Step |
Hyp |
Ref |
Expression |
1 |
|
lkrlsp.d |
⊢ 𝐷 = ( Scalar ‘ 𝑊 ) |
2 |
|
lkrlsp.o |
⊢ 0 = ( 0g ‘ 𝐷 ) |
3 |
|
lkrlsp.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
4 |
|
lkrlsp.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
5 |
|
lkrlsp.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
6 |
|
lkrlsp.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
7 |
|
lkrlsp.k |
⊢ 𝐾 = ( LKer ‘ 𝑊 ) |
8 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
9 |
8
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → 𝑊 ∈ LMod ) |
10 |
|
simp2r |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → 𝐺 ∈ 𝐹 ) |
11 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
12 |
6 7 11
|
lkrlss |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
13 |
9 10 12
|
syl2anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
14 |
|
simp2l |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → 𝑋 ∈ 𝑉 ) |
15 |
3 11 4
|
lspsncl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
16 |
9 14 15
|
syl2anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
17 |
11 5
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) → ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
18 |
9 13 16 17
|
syl3anc |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑊 ) ) |
19 |
3 11
|
lssss |
⊢ ( ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑊 ) → ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ 𝑉 ) |
20 |
18 19
|
syl |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ⊆ 𝑉 ) |
21 |
|
simpl1 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝑊 ∈ LVec ) |
22 |
21 8
|
syl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝑊 ∈ LMod ) |
23 |
|
simpr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝑢 ∈ 𝑉 ) |
24 |
1
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝐷 ∈ Ring ) |
25 |
22 24
|
syl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝐷 ∈ Ring ) |
26 |
|
simpl2r |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝐺 ∈ 𝐹 ) |
27 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
28 |
1 27 3 6
|
lflcl |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑢 ) ∈ ( Base ‘ 𝐷 ) ) |
29 |
21 26 23 28
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑢 ) ∈ ( Base ‘ 𝐷 ) ) |
30 |
1
|
lvecdrng |
⊢ ( 𝑊 ∈ LVec → 𝐷 ∈ DivRing ) |
31 |
21 30
|
syl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝐷 ∈ DivRing ) |
32 |
|
simpl2l |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 ) |
33 |
1 27 3 6
|
lflcl |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) |
34 |
21 26 32 33
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) |
35 |
|
simpl3 |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑋 ) ≠ 0 ) |
36 |
|
eqid |
⊢ ( invr ‘ 𝐷 ) = ( invr ‘ 𝐷 ) |
37 |
27 2 36
|
drnginvrcl |
⊢ ( ( 𝐷 ∈ DivRing ∧ ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐷 ) ) |
38 |
31 34 35 37
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐷 ) ) |
39 |
|
eqid |
⊢ ( .r ‘ 𝐷 ) = ( .r ‘ 𝐷 ) |
40 |
27 39
|
ringcl |
⊢ ( ( 𝐷 ∈ Ring ∧ ( 𝐺 ‘ 𝑢 ) ∈ ( Base ‘ 𝐷 ) ∧ ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ∈ ( Base ‘ 𝐷 ) ) |
41 |
25 29 38 40
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ∈ ( Base ‘ 𝐷 ) ) |
42 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
43 |
3 1 42 27
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ∈ ( Base ‘ 𝐷 ) ∧ 𝑋 ∈ 𝑉 ) → ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ 𝑉 ) |
44 |
22 41 32 43
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ 𝑉 ) |
45 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
46 |
|
eqid |
⊢ ( -g ‘ 𝑊 ) = ( -g ‘ 𝑊 ) |
47 |
3 45 46
|
lmodvnpcan |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑢 ∈ 𝑉 ∧ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ 𝑉 ) → ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ( +g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) = 𝑢 ) |
48 |
22 23 44 47
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ( +g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) = 𝑢 ) |
49 |
11
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
50 |
22 49
|
syl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( LSubSp ‘ 𝑊 ) ⊆ ( SubGrp ‘ 𝑊 ) ) |
51 |
13
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐾 ‘ 𝐺 ) ∈ ( LSubSp ‘ 𝑊 ) ) |
52 |
50 51
|
sseldd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐾 ‘ 𝐺 ) ∈ ( SubGrp ‘ 𝑊 ) ) |
53 |
16
|
adantr |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑊 ) ) |
54 |
50 53
|
sseldd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) |
55 |
3 46
|
lmodvsubcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑢 ∈ 𝑉 ∧ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ 𝑉 ) → ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ 𝑉 ) |
56 |
22 23 44 55
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ 𝑉 ) |
57 |
|
eqid |
⊢ ( -g ‘ 𝐷 ) = ( -g ‘ 𝐷 ) |
58 |
1 57 3 46 6
|
lflsub |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑢 ∈ 𝑉 ∧ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ 𝑉 ) ) → ( 𝐺 ‘ ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) = ( ( 𝐺 ‘ 𝑢 ) ( -g ‘ 𝐷 ) ( 𝐺 ‘ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) ) |
59 |
22 26 23 44 58
|
syl112anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) = ( ( 𝐺 ‘ 𝑢 ) ( -g ‘ 𝐷 ) ( 𝐺 ‘ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) ) |
60 |
1 27 39 3 42 6
|
lflmul |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ∈ ( Base ‘ 𝐷 ) ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) = ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) |
61 |
22 26 41 32 60
|
syl112anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) = ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) |
62 |
27 39
|
ringass |
⊢ ( ( 𝐷 ∈ Ring ∧ ( ( 𝐺 ‘ 𝑢 ) ∈ ( Base ‘ 𝐷 ) ∧ ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐷 ) ∧ ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) ) → ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) ) |
63 |
25 29 38 34 62
|
syl13anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) ) |
64 |
|
eqid |
⊢ ( 1r ‘ 𝐷 ) = ( 1r ‘ 𝐷 ) |
65 |
27 2 39 64 36
|
drnginvrl |
⊢ ( ( 𝐷 ∈ DivRing ∧ ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( 1r ‘ 𝐷 ) ) |
66 |
31 34 35 65
|
syl3anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( 1r ‘ 𝐷 ) ) |
67 |
66
|
oveq2d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) = ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( 1r ‘ 𝐷 ) ) ) |
68 |
27 39 64
|
ringridm |
⊢ ( ( 𝐷 ∈ Ring ∧ ( 𝐺 ‘ 𝑢 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( 1r ‘ 𝐷 ) ) = ( 𝐺 ‘ 𝑢 ) ) |
69 |
25 29 68
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( 1r ‘ 𝐷 ) ) = ( 𝐺 ‘ 𝑢 ) ) |
70 |
67 69
|
eqtrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) = ( 𝐺 ‘ 𝑢 ) ) |
71 |
61 63 70
|
3eqtrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) = ( 𝐺 ‘ 𝑢 ) ) |
72 |
71
|
oveq2d |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝐺 ‘ 𝑢 ) ( -g ‘ 𝐷 ) ( 𝐺 ‘ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) = ( ( 𝐺 ‘ 𝑢 ) ( -g ‘ 𝐷 ) ( 𝐺 ‘ 𝑢 ) ) ) |
73 |
1
|
lmodfgrp |
⊢ ( 𝑊 ∈ LMod → 𝐷 ∈ Grp ) |
74 |
22 73
|
syl |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝐷 ∈ Grp ) |
75 |
27 2 57
|
grpsubid |
⊢ ( ( 𝐷 ∈ Grp ∧ ( 𝐺 ‘ 𝑢 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝐺 ‘ 𝑢 ) ( -g ‘ 𝐷 ) ( 𝐺 ‘ 𝑢 ) ) = 0 ) |
76 |
74 29 75
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝐺 ‘ 𝑢 ) ( -g ‘ 𝐷 ) ( 𝐺 ‘ 𝑢 ) ) = 0 ) |
77 |
59 72 76
|
3eqtrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝐺 ‘ ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) = 0 ) |
78 |
3 1 2 6 7
|
ellkr |
⊢ ( ( 𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ) → ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ ( 𝐾 ‘ 𝐺 ) ↔ ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ 𝑉 ∧ ( 𝐺 ‘ ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) = 0 ) ) ) |
79 |
21 26 78
|
syl2anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ ( 𝐾 ‘ 𝐺 ) ↔ ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ 𝑉 ∧ ( 𝐺 ‘ ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ) = 0 ) ) ) |
80 |
56 77 79
|
mpbir2and |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ ( 𝐾 ‘ 𝐺 ) ) |
81 |
3 42 1 27 4 22 41 32
|
lspsneli |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) |
82 |
45 5
|
lsmelvali |
⊢ ( ( ( ( 𝐾 ‘ 𝐺 ) ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝑊 ) ) ∧ ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ ( 𝐾 ‘ 𝐺 ) ∧ ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ∈ ( 𝑁 ‘ { 𝑋 } ) ) ) → ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ( +g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
83 |
52 54 80 81 82
|
syl22anc |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → ( ( 𝑢 ( -g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ( +g ‘ 𝑊 ) ( ( ( 𝐺 ‘ 𝑢 ) ( .r ‘ 𝐷 ) ( ( invr ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑋 ) ) ) ( ·𝑠 ‘ 𝑊 ) 𝑋 ) ) ∈ ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
84 |
48 83
|
eqeltrrd |
⊢ ( ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) ∧ 𝑢 ∈ 𝑉 ) → 𝑢 ∈ ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) ) |
85 |
20 84
|
eqelssd |
⊢ ( ( 𝑊 ∈ LVec ∧ ( 𝑋 ∈ 𝑉 ∧ 𝐺 ∈ 𝐹 ) ∧ ( 𝐺 ‘ 𝑋 ) ≠ 0 ) → ( ( 𝐾 ‘ 𝐺 ) ⊕ ( 𝑁 ‘ { 𝑋 } ) ) = 𝑉 ) |