Step |
Hyp |
Ref |
Expression |
1 |
|
lsatcv0.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
2 |
|
lsatcv0.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
3 |
|
lsatcv0.c |
⊢ 𝐶 = ( ⋖L ‘ 𝑊 ) |
4 |
|
lsatcv0.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
5 |
|
lsatcv0.q |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
6 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
7 |
4 6
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
8 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
9 |
8 2 7 5
|
lsatlssel |
⊢ ( 𝜑 → 𝑄 ∈ ( LSubSp ‘ 𝑊 ) ) |
10 |
1 8
|
lss0ss |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑄 ∈ ( LSubSp ‘ 𝑊 ) ) → { 0 } ⊆ 𝑄 ) |
11 |
7 9 10
|
syl2anc |
⊢ ( 𝜑 → { 0 } ⊆ 𝑄 ) |
12 |
1 2 7 5
|
lsatn0 |
⊢ ( 𝜑 → 𝑄 ≠ { 0 } ) |
13 |
12
|
necomd |
⊢ ( 𝜑 → { 0 } ≠ 𝑄 ) |
14 |
|
df-pss |
⊢ ( { 0 } ⊊ 𝑄 ↔ ( { 0 } ⊆ 𝑄 ∧ { 0 } ≠ 𝑄 ) ) |
15 |
11 13 14
|
sylanbrc |
⊢ ( 𝜑 → { 0 } ⊊ 𝑄 ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
17 |
|
eqid |
⊢ ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 ) |
18 |
16 17 1 2
|
islsat |
⊢ ( 𝑊 ∈ LMod → ( 𝑄 ∈ 𝐴 ↔ ∃ 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) |
19 |
7 18
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ↔ ∃ 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) |
20 |
5 19
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) |
21 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) ) → 𝑊 ∈ LVec ) |
22 |
|
eldifi |
⊢ ( 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) → 𝑥 ∈ ( Base ‘ 𝑊 ) ) |
23 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) ) → 𝑥 ∈ ( Base ‘ 𝑊 ) ) |
24 |
16 1 8 17 21 23
|
lspsncv0 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) ) → ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) |
25 |
24
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) → ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) ) |
26 |
|
psseq2 |
⊢ ( 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) → ( 𝑠 ⊊ 𝑄 ↔ 𝑠 ⊊ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) |
27 |
26
|
anbi2d |
⊢ ( 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) → ( ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ↔ ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) ) |
28 |
27
|
rexbidv |
⊢ ( 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) → ( ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ↔ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) ) |
29 |
28
|
notbid |
⊢ ( 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) → ( ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ↔ ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) ) ) |
30 |
29
|
biimprcd |
⊢ ( ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) ) → ( 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) → ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ) ) |
31 |
25 30
|
syl6 |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) → ( 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) → ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ) ) ) |
32 |
31
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( ( Base ‘ 𝑊 ) ∖ { 0 } ) 𝑄 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑥 } ) → ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ) ) |
33 |
20 32
|
mpd |
⊢ ( 𝜑 → ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ) |
34 |
1 8
|
lsssn0 |
⊢ ( 𝑊 ∈ LMod → { 0 } ∈ ( LSubSp ‘ 𝑊 ) ) |
35 |
7 34
|
syl |
⊢ ( 𝜑 → { 0 } ∈ ( LSubSp ‘ 𝑊 ) ) |
36 |
8 3 4 35 9
|
lcvbr |
⊢ ( 𝜑 → ( { 0 } 𝐶 𝑄 ↔ ( { 0 } ⊊ 𝑄 ∧ ¬ ∃ 𝑠 ∈ ( LSubSp ‘ 𝑊 ) ( { 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄 ) ) ) ) |
37 |
15 33 36
|
mpbir2and |
⊢ ( 𝜑 → { 0 } 𝐶 𝑄 ) |