Step |
Hyp |
Ref |
Expression |
1 |
|
lsatcmp2.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
2 |
|
lsatcmp2.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
3 |
|
lsatcmp2.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
4 |
|
lsatcmp2.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝐴 ) |
5 |
|
lsatcmp2.u |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐴 ∨ 𝑈 = { 0 } ) ) |
6 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → 𝑇 ⊆ 𝑈 ) |
7 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → 𝑊 ∈ LVec ) |
8 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → 𝑇 ∈ 𝐴 ) |
9 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
10 |
3 9
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → 𝑊 ∈ LMod ) |
12 |
1 2 11 8 6
|
lsatssn0 |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → 𝑈 ≠ { 0 } ) |
13 |
5
|
ord |
⊢ ( 𝜑 → ( ¬ 𝑈 ∈ 𝐴 → 𝑈 = { 0 } ) ) |
14 |
13
|
necon1ad |
⊢ ( 𝜑 → ( 𝑈 ≠ { 0 } → 𝑈 ∈ 𝐴 ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → ( 𝑈 ≠ { 0 } → 𝑈 ∈ 𝐴 ) ) |
16 |
12 15
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → 𝑈 ∈ 𝐴 ) |
17 |
2 7 8 16
|
lsatcmp |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → ( 𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈 ) ) |
18 |
6 17
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑇 ⊆ 𝑈 ) → 𝑇 = 𝑈 ) |
19 |
18
|
ex |
⊢ ( 𝜑 → ( 𝑇 ⊆ 𝑈 → 𝑇 = 𝑈 ) ) |
20 |
|
eqimss |
⊢ ( 𝑇 = 𝑈 → 𝑇 ⊆ 𝑈 ) |
21 |
19 20
|
impbid1 |
⊢ ( 𝜑 → ( 𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈 ) ) |