Step |
Hyp |
Ref |
Expression |
1 |
|
sraassab.a |
|- A = ( ( subringAlg ` W ) ` S ) |
2 |
|
sraassab.z |
|- Z = ( Cntr ` ( mulGrp ` W ) ) |
3 |
|
sraassab.w |
|- ( ph -> W e. Ring ) |
4 |
|
sraassab.s |
|- ( ph -> S e. ( SubRing ` W ) ) |
5 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
6 |
5
|
subrgss |
|- ( S e. ( SubRing ` W ) -> S C_ ( Base ` W ) ) |
7 |
4 6
|
syl |
|- ( ph -> S C_ ( Base ` W ) ) |
8 |
7
|
adantr |
|- ( ( ph /\ A e. AssAlg ) -> S C_ ( Base ` W ) ) |
9 |
8
|
sselda |
|- ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) -> y e. ( Base ` W ) ) |
10 |
|
simpllr |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> A e. AssAlg ) |
11 |
|
eqid |
|- ( W |`s S ) = ( W |`s S ) |
12 |
11
|
subrgbas |
|- ( S e. ( SubRing ` W ) -> S = ( Base ` ( W |`s S ) ) ) |
13 |
4 12
|
syl |
|- ( ph -> S = ( Base ` ( W |`s S ) ) ) |
14 |
1
|
a1i |
|- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) |
15 |
14 7
|
srasca |
|- ( ph -> ( W |`s S ) = ( Scalar ` A ) ) |
16 |
15
|
fveq2d |
|- ( ph -> ( Base ` ( W |`s S ) ) = ( Base ` ( Scalar ` A ) ) ) |
17 |
13 16
|
eqtrd |
|- ( ph -> S = ( Base ` ( Scalar ` A ) ) ) |
18 |
17
|
eqimssd |
|- ( ph -> S C_ ( Base ` ( Scalar ` A ) ) ) |
19 |
18
|
sselda |
|- ( ( ph /\ y e. S ) -> y e. ( Base ` ( Scalar ` A ) ) ) |
20 |
19
|
ad4ant13 |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> y e. ( Base ` ( Scalar ` A ) ) ) |
21 |
14 7
|
srabase |
|- ( ph -> ( Base ` W ) = ( Base ` A ) ) |
22 |
21
|
eqimssd |
|- ( ph -> ( Base ` W ) C_ ( Base ` A ) ) |
23 |
22
|
ad2antrr |
|- ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) -> ( Base ` W ) C_ ( Base ` A ) ) |
24 |
23
|
sselda |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> x e. ( Base ` A ) ) |
25 |
|
eqid |
|- ( 1r ` W ) = ( 1r ` W ) |
26 |
5 25
|
ringidcl |
|- ( W e. Ring -> ( 1r ` W ) e. ( Base ` W ) ) |
27 |
3 26
|
syl |
|- ( ph -> ( 1r ` W ) e. ( Base ` W ) ) |
28 |
27 21
|
eleqtrd |
|- ( ph -> ( 1r ` W ) e. ( Base ` A ) ) |
29 |
28
|
ad3antrrr |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( 1r ` W ) e. ( Base ` A ) ) |
30 |
|
eqid |
|- ( Base ` A ) = ( Base ` A ) |
31 |
|
eqid |
|- ( Scalar ` A ) = ( Scalar ` A ) |
32 |
|
eqid |
|- ( Base ` ( Scalar ` A ) ) = ( Base ` ( Scalar ` A ) ) |
33 |
|
eqid |
|- ( .s ` A ) = ( .s ` A ) |
34 |
|
eqid |
|- ( .r ` A ) = ( .r ` A ) |
35 |
30 31 32 33 34
|
assaassr |
|- ( ( A e. AssAlg /\ ( y e. ( Base ` ( Scalar ` A ) ) /\ x e. ( Base ` A ) /\ ( 1r ` W ) e. ( Base ` A ) ) ) -> ( x ( .r ` A ) ( y ( .s ` A ) ( 1r ` W ) ) ) = ( y ( .s ` A ) ( x ( .r ` A ) ( 1r ` W ) ) ) ) |
36 |
10 20 24 29 35
|
syl13anc |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( x ( .r ` A ) ( y ( .s ` A ) ( 1r ` W ) ) ) = ( y ( .s ` A ) ( x ( .r ` A ) ( 1r ` W ) ) ) ) |
37 |
14 7
|
sramulr |
|- ( ph -> ( .r ` W ) = ( .r ` A ) ) |
38 |
37
|
ad3antrrr |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( .r ` W ) = ( .r ` A ) ) |
39 |
38
|
oveqd |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( x ( .r ` W ) ( y ( .s ` A ) ( 1r ` W ) ) ) = ( x ( .r ` A ) ( y ( .s ` A ) ( 1r ` W ) ) ) ) |
40 |
14 7
|
sravsca |
|- ( ph -> ( .r ` W ) = ( .s ` A ) ) |
41 |
40
|
ad3antrrr |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( .r ` W ) = ( .s ` A ) ) |
42 |
41
|
oveqd |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( y ( .r ` W ) ( 1r ` W ) ) = ( y ( .s ` A ) ( 1r ` W ) ) ) |
43 |
|
eqid |
|- ( .r ` W ) = ( .r ` W ) |
44 |
3
|
ad3antrrr |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> W e. Ring ) |
45 |
9
|
adantr |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> y e. ( Base ` W ) ) |
46 |
5 43 25 44 45
|
ringridmd |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( y ( .r ` W ) ( 1r ` W ) ) = y ) |
47 |
42 46
|
eqtr3d |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( y ( .s ` A ) ( 1r ` W ) ) = y ) |
48 |
47
|
oveq2d |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( x ( .r ` W ) ( y ( .s ` A ) ( 1r ` W ) ) ) = ( x ( .r ` W ) y ) ) |
49 |
39 48
|
eqtr3d |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( x ( .r ` A ) ( y ( .s ` A ) ( 1r ` W ) ) ) = ( x ( .r ` W ) y ) ) |
50 |
41
|
oveqd |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( y ( .r ` W ) ( x ( .r ` A ) ( 1r ` W ) ) ) = ( y ( .s ` A ) ( x ( .r ` A ) ( 1r ` W ) ) ) ) |
51 |
38
|
oveqd |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( x ( .r ` W ) ( 1r ` W ) ) = ( x ( .r ` A ) ( 1r ` W ) ) ) |
52 |
|
simpr |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> x e. ( Base ` W ) ) |
53 |
5 43 25 44 52
|
ringridmd |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( x ( .r ` W ) ( 1r ` W ) ) = x ) |
54 |
51 53
|
eqtr3d |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( x ( .r ` A ) ( 1r ` W ) ) = x ) |
55 |
54
|
oveq2d |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( y ( .r ` W ) ( x ( .r ` A ) ( 1r ` W ) ) ) = ( y ( .r ` W ) x ) ) |
56 |
50 55
|
eqtr3d |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( y ( .s ` A ) ( x ( .r ` A ) ( 1r ` W ) ) ) = ( y ( .r ` W ) x ) ) |
57 |
36 49 56
|
3eqtr3rd |
|- ( ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) /\ x e. ( Base ` W ) ) -> ( y ( .r ` W ) x ) = ( x ( .r ` W ) y ) ) |
58 |
57
|
ralrimiva |
|- ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) -> A. x e. ( Base ` W ) ( y ( .r ` W ) x ) = ( x ( .r ` W ) y ) ) |
59 |
|
eqid |
|- ( mulGrp ` W ) = ( mulGrp ` W ) |
60 |
59 5
|
mgpbas |
|- ( Base ` W ) = ( Base ` ( mulGrp ` W ) ) |
61 |
59 43
|
mgpplusg |
|- ( .r ` W ) = ( +g ` ( mulGrp ` W ) ) |
62 |
60 61 2
|
elcntr |
|- ( y e. Z <-> ( y e. ( Base ` W ) /\ A. x e. ( Base ` W ) ( y ( .r ` W ) x ) = ( x ( .r ` W ) y ) ) ) |
63 |
9 58 62
