R ) } ) C_ ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |
60 |
55 59
|
eqsstrrd |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` L ) } ) C_ ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |
61 |
60
|
iunssd |
|- ( ph -> U_ s e. S ( s \ { ( 0g ` L ) } ) C_ ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |
62 |
44 61
|
sstrd |
|- ( ph -> |^|_ s e. S ( s \ { ( 0g ` L ) } ) C_ ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |
63 |
32 62
|
eqsstrrid |
|- ( ph -> |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) C_ ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |
64 |
|
ressabs |
|- ( ( ( ( Base ` R ) \ { ( 0g ` R ) } ) e. _V /\ |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) C_ ( ( Base ` R ) \ { ( 0g ` R ) } ) ) -> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) = ( ( mulGrp ` R ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) ) |
65 |
42 63 64
|
sylancr |
|- ( ph -> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) = ( ( mulGrp ` R ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) ) |
66 |
18 50 38
|
drngmgp |
|- ( R e. DivRing -> ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp ) |
67 |
2 66
|
syl |
|- ( ph -> ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp ) |
68 |
67
|
adantr |
|- ( ( ph /\ s e. S ) -> ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp ) |
69 |
60 41
|
sseqtrdi |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` L ) } ) C_ ( Base ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
70 |
|
ressabs |
|- ( ( ( ( Base ` R ) \ { ( 0g ` R ) } ) e. _V /\ ( s \ { ( 0g ` L ) } ) C_ ( ( Base ` R ) \ { ( 0g ` R ) } ) ) -> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s ( s \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( s \ { ( 0g ` L ) } ) ) ) |
71 |
42 60 70
|
sylancr |
|- ( ( ph /\ s e. S ) -> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s ( s \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( s \ { ( 0g ` L ) } ) ) ) |
72 |
|
eqid |
|- ( R |`s s ) = ( R |`s s ) |
73 |
72 13
|
mgpress |
|- ( ( R e. DivRing /\ s e. S ) -> ( ( mulGrp ` R ) |`s s ) = ( mulGrp ` ( R |`s s ) ) ) |
74 |
2 73
|
sylan |
|- ( ( ph /\ s e. S ) -> ( ( mulGrp ` R ) |`s s ) = ( mulGrp ` ( R |`s s ) ) ) |
75 |
55
|
eqcomd |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` L ) } ) = ( s \ { ( 0g ` R ) } ) ) |
76 |
74 75
|
oveq12d |
|- ( ( ph /\ s e. S ) -> ( ( ( mulGrp ` R ) |`s s ) |`s ( s \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` ( R |`s s ) ) |`s ( s \ { ( 0g ` R ) } ) ) ) |
77 |
|
simpr |
|- ( ( ph /\ s e. S ) -> s e. S ) |
78 |
|
difssd |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` L ) } ) C_ s ) |
79 |
|
ressabs |
|- ( ( s e. S /\ ( s \ { ( 0g ` L ) } ) C_ s ) -> ( ( ( mulGrp ` R ) |`s s ) |`s ( s \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( s \ { ( 0g ` L ) } ) ) ) |
80 |
77 78 79
|
syl2anc |
|- ( ( ph /\ s e. S ) -> ( ( ( mulGrp ` R ) |`s s ) |`s ( s \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` R ) |`s ( s \ { ( 0g ` L ) } ) ) ) |
81 |
76 80
|
eqtr3d |
|- ( ( ph /\ s e. S ) -> ( ( mulGrp ` ( R |`s s ) ) |`s ( s \ { ( 0g ` R ) } ) ) = ( ( mulGrp ` R ) |`s ( s \ { ( 0g ` L ) } ) ) ) |
82 |
72 18
|
ressbas2 |
|- ( s C_ ( Base ` R ) -> s = ( Base ` ( R |`s s ) ) ) |
83 |
56 57 82
|
3syl |
|- ( ( ph /\ s e. S ) -> s = ( Base ` ( R |`s s ) ) ) |
84 |
72 50
|
subrg0 |
|- ( s e. ( SubRing ` R ) -> ( 0g ` R ) = ( 0g ` ( R |`s s ) ) ) |
85 |
56 84
|
syl |
|- ( ( ph /\ s e. S ) -> ( 0g ` R ) = ( 0g ` ( R |`s s ) ) ) |
86 |
85
|
sneqd |
|- ( ( ph /\ s e. S ) -> { ( 0g ` R ) } = { ( 0g ` ( R |`s s ) ) } ) |
87 |
83 86
|
difeq12d |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` R ) } ) = ( ( Base ` ( R |`s s ) ) \ { ( 0g ` ( R |`s s ) ) } ) ) |
88 |
87
|
oveq2d |
|- ( ( ph /\ s e. S ) -> ( ( mulGrp ` ( R |`s s ) ) |`s ( s \ { ( 0g ` R ) } ) ) = ( ( mulGrp ` ( R |`s s ) ) |`s ( ( Base ` ( R |`s s ) ) \ { ( 0g ` ( R |`s s ) ) } ) ) ) |
89 |
|
eqid |
|- ( Base ` ( R |`s s ) ) = ( Base ` ( R |`s s ) ) |
90 |
|
eqid |
|- ( 0g ` ( R |`s s ) ) = ( 0g ` ( R |`s s ) ) |
91 |
|
eqid |
|- ( ( mulGrp ` ( R |`s s ) ) |`s ( ( Base ` ( R |`s s ) ) \ { ( 0g ` ( R |`s s ) ) } ) ) = ( ( mulGrp ` ( R |`s s ) ) |`s ( ( Base ` ( R |`s s ) ) \ { ( 0g ` ( R |`s s ) ) } ) ) |
92 |
89 90 91
|
drngmgp |
|- ( ( R |`s s ) e. DivRing -> ( ( mulGrp ` ( R |`s s ) ) |`s ( ( Base ` ( R |`s s ) ) \ { ( 0g ` ( R |`s s ) ) } ) ) e. Grp ) |
93 |
5 92
|
syl |
|- ( ( ph /\ s e. S ) -> ( ( mulGrp ` ( R |`s s ) ) |`s ( ( Base ` ( R |`s s ) ) \ { ( 0g ` ( R |`s s ) ) } ) ) e. Grp ) |
