Step |
Hyp |
Ref |
Expression |
1 |
|
lsatcvat.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
2 |
|
lsatcvat.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lsatcvat.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
4 |
|
lsatcvat.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
5 |
|
lsatcvat.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
6 |
|
lsatcvat.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
7 |
|
lsatcvat.q |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
8 |
|
lsatcvat.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
9 |
|
lsatcvat.n |
⊢ ( 𝜑 → 𝑈 ≠ { 0 } ) |
10 |
|
lsatcvat.l |
⊢ ( 𝜑 → 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) ) |
11 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → 𝑊 ∈ LVec ) |
12 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → 𝑈 ∈ 𝑆 ) |
13 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → 𝑄 ∈ 𝐴 ) |
14 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → 𝑅 ∈ 𝐴 ) |
15 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → 𝑈 ≠ { 0 } ) |
16 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → ¬ 𝑄 ⊆ 𝑈 ) |
18 |
1 2 3 4 11 12 13 14 15 16 17
|
lsatcvatlem |
⊢ ( ( 𝜑 ∧ ¬ 𝑄 ⊆ 𝑈 ) → 𝑈 ∈ 𝐴 ) |
19 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → 𝑊 ∈ LVec ) |
20 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → 𝑈 ∈ 𝑆 ) |
21 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → 𝑅 ∈ 𝐴 ) |
22 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → 𝑄 ∈ 𝐴 ) |
23 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → 𝑈 ≠ { 0 } ) |
24 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
25 |
5 24
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
26 |
|
lmodabl |
⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Abel ) |
27 |
25 26
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ Abel ) |
28 |
2
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
29 |
25 28
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
30 |
2 4 25 7
|
lsatlssel |
⊢ ( 𝜑 → 𝑄 ∈ 𝑆 ) |
31 |
29 30
|
sseldd |
⊢ ( 𝜑 → 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ) |
32 |
2 4 25 8
|
lsatlssel |
⊢ ( 𝜑 → 𝑅 ∈ 𝑆 ) |
33 |
29 32
|
sseldd |
⊢ ( 𝜑 → 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ) |
34 |
3
|
lsmcom |
⊢ ( ( 𝑊 ∈ Abel ∧ 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ) → ( 𝑄 ⊕ 𝑅 ) = ( 𝑅 ⊕ 𝑄 ) ) |
35 |
27 31 33 34
|
syl3anc |
⊢ ( 𝜑 → ( 𝑄 ⊕ 𝑅 ) = ( 𝑅 ⊕ 𝑄 ) ) |
36 |
35
|
psseq2d |
⊢ ( 𝜑 → ( 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) ↔ 𝑈 ⊊ ( 𝑅 ⊕ 𝑄 ) ) ) |
37 |
10 36
|
mpbid |
⊢ ( 𝜑 → 𝑈 ⊊ ( 𝑅 ⊕ 𝑄 ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → 𝑈 ⊊ ( 𝑅 ⊕ 𝑄 ) ) |
39 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → ¬ 𝑅 ⊆ 𝑈 ) |
40 |
1 2 3 4 19 20 21 22 23 38 39
|
lsatcvatlem |
⊢ ( ( 𝜑 ∧ ¬ 𝑅 ⊆ 𝑈 ) → 𝑈 ∈ 𝐴 ) |
41 |
29 6
|
sseldd |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ) |
42 |
3
|
lsmlub |
⊢ ( ( 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝑊 ) ) → ( ( 𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈 ) ↔ ( 𝑄 ⊕ 𝑅 ) ⊆ 𝑈 ) ) |
43 |
31 33 41 42
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈 ) ↔ ( 𝑄 ⊕ 𝑅 ) ⊆ 𝑈 ) ) |
44 |
|
ssnpss |
⊢ ( ( 𝑄 ⊕ 𝑅 ) ⊆ 𝑈 → ¬ 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) ) |
45 |
43 44
|
syl6bi |
⊢ ( 𝜑 → ( ( 𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈 ) → ¬ 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) ) ) |
46 |
45
|
con2d |
⊢ ( 𝜑 → ( 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) → ¬ ( 𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈 ) ) ) |
47 |
|
ianor |
⊢ ( ¬ ( 𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈 ) ↔ ( ¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈 ) ) |
48 |
46 47
|
syl6ib |
⊢ ( 𝜑 → ( 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) → ( ¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈 ) ) ) |
49 |
10 48
|
mpd |
⊢ ( 𝜑 → ( ¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈 ) ) |
50 |
18 40 49
|
mpjaodan |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |