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 |
|
lsatcvat.m |
⊢ ( 𝜑 → ¬ 𝑄 ⊆ 𝑈 ) |
12 |
|
lveclmod |
⊢ ( 𝑊 ∈ LVec → 𝑊 ∈ LMod ) |
13 |
5 12
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
14 |
2 1 4 13 6 9
|
lssatomic |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝑈 ) |
15 |
|
eqid |
⊢ ( ⋖L ‘ 𝑊 ) = ( ⋖L ‘ 𝑊 ) |
16 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑊 ∈ LVec ) |
17 |
13
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑊 ∈ LMod ) |
18 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ∈ 𝐴 ) |
19 |
2 4 17 18
|
lsatlssel |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ∈ 𝑆 ) |
20 |
2 4 13 7
|
lsatlssel |
⊢ ( 𝜑 → 𝑄 ∈ 𝑆 ) |
21 |
20
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑄 ∈ 𝑆 ) |
22 |
2 3
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆 ) → ( 𝑄 ⊕ 𝑥 ) ∈ 𝑆 ) |
23 |
17 21 19 22
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( 𝑄 ⊕ 𝑥 ) ∈ 𝑆 ) |
24 |
6
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑈 ∈ 𝑆 ) |
25 |
11
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ¬ 𝑄 ⊆ 𝑈 ) |
26 |
|
sseq1 |
⊢ ( 𝑥 = 𝑄 → ( 𝑥 ⊆ 𝑈 ↔ 𝑄 ⊆ 𝑈 ) ) |
27 |
26
|
biimpcd |
⊢ ( 𝑥 ⊆ 𝑈 → ( 𝑥 = 𝑄 → 𝑄 ⊆ 𝑈 ) ) |
28 |
27
|
necon3bd |
⊢ ( 𝑥 ⊆ 𝑈 → ( ¬ 𝑄 ⊆ 𝑈 → 𝑥 ≠ 𝑄 ) ) |
29 |
28
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( ¬ 𝑄 ⊆ 𝑈 → 𝑥 ≠ 𝑄 ) ) |
30 |
25 29
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ≠ 𝑄 ) |
31 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑄 ∈ 𝐴 ) |
32 |
1 4 16 18 31
|
lsatnem0 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( 𝑥 ≠ 𝑄 ↔ ( 𝑥 ∩ 𝑄 ) = { 0 } ) ) |
33 |
30 32
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( 𝑥 ∩ 𝑄 ) = { 0 } ) |
34 |
2 3 1 4 15 16 19 31
|
lcvp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( ( 𝑥 ∩ 𝑄 ) = { 0 } ↔ 𝑥 ( ⋖L ‘ 𝑊 ) ( 𝑥 ⊕ 𝑄 ) ) ) |
35 |
33 34
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ( ⋖L ‘ 𝑊 ) ( 𝑥 ⊕ 𝑄 ) ) |
36 |
|
lmodabl |
⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Abel ) |
37 |
17 36
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑊 ∈ Abel ) |
38 |
2
|
lsssssubg |
⊢ ( 𝑊 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
39 |
17 38
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) ) |
40 |
39 19
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ∈ ( SubGrp ‘ 𝑊 ) ) |
41 |
39 21
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ) |
42 |
3
|
lsmcom |
⊢ ( ( 𝑊 ∈ Abel ∧ 𝑥 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ) → ( 𝑥 ⊕ 𝑄 ) = ( 𝑄 ⊕ 𝑥 ) ) |
43 |
37 40 41 42
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( 𝑥 ⊕ 𝑄 ) = ( 𝑄 ⊕ 𝑥 ) ) |
44 |
35 43
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ( ⋖L ‘ 𝑊 ) ( 𝑄 ⊕ 𝑥 ) ) |
45 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ⊆ 𝑈 ) |
46 |
10
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑈 ⊊ ( 𝑄 ⊕ 𝑅 ) ) |
47 |
3
|
lsmub1 |
⊢ ( ( 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝑊 ) ) → 𝑄 ⊆ ( 𝑄 ⊕ 𝑥 ) ) |
48 |
41 40 47
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑄 ⊆ ( 𝑄 ⊕ 𝑥 ) ) |
49 |
8
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑅 ∈ 𝐴 ) |
50 |
10
|
pssssd |
⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑄 ⊕ 𝑅 ) ) |
51 |
50
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑈 ⊆ ( 𝑄 ⊕ 𝑅 ) ) |
52 |
45 51
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑥 ⊆ ( 𝑄 ⊕ 𝑅 ) ) |
53 |
3 4 16 18 49 31 52 30
|
lsatexch1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑅 ⊆ ( 𝑄 ⊕ 𝑥 ) ) |
54 |
2 4 13 8
|
lsatlssel |
⊢ ( 𝜑 → 𝑅 ∈ 𝑆 ) |
55 |
54
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑅 ∈ 𝑆 ) |
56 |
39 55
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ) |
57 |
39 23
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( 𝑄 ⊕ 𝑥 ) ∈ ( SubGrp ‘ 𝑊 ) ) |
58 |
3
|
lsmlub |
⊢ ( ( 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑅 ∈ ( SubGrp ‘ 𝑊 ) ∧ ( 𝑄 ⊕ 𝑥 ) ∈ ( SubGrp ‘ 𝑊 ) ) → ( ( 𝑄 ⊆ ( 𝑄 ⊕ 𝑥 ) ∧ 𝑅 ⊆ ( 𝑄 ⊕ 𝑥 ) ) ↔ ( 𝑄 ⊕ 𝑅 ) ⊆ ( 𝑄 ⊕ 𝑥 ) ) ) |
59 |
41 56 57 58
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( ( 𝑄 ⊆ ( 𝑄 ⊕ 𝑥 ) ∧ 𝑅 ⊆ ( 𝑄 ⊕ 𝑥 ) ) ↔ ( 𝑄 ⊕ 𝑅 ) ⊆ ( 𝑄 ⊕ 𝑥 ) ) ) |
60 |
48 53 59
|
mpbi2and |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → ( 𝑄 ⊕ 𝑅 ) ⊆ ( 𝑄 ⊕ 𝑥 ) ) |
61 |
46 60
|
psssstrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑈 ⊊ ( 𝑄 ⊕ 𝑥 ) ) |
62 |
2 15 16 19 23 24 44 45 61
|
lcvnbtwn3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑈 = 𝑥 ) |
63 |
62 18
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑈 ) → 𝑈 ∈ 𝐴 ) |
64 |
63
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝑈 → 𝑈 ∈ 𝐴 ) ) |
65 |
14 64
|
mpd |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |