Step |
Hyp |
Ref |
Expression |
1 |
|
trljco.j |
|- .\/ = ( join ` K ) |
2 |
|
trljco.h |
|- H = ( LHyp ` K ) |
3 |
|
trljco.t |
|- T = ( ( LTrn ` K ) ` W ) |
4 |
|
trljco.r |
|- R = ( ( trL ` K ) ` W ) |
5 |
|
coeq1 |
|- ( F = ( _I |` ( Base ` K ) ) -> ( F o. G ) = ( ( _I |` ( Base ` K ) ) o. G ) ) |
6 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
7 |
6 2 3
|
ltrn1o |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> G : ( Base ` K ) -1-1-onto-> ( Base ` K ) ) |
8 |
7
|
3adant2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> G : ( Base ` K ) -1-1-onto-> ( Base ` K ) ) |
9 |
|
f1of |
|- ( G : ( Base ` K ) -1-1-onto-> ( Base ` K ) -> G : ( Base ` K ) --> ( Base ` K ) ) |
10 |
|
fcoi2 |
|- ( G : ( Base ` K ) --> ( Base ` K ) -> ( ( _I |` ( Base ` K ) ) o. G ) = G ) |
11 |
8 9 10
|
3syl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( _I |` ( Base ` K ) ) o. G ) = G ) |
12 |
5 11
|
sylan9eqr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> ( F o. G ) = G ) |
13 |
12
|
fveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> ( R ` ( F o. G ) ) = ( R ` G ) ) |
14 |
13
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) |
15 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> K e. HL ) |
16 |
15
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> K e. Lat ) |
17 |
6 2 3 4
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) ) |
18 |
17
|
3adant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` F ) e. ( Base ` K ) ) |
19 |
6 1
|
latjidm |
|- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` F ) ) = ( R ` F ) ) |
20 |
16 18 19
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` F ) ) = ( R ` F ) ) |
21 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
22 |
15 21
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> K e. OL ) |
23 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
24 |
6 1 23
|
olj01 |
|- ( ( K e. OL /\ ( R ` F ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( 0. ` K ) ) = ( R ` F ) ) |
25 |
22 18 24
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( 0. ` K ) ) = ( R ` F ) ) |
26 |
20 25
|
eqtr4d |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` F ) ) = ( ( R ` F ) .\/ ( 0. ` K ) ) ) |
27 |
26
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ G = ( _I |` ( Base ` K ) ) ) -> ( ( R ` F ) .\/ ( R ` F ) ) = ( ( R ` F ) .\/ ( 0. ` K ) ) ) |
28 |
|
coeq2 |
|- ( G = ( _I |` ( Base ` K ) ) -> ( F o. G ) = ( F o. ( _I |` ( Base ` K ) ) ) ) |
29 |
6 2 3
|
ltrn1o |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) ) |
30 |
29
|
3adant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) ) |
31 |
|
f1of |
|- ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) -> F : ( Base ` K ) --> ( Base ` K ) ) |
32 |
|
fcoi1 |
|- ( F : ( Base ` K ) --> ( Base ` K ) -> ( F o. ( _I |` ( Base ` K ) ) ) = F ) |
33 |
30 31 32
|
3syl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. ( _I |` ( Base ` K ) ) ) = F ) |
34 |
28 33
|
sylan9eqr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ G = ( _I |` ( Base ` K ) ) ) -> ( F o. G ) = F ) |
35 |
34
|
fveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ G = ( _I |` ( Base ` K ) ) ) -> ( R ` ( F o. G ) ) = ( R ` F ) ) |
36 |
35
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ G = ( _I |` ( Base ` K ) ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` F ) ) ) |
37 |
6 23 2 3 4
|
trlid0b |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( G = ( _I |` ( Base ` K ) ) <-> ( R ` G ) = ( 0. ` K ) ) ) |
38 |
37
|
3adant2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( G = ( _I |` ( Base ` K ) ) <-> ( R ` G ) = ( 0. ` K ) ) ) |
39 |
38
|
biimpa |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ G = ( _I |` ( Base ` K ) ) ) -> ( R ` G ) = ( 0. ` K ) ) |
40 |
39
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ G = ( _I |` ( Base ` K ) ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) = ( ( R ` F ) .\/ ( 0. ` K ) ) ) |
41 |
27 36 40
|
3eqtr4d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ G = ( _I |` ( Base ` K ) ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) |
42 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
43 |
16
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> K e. Lat ) |
44 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( K e. HL /\ W e. H ) ) |
45 |
2 3
|
ltrnco |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) e. T ) |
46 |
6 2 3 4
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F o. G ) e. T ) -> ( R ` ( F o. G ) ) e. ( Base ` K ) ) |
47 |
44 45 46
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` ( F o. G ) ) e. ( Base ` K ) ) |
48 |
6 1
|
latjcl |
|- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` ( F o. G ) ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) e. ( Base ` K ) ) |
49 |
16 18 47 48
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) e. ( Base ` K ) ) |
50 |
49
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) e. ( Base ` K ) ) |
51 |
6 2 3 4
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) ) |
