Step |
Hyp |
Ref |
Expression |
1 |
|
lkrlspeq.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
2 |
|
lkrlspeq.l |
⊢ 𝐿 = ( LKer ‘ 𝑊 ) |
3 |
|
lkrlspeq.d |
⊢ 𝐷 = ( LDual ‘ 𝑊 ) |
4 |
|
lkrlspeq.o |
⊢ 0 = ( 0g ‘ 𝐷 ) |
5 |
|
lkrlspeq.j |
⊢ 𝑁 = ( LSpan ‘ 𝐷 ) |
6 |
|
lkrlspeq.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
7 |
|
lkrlspeq.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝐹 ) |
8 |
|
lkrlspeq.g |
⊢ ( 𝜑 → 𝐺 ∈ ( ( 𝑁 ‘ { 𝐻 } ) ∖ { 0 } ) ) |
9 |
8
|
eldifad |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝑁 ‘ { 𝐻 } ) ) |
10 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
11 |
6 10
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
12 |
3 11
|
lduallmod |
⊢ ( 𝜑 → 𝐷 ∈ LMod ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
14 |
1 3 13 6 7
|
ldualelvbase |
⊢ ( 𝜑 → 𝐻 ∈ ( Base ‘ 𝐷 ) ) |
15 |
|
eqid |
⊢ ( Scalar ‘ 𝐷 ) = ( Scalar ‘ 𝐷 ) |
16 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝐷 ) ) |
17 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 ) |
18 |
15 16 13 17 5
|
lspsnel |
⊢ ( ( 𝐷 ∈ LMod ∧ 𝐻 ∈ ( Base ‘ 𝐷 ) ) → ( 𝐺 ∈ ( 𝑁 ‘ { 𝐻 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) ) |
19 |
12 14 18
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝑁 ‘ { 𝐻 } ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) ) |
20 |
9 19
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) |
21 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
22 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
23 |
21 22 3 15 16 6
|
ldualsbase |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
24 |
23
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ↔ ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) ) |
25 |
20 24
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) |
26 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
27 |
6
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝑊 ∈ LVec ) |
28 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
29 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) |
30 |
|
eldifsni |
⊢ ( 𝐺 ∈ ( ( 𝑁 ‘ { 𝐻 } ) ∖ { 0 } ) → 𝐺 ≠ 0 ) |
31 |
8 30
|
syl |
⊢ ( 𝜑 → 𝐺 ≠ 0 ) |
32 |
31
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝐺 ≠ 0 ) |
33 |
29 32
|
eqnetrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ≠ 0 ) |
34 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝐷 ) ) = ( 0g ‘ ( Scalar ‘ 𝐷 ) ) |
35 |
21 26 3 15 34 11
|
ldual0 |
⊢ ( 𝜑 → ( 0g ‘ ( Scalar ‘ 𝐷 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
36 |
35
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( 0g ‘ ( Scalar ‘ 𝐷 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
37 |
36
|
eqeq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( 𝑘 = ( 0g ‘ ( Scalar ‘ 𝐷 ) ) ↔ 𝑘 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
38 |
|
orc |
⊢ ( 𝑘 = ( 0g ‘ ( Scalar ‘ 𝐷 ) ) → ( 𝑘 = ( 0g ‘ ( Scalar ‘ 𝐷 ) ) ∨ 𝐻 = 0 ) ) |
39 |
37 38
|
syl6bir |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( 𝑘 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) → ( 𝑘 = ( 0g ‘ ( Scalar ‘ 𝐷 ) ) ∨ 𝐻 = 0 ) ) ) |
40 |
3 6
|
lduallvec |
⊢ ( 𝜑 → 𝐷 ∈ LVec ) |
41 |
40
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝐷 ∈ LVec ) |
42 |
23
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
43 |
28 42
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ) |
44 |
14
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝐻 ∈ ( Base ‘ 𝐷 ) ) |
45 |
13 17 15 16 34 4 41 43 44
|
lvecvs0or |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) = 0 ↔ ( 𝑘 = ( 0g ‘ ( Scalar ‘ 𝐷 ) ) ∨ 𝐻 = 0 ) ) ) |
46 |
39 45
|
sylibrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( 𝑘 = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) → ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) = 0 ) ) |
47 |
46
|
necon3d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ≠ 0 → 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
48 |
33 47
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
49 |
|
eldifsn |
⊢ ( 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ↔ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑘 ≠ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
50 |
28 48 49
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝑘 ∈ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ) |
51 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → 𝐻 ∈ 𝐹 ) |
52 |
21 22 26 1 2 3 17 27 50 51 29
|
lkreqN |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) → ( 𝐿 ‘ 𝐺 ) = ( 𝐿 ‘ 𝐻 ) ) |
53 |
52
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) 𝐺 = ( 𝑘 ( ·𝑠 ‘ 𝐷 ) 𝐻 ) → ( 𝐿 ‘ 𝐺 ) = ( 𝐿 ‘ 𝐻 ) ) ) |
54 |
25 53
|
mpd |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( 𝐿 ‘ 𝐻 ) ) |