94 |
88 93
|
eqeltrd |
|- ( ( ph /\ s e. S ) -> ( ( mulGrp ` ( R |`s s ) ) |`s ( s \ { ( 0g ` R ) } ) ) e. Grp ) |
95 |
81 94
|
eqeltrrd |
|- ( ( ph /\ s e. S ) -> ( ( mulGrp ` R ) |`s ( s \ { ( 0g ` L ) } ) ) e. Grp ) |
96 |
71 95
|
eqeltrd |
|- ( ( ph /\ s e. S ) -> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s ( s \ { ( 0g ` L ) } ) ) e. Grp ) |
97 |
|
eqid |
|- ( Base ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) = ( Base ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) |
98 |
97
|
issubg |
|- ( ( s \ { ( 0g ` L ) } ) e. ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) <-> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) e. Grp /\ ( s \ { ( 0g ` L ) } ) C_ ( Base ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) /\ ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s ( s \ { ( 0g ` L ) } ) ) e. Grp ) ) |
99 |
68 69 96 98
|
syl3anbrc |
|- ( ( ph /\ s e. S ) -> ( s \ { ( 0g ` L ) } ) e. ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
100 |
99
|
ralrimiva |
|- ( ph -> A. s e. S ( s \ { ( 0g ` L ) } ) e. ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
101 |
|
eqid |
|- ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) = ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) |
102 |
101
|
rnmptss |
|- ( A. s e. S ( s \ { ( 0g ` L ) } ) e. ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) -> ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) C_ ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
103 |
100 102
|
syl |
|- ( ph -> ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) C_ ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
104 |
|
dmmptg |
|- ( A. s e. S ( s \ { ( 0g ` L ) } ) e. _V -> dom ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) = S ) |
105 |
|
difexg |
|- ( s e. S -> ( s \ { ( 0g ` L ) } ) e. _V ) |
106 |
104 105
|
mprg |
|- dom ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) = S |
107 |
106
|
a1i |
|- ( ph -> dom ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) = S ) |
108 |
107 4
|
eqnetrd |
|- ( ph -> dom ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) =/= (/) ) |
109 |
|
dm0rn0 |
|- ( dom ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) = (/) <-> ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) = (/) ) |
110 |
109
|
necon3bii |
|- ( dom ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) =/= (/) <-> ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) =/= (/) ) |
111 |
108 110
|
sylib |
|- ( ph -> ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) =/= (/) ) |
112 |
|
subgint |
|- ( ( ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) C_ ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) /\ ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) =/= (/) ) -> |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) e. ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
113 |
103 111 112
|
syl2anc |
|- ( ph -> |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) e. ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) ) |
114 |
|
eqid |
|- ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) = ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) |
115 |
114
|
subggrp |
|- ( |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) e. ( SubGrp ` ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) ) -> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) e. Grp ) |
116 |
113 115
|
syl |
|- ( ph -> ( ( ( mulGrp ` R ) |`s ( ( Base ` R ) \ { ( 0g ` R ) } ) ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) e. Grp ) |
117 |
65 116
|
eqeltrrd |
|- ( ph -> ( ( mulGrp ` R ) |`s |^| ran ( s e. S |-> ( s \ { ( 0g ` L ) } ) ) ) e. Grp ) |
118 |
36 117
|
eqeltrrd |
|- ( ph -> ( ( mulGrp ` R ) |`s ( |^|_ s e. S s \ { ( 0g ` L ) } ) ) e. Grp ) |
119 |
29 118
|
eqeltrd |
|- ( ph -> ( ( mulGrp ` R ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) e. Grp ) |
120 |
25 119
|
eqeltrd |
|- ( ph -> ( ( mulGrp ` ( R |`s |^| S ) ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) e. Grp ) |
121 |
11 120
|
eqeltrid |
|- ( ph -> ( ( mulGrp ` L ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) e. Grp ) |
122 |
|
eqid |
|- ( Base ` L ) = ( Base ` L ) |
123 |
|
eqid |
|- ( 0g ` L ) = ( 0g ` L ) |
124 |
|
eqid |
|- ( ( mulGrp ` L ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) = ( ( mulGrp ` L ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) |
125 |
122 123 124
|
isdrng2 |
|- ( L e. DivRing <-> ( L e. Ring /\ ( ( mulGrp ` L ) |`s ( ( Base ` L ) \ { ( 0g ` L ) } ) ) e. Grp ) ) |
126 |
9 121 125
|
sylanbrc |
|- ( ph -> L e. DivRing ) |