Step |
Hyp |
Ref |
Expression |
1 |
|
pgpfac1.k |
|- K = ( mrCls ` ( SubGrp ` G ) ) |
2 |
|
pgpfac1.s |
|- S = ( K ` { A } ) |
3 |
|
pgpfac1.b |
|- B = ( Base ` G ) |
4 |
|
pgpfac1.o |
|- O = ( od ` G ) |
5 |
|
pgpfac1.e |
|- E = ( gEx ` G ) |
6 |
|
pgpfac1.z |
|- .0. = ( 0g ` G ) |
7 |
|
pgpfac1.l |
|- .(+) = ( LSSum ` G ) |
8 |
|
pgpfac1.p |
|- ( ph -> P pGrp G ) |
9 |
|
pgpfac1.g |
|- ( ph -> G e. Abel ) |
10 |
|
pgpfac1.n |
|- ( ph -> B e. Fin ) |
11 |
|
pgpfac1.oe |
|- ( ph -> ( O ` A ) = E ) |
12 |
|
pgpfac1.u |
|- ( ph -> U e. ( SubGrp ` G ) ) |
13 |
|
pgpfac1.au |
|- ( ph -> A e. U ) |
14 |
|
pgpfac1.w |
|- ( ph -> W e. ( SubGrp ` G ) ) |
15 |
|
pgpfac1.i |
|- ( ph -> ( S i^i W ) = { .0. } ) |
16 |
|
pgpfac1.ss |
|- ( ph -> ( S .(+) W ) C_ U ) |
17 |
|
pgpfac1.2 |
|- ( ph -> A. w e. ( SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) ) |
18 |
16
|
adantr |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) C_ U ) |
19 |
|
ablgrp |
|- ( G e. Abel -> G e. Grp ) |
20 |
3
|
subgacs |
|- ( G e. Grp -> ( SubGrp ` G ) e. ( ACS ` B ) ) |
21 |
|
acsmre |
|- ( ( SubGrp ` G ) e. ( ACS ` B ) -> ( SubGrp ` G ) e. ( Moore ` B ) ) |
22 |
9 19 20 21
|
4syl |
|- ( ph -> ( SubGrp ` G ) e. ( Moore ` B ) ) |
23 |
22
|
adantr |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( SubGrp ` G ) e. ( Moore ` B ) ) |
24 |
|
eldifi |
|- ( C e. ( U \ ( S .(+) W ) ) -> C e. U ) |
25 |
24
|
adantl |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. U ) |
26 |
25
|
snssd |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> { C } C_ U ) |
27 |
12
|
adantr |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> U e. ( SubGrp ` G ) ) |
28 |
1
|
mrcsscl |
|- ( ( ( SubGrp ` G ) e. ( Moore ` B ) /\ { C } C_ U /\ U e. ( SubGrp ` G ) ) -> ( K ` { C } ) C_ U ) |
29 |
23 26 27 28
|
syl3anc |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( K ` { C } ) C_ U ) |
30 |
3
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ B ) |
31 |
12 30
|
syl |
|- ( ph -> U C_ B ) |
32 |
31 13
|
sseldd |
|- ( ph -> A e. B ) |
33 |
1
|
mrcsncl |
|- ( ( ( SubGrp ` G ) e. ( Moore ` B ) /\ A e. B ) -> ( K ` { A } ) e. ( SubGrp ` G ) ) |
34 |
22 32 33
|
syl2anc |
|- ( ph -> ( K ` { A } ) e. ( SubGrp ` G ) ) |
35 |
2 34
|
eqeltrid |
|- ( ph -> S e. ( SubGrp ` G ) ) |
36 |
7
|
lsmsubg2 |
|- ( ( G e. Abel /\ S e. ( SubGrp ` G ) /\ W e. ( SubGrp ` G ) ) -> ( S .(+) W ) e. ( SubGrp ` G ) ) |
37 |
9 35 14 36
|
syl3anc |
|- ( ph -> ( S .(+) W ) e. ( SubGrp ` G ) ) |
38 |
37
|
adantr |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) e. ( SubGrp ` G ) ) |
39 |
31
|
sselda |
|- ( ( ph /\ C e. U ) -> C e. B ) |
40 |
24 39
|
sylan2 |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. B ) |
41 |
1
|
mrcsncl |
|- ( ( ( SubGrp ` G ) e. ( Moore ` B ) /\ C e. B ) -> ( K ` { C } ) e. ( SubGrp ` G ) ) |
42 |
23 40 41
|
syl2anc |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( K ` { C } ) e. ( SubGrp ` G ) ) |
43 |
7
|
lsmlub |
|- ( ( ( S .(+) W ) e. ( SubGrp ` G ) /\ ( K ` { C } ) e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( ( ( S .(+) W ) C_ U /\ ( K ` { C } ) C_ U ) <-> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U ) ) |
44 |
38 42 27 43
|
syl3anc |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( S .(+) W ) C_ U /\ ( K ` { C } ) C_ U ) <-> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U ) ) |
45 |
18 29 44
|
mpbi2and |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U ) |
46 |
7
|
lsmub1 |
|- ( ( ( S .(+) W ) e. ( SubGrp ` G ) /\ ( K ` { C } ) e. ( SubGrp ` G ) ) -> ( S .(+) W ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) |
47 |
38 42 46
|
syl2anc |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) |
48 |
7
|
lsmub2 |
|- ( ( ( S .(+) W ) e. ( SubGrp ` G ) /\ ( K ` { C } ) e. ( SubGrp ` G ) ) -> ( K ` { C } ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) |
49 |
38 42 48
|
syl2anc |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( K ` { C } ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) |
50 |
40
|
snssd |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> { C } C_ B ) |
51 |
23 1 50
|
mrcssidd |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> { C } C_ ( K ` { C } ) ) |
52 |
|
snssg |
|- ( C e. B -> ( C e. ( K ` { C } ) <-> { C } C_ ( K ` { C } ) ) ) |
53 |
40 52
|
syl |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( C e. ( K ` { C } ) <-> { C } C_ ( K ` { C } ) ) ) |
54 |
51 53
|
mpbird |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. ( K ` { C } ) ) |
55 |
49 54
|
sseldd |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) |
56 |
|
eldifn |
|- ( C e. ( U \ ( S .(+) W ) ) -> -. C e. ( S .(+) W ) ) |
57 |
56
|
adantl |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> -. C e. ( S .(+) W ) ) |
58 |
47 55 57
|
ssnelpssd |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) |
59 |
7
|
lsmub1 |
|- ( ( S e. ( SubGrp ` G ) /\ W e. ( SubGrp ` G ) ) -> S C_ ( S .(+) W ) ) |
60 |
35 14 59
|
syl2anc |
|- ( ph -> S C_ ( S .(+) W ) ) |
61 |
32
|
snssd |
|- ( ph -> { A } C_ B ) |
62 |
22 1 61
|
mrcssidd |
|- ( ph -> { A } C_ ( K ` { A } ) ) |
63 |
62 2
|
sseqtrrdi |
|- ( ph -> { A } C_ S ) |
64 |
|
snssg |
|- ( A e. U -> ( A e. S <-> { A } C_ S ) ) |
65 |
13 64
|
syl |
|- ( ph -> ( A e. S <-> { A } C_ S ) ) |
66 |
63 65
|
mpbird |
|- ( ph -> A e. S ) |
67 |
60 66
|
sseldd |
|- ( ph -> A e. ( S .(+) W ) ) |
68 |
67
|
adantr |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> A e. ( S .(+) W ) ) |
69 |
47 68
|
sseldd |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) |
70 |
|
psseq1 |
|- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( w C. U <-> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U ) ) |
71 |
|
eleq2 |
|- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( A e. w <-> A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) |
72 |
70 71
|
anbi12d |
|- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( ( w C. U /\ A e. w ) <-> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U /\ A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) ) |
73 |
|
psseq2 |
|- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( ( S .(+) W ) C. w <-> ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) |
74 |
73
|
notbid |
|- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( -. ( S .(+) W ) C. w <-> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) |
75 |
72 74
|
imbi12d |
|- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) <-> ( ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U /\ A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) -> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) ) |
76 |
17
|
adantr |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> A. w e. ( SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) ) |
77 |
9
|
adantr |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> G e. Abel ) |
78 |
7
|
lsmsubg2 |
|- ( ( G e. Abel /\ ( S .(+) W ) e. ( SubGrp ` G ) /\ ( K ` { C } ) e. ( SubGrp ` G ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) e. ( SubGrp ` G ) ) |
79 |
77 38 42 78
|
syl3anc |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) e. ( SubGrp ` G ) ) |
80 |
75 76 79
|
rspcdva |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U /\ A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) -> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) |
81 |
69 80
|
mpan2d |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U -> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) |
82 |
58 81
|
mt2d |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> -. ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U ) |
83 |
|
npss |
|- ( -. ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U <-> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) = U ) ) |
84 |
82 83
|
sylib |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) = U ) ) |
85 |
45 84
|
mpd |
|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) = U ) |