52 |
51
|
3adant2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) ) |
53 |
6 1
|
latjcl |
|- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) ) |
54 |
16 18 52 53
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) ) |
55 |
54
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) ) |
56 |
6 42 1
|
latlej1 |
|- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( R ` F ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) |
57 |
16 18 52 56
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` F ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) |
58 |
42 1 2 3 4
|
trlco |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` ( F o. G ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) |
59 |
6 42 1
|
latjle12 |
|- ( ( K e. Lat /\ ( ( R ` F ) e. ( Base ` K ) /\ ( R ` ( F o. G ) ) e. ( Base ` K ) /\ ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) ) ) -> ( ( ( R ` F ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) /\ ( R ` ( F o. G ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) <-> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) ) |
60 |
16 18 47 54 59
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( ( R ` F ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) /\ ( R ` ( F o. G ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) <-> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) ) |
61 |
57 58 60
|
mpbi2and |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) |
62 |
61
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) |
63 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( R ` F ) = ( R ` G ) ) |
64 |
63
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( R ` F ) .\/ ( R ` F ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) |
65 |
6 42 1
|
latlej1 |
|- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` ( F o. G ) ) e. ( Base ` K ) ) -> ( R ` F ) ( le ` K ) ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ) |
66 |
16 18 47 65
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` F ) ( le ` K ) ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ) |
67 |
20 66
|
eqbrtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` F ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ) |
68 |
67
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( R ` F ) .\/ ( R ` F ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ) |
69 |
64 68
|
eqbrtrrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ) |
70 |
6 42 43 50 55 62 69
|
latasymd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) |
71 |
61
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) ) |
72 |
|
simpl1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. HL ) |
73 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) ) |
74 |
|
simpl2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> F e. T ) |
75 |
|
simpr1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> F =/= ( _I |` ( Base ` K ) ) ) |
76 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
77 |
6 76 2 3 4
|
trlnidat |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` ( Base ` K ) ) ) -> ( R ` F ) e. ( Atoms ` K ) ) |
78 |
73 74 75 77
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) e. ( Atoms ` K ) ) |
79 |
|
simpl3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> G e. T ) |
80 |
74 79
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F e. T /\ G e. T ) ) |
81 |
|
simpr3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` G ) ) |
82 |
76 2 3 4
|
trlcoat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( R ` ( F o. G ) ) e. ( Atoms ` K ) ) |
83 |
73 80 81 82
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` ( F o. G ) ) e. ( Atoms ` K ) ) |
84 |
|
simpr2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> G =/= ( _I |` ( Base ` K ) ) ) |
85 |
6 2 3 4
|
trlcone |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` F ) =/= ( R ` G ) /\ G =/= ( _I |` ( Base ` K ) ) ) ) -> ( R ` F ) =/= ( R ` ( F o. G ) ) ) |
86 |
73 80 81 84 85
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` ( F o. G ) ) ) |
87 |
6 76 2 3 4
|
trlnidat |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` ( Base ` K ) ) ) -> ( R ` G ) e. ( Atoms ` K ) ) |
88 |
73 79 84 87
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) e. ( Atoms ` K ) ) |
89 |
42 1 76
|
ps-1 |
|- ( ( K e. HL /\ ( ( R ` F ) e. ( Atoms ` K ) /\ ( R ` ( F o. G ) ) e. ( Atoms ` K ) /\ ( R ` F ) =/= ( R ` ( F o. G ) ) ) /\ ( ( R ` F ) e. ( Atoms ` K ) /\ ( R ` G ) e. ( Atoms ` K ) ) ) -> ( ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) <-> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) ) |
90 |
72 78 83 86 78 88 89
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) ( le ` K ) ( ( R ` F ) .\/ ( R ` G ) ) <-> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) ) |
91 |
71 90
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ G =/= ( _I |` ( Base ` K ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) |
92 |
14 41 70 91
|
pm2.61da3ne |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) ) |