Step |
Hyp |
Ref |
Expression |
1 |
|
dochkr1OLD.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochkr1OLD.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochkr1OLD.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochkr1OLD.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochkr1OLD.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
6 |
|
dochkr1OLD.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
7 |
|
dochkr1OLD.i |
⊢ 1 = ( 1r ‘ 𝑅 ) |
8 |
|
dochkr1OLD.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
9 |
|
dochkr1OLD.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
10 |
|
dochkr1OLD.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
11 |
|
dochkr1OLD.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
12 |
|
dochkr1OLD.n |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ) |
13 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
14 |
|
eqid |
⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 ) |
15 |
1 3 10
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
16 |
1 2 3 4 14 8 9 10 11
|
dochkrsat2 |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ≠ 𝑉 ↔ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ) ) |
17 |
12 16
|
mpbid |
⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ) |
18 |
13 14 15 17
|
lsateln0 |
⊢ ( 𝜑 → ∃ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) 𝑧 ≠ ( 0g ‘ 𝑈 ) ) |
19 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∧ 𝑧 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
20 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∧ 𝑧 ≠ ( 0g ‘ 𝑈 ) ) → 𝐺 ∈ 𝐹 ) |
21 |
|
eldifsn |
⊢ ( 𝑧 ∈ ( ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∖ { ( 0g ‘ 𝑈 ) } ) ↔ ( 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ 𝑧 ≠ ( 0g ‘ 𝑈 ) ) ) |
22 |
21
|
biimpri |
⊢ ( ( 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ 𝑧 ≠ ( 0g ‘ 𝑈 ) ) → 𝑧 ∈ ( ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∖ { ( 0g ‘ 𝑈 ) } ) ) |
23 |
22
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∧ 𝑧 ≠ ( 0g ‘ 𝑈 ) ) → 𝑧 ∈ ( ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∖ { ( 0g ‘ 𝑈 ) } ) ) |
24 |
1 2 3 4 5 6 13 8 9 19 20 23
|
dochfln0 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∧ 𝑧 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ 𝑧 ) ≠ 0 ) |
25 |
24
|
ex |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) → ( 𝑧 ≠ ( 0g ‘ 𝑈 ) → ( 𝐺 ‘ 𝑧 ) ≠ 0 ) ) |
26 |
25
|
reximdva |
⊢ ( 𝜑 → ( ∃ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) 𝑧 ≠ ( 0g ‘ 𝑈 ) → ∃ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ( 𝐺 ‘ 𝑧 ) ≠ 0 ) ) |
27 |
18 26
|
mpd |
⊢ ( 𝜑 → ∃ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ( 𝐺 ‘ 𝑧 ) ≠ 0 ) |
28 |
4 8 9 15 11
|
lkrssv |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ 𝑉 ) |
29 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
30 |
1 3 4 29 2
|
dochlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐿 ‘ 𝐺 ) ⊆ 𝑉 ) → ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
31 |
10 28 30
|
syl2anc |
⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
32 |
15 31
|
jca |
⊢ ( 𝜑 → ( 𝑈 ∈ LMod ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( 𝑈 ∈ LMod ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) ) |
34 |
1 3 10
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
35 |
34
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → 𝑈 ∈ LVec ) |
36 |
5
|
lvecdrng |
⊢ ( 𝑈 ∈ LVec → 𝑅 ∈ DivRing ) |
37 |
35 36
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → 𝑅 ∈ DivRing ) |
38 |
15
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → 𝑈 ∈ LMod ) |
39 |
11
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → 𝐺 ∈ 𝐹 ) |
40 |
1 3 4 2
|
dochssv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐿 ‘ 𝐺 ) ⊆ 𝑉 ) → ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑉 ) |
41 |
10 28 40
|
syl2anc |
⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ⊆ 𝑉 ) |
42 |
41
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) → 𝑧 ∈ 𝑉 ) |
43 |
42
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → 𝑧 ∈ 𝑉 ) |
44 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
45 |
5 44 4 8
|
lflcl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑧 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) |
46 |
38 39 43 45
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) |
47 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( 𝐺 ‘ 𝑧 ) ≠ 0 ) |
48 |
|
eqid |
⊢ ( invr ‘ 𝑅 ) = ( invr ‘ 𝑅 ) |
49 |
44 6 48
|
drnginvrcl |
⊢ ( ( 𝑅 ∈ DivRing ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ) |
50 |
37 46 47 49
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ) |
51 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
52 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 ) |
53 |
5 52 44 29
|
lssvscl |
⊢ ( ( ( 𝑈 ∈ LMod ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) ∧ ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ) → ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
54 |
33 50 51 53
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) |
55 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
56 |
5 44 55 4 52 8
|
lflmul |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ) = ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) ) |
57 |
38 39 50 43 56
|
syl112anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( 𝐺 ‘ ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ) = ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) ) |
58 |
44 6 55 7 48
|
drnginvrl |
⊢ ( ( 𝑅 ∈ DivRing ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) = 1 ) |
59 |
37 46 47 58
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ 𝑧 ) ) = 1 ) |
60 |
57 59
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ( 𝐺 ‘ ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ) = 1 ) |
61 |
|
fveqeq2 |
⊢ ( 𝑥 = ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) → ( ( 𝐺 ‘ 𝑥 ) = 1 ↔ ( 𝐺 ‘ ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ) = 1 ) ) |
62 |
61
|
rspcev |
⊢ ( ( ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ ( ( ( invr ‘ 𝑅 ) ‘ ( 𝐺 ‘ 𝑧 ) ) ( ·𝑠 ‘ 𝑈 ) 𝑧 ) ) = 1 ) → ∃ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ( 𝐺 ‘ 𝑥 ) = 1 ) |
63 |
54 60 62
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∧ ( 𝐺 ‘ 𝑧 ) ≠ 0 ) → ∃ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ( 𝐺 ‘ 𝑥 ) = 1 ) |
64 |
63
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑧 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ( 𝐺 ‘ 𝑧 ) ≠ 0 → ∃ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ( 𝐺 ‘ 𝑥 ) = 1 ) ) |
65 |
27 64
|
mpd |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ( 𝐺 ‘ 𝑥 ) = 1 ) |