|
sylanbrc |
|- ( ( ( ph /\ A e. AssAlg ) /\ y e. S ) -> y e. Z ) |
64 |
63
|
ex |
|- ( ( ph /\ A e. AssAlg ) -> ( y e. S -> y e. Z ) ) |
65 |
64
|
ssrdv |
|- ( ( ph /\ A e. AssAlg ) -> S C_ Z ) |
66 |
21
|
adantr |
|- ( ( ph /\ S C_ Z ) -> ( Base ` W ) = ( Base ` A ) ) |
67 |
15
|
adantr |
|- ( ( ph /\ S C_ Z ) -> ( W |`s S ) = ( Scalar ` A ) ) |
68 |
13
|
adantr |
|- ( ( ph /\ S C_ Z ) -> S = ( Base ` ( W |`s S ) ) ) |
69 |
40
|
adantr |
|- ( ( ph /\ S C_ Z ) -> ( .r ` W ) = ( .s ` A ) ) |
70 |
37
|
adantr |
|- ( ( ph /\ S C_ Z ) -> ( .r ` W ) = ( .r ` A ) ) |
71 |
1
|
sralmod |
|- ( S e. ( SubRing ` W ) -> A e. LMod ) |
72 |
4 71
|
syl |
|- ( ph -> A e. LMod ) |
73 |
72
|
adantr |
|- ( ( ph /\ S C_ Z ) -> A e. LMod ) |
74 |
1 5
|
sraring |
|- ( ( W e. Ring /\ S C_ ( Base ` W ) ) -> A e. Ring ) |
75 |
3 7 74
|
syl2anc |
|- ( ph -> A e. Ring ) |
76 |
75
|
adantr |
|- ( ( ph /\ S C_ Z ) -> A e. Ring ) |
77 |
3
|
ad2antrr |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> W e. Ring ) |
78 |
7
|
adantr |
|- ( ( ph /\ S C_ Z ) -> S C_ ( Base ` W ) ) |
79 |
78
|
sselda |
|- ( ( ( ph /\ S C_ Z ) /\ x e. S ) -> x e. ( Base ` W ) ) |
80 |
79
|
3ad2antr1 |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> x e. ( Base ` W ) ) |
81 |
|
simpr2 |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> y e. ( Base ` W ) ) |
82 |
|
simpr3 |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> z e. ( Base ` W ) ) |
83 |
5 43 77 80 81 82
|
ringassd |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( .r ` W ) y ) ( .r ` W ) z ) = ( x ( .r ` W ) ( y ( .r ` W ) z ) ) ) |
84 |
|
ssel2 |
|- ( ( S C_ Z /\ x e. S ) -> x e. Z ) |
85 |
84
|
ad2ant2lr |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) ) ) -> x e. Z ) |
86 |
|
simprr |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) ) ) -> y e. ( Base ` W ) ) |
87 |
60 61 2
|
cntri |
|- ( ( x e. Z /\ y e. ( Base ` W ) ) -> ( x ( .r ` W ) y ) = ( y ( .r ` W ) x ) ) |
88 |
85 86 87
|
syl2anc |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) ) ) -> ( x ( .r ` W ) y ) = ( y ( .r ` W ) x ) ) |
89 |
88
|
3adantr3 |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( x ( .r ` W ) y ) = ( y ( .r ` W ) x ) ) |
90 |
89
|
oveq1d |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( .r ` W ) y ) ( .r ` W ) z ) = ( ( y ( .r ` W ) x ) ( .r ` W ) z ) ) |
91 |
5 43 77 81 80 82
|
ringassd |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( y ( .r ` W ) x ) ( .r ` W ) z ) = ( y ( .r ` W ) ( x ( .r ` W ) z ) ) ) |
92 |
90 83 91
|
3eqtr3rd |
|- ( ( ( ph /\ S C_ Z ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( y ( .r ` W ) ( x ( .r ` W ) z ) ) = ( x ( .r ` W ) ( y ( .r ` W ) z ) ) ) |
93 |
66 67 68 69 70 73 76 83 92
|
isassad |
|- ( ( ph /\ S C_ Z ) -> A e. AssAlg ) |
94 |
65 93
|
impbida |
|- ( ph -> ( A e. AssAlg <-> S C_ Z ) ) |