Metamath Proof Explorer


Theorem lsatcvatlem

Description: Lemma for lsatcvat . (Contributed by NM, 10-Jan-2015)

Ref Expression
Hypotheses lsatcvat.o 0 = ( 0g𝑊 )
lsatcvat.s 𝑆 = ( LSubSp ‘ 𝑊 )
lsatcvat.p = ( LSSum ‘ 𝑊 )
lsatcvat.a 𝐴 = ( LSAtoms ‘ 𝑊 )
lsatcvat.w ( 𝜑𝑊 ∈ LVec )
lsatcvat.u ( 𝜑𝑈𝑆 )
lsatcvat.q ( 𝜑𝑄𝐴 )
lsatcvat.r ( 𝜑𝑅𝐴 )
lsatcvat.n ( 𝜑𝑈 ≠ { 0 } )
lsatcvat.l ( 𝜑𝑈 ⊊ ( 𝑄 𝑅 ) )
lsatcvat.m ( 𝜑 → ¬ 𝑄𝑈 )
Assertion lsatcvatlem ( 𝜑𝑈𝐴 )

Proof

Step Hyp Ref Expression
1 lsatcvat.o 0 = ( 0g𝑊 )
2 lsatcvat.s 𝑆 = ( LSubSp ‘ 𝑊 )
3 lsatcvat.p = ( LSSum ‘ 𝑊 )
4 lsatcvat.a 𝐴 = ( LSAtoms ‘ 𝑊 )
5 lsatcvat.w ( 𝜑𝑊 ∈ LVec )
6 lsatcvat.u ( 𝜑𝑈𝑆 )
7 lsatcvat.q ( 𝜑𝑄𝐴 )
8 lsatcvat.r ( 𝜑𝑅𝐴 )
9 lsatcvat.n ( 𝜑𝑈 ≠ { 0 } )
10 lsatcvat.l ( 𝜑𝑈 ⊊ ( 𝑄 𝑅 ) )
11 lsatcvat.m ( 𝜑 → ¬ 𝑄𝑈 )
12 lveclmod ( 𝑊 ∈ LVec → 𝑊 ∈ LMod )
13 5 12 syl ( 𝜑𝑊 ∈ LMod )
14 2 1 4 13 6 9 lssatomic ( 𝜑 → ∃ 𝑥𝐴 𝑥𝑈 )
15 eqid ( ⋖L𝑊 ) = ( ⋖L𝑊 )
16 5 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑊 ∈ LVec )
17 13 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑊 ∈ LMod )
18 simp2 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥𝐴 )
19 2 4 17 18 lsatlssel ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥𝑆 )
20 2 4 13 7 lsatlssel ( 𝜑𝑄𝑆 )
21 20 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑄𝑆 )
22 2 3 lsmcl ( ( 𝑊 ∈ LMod ∧ 𝑄𝑆𝑥𝑆 ) → ( 𝑄 𝑥 ) ∈ 𝑆 )
23 17 21 19 22 syl3anc ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( 𝑄 𝑥 ) ∈ 𝑆 )
24 6 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑈𝑆 )
25 11 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ¬ 𝑄𝑈 )
26 sseq1 ( 𝑥 = 𝑄 → ( 𝑥𝑈𝑄𝑈 ) )
27 26 biimpcd ( 𝑥𝑈 → ( 𝑥 = 𝑄𝑄𝑈 ) )
28 27 necon3bd ( 𝑥𝑈 → ( ¬ 𝑄𝑈𝑥𝑄 ) )
29 28 3ad2ant3 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( ¬ 𝑄𝑈𝑥𝑄 ) )
30 25 29 mpd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥𝑄 )
31 7 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑄𝐴 )
32 1 4 16 18 31 lsatnem0 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( 𝑥𝑄 ↔ ( 𝑥𝑄 ) = { 0 } ) )
33 30 32 mpbid ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( 𝑥𝑄 ) = { 0 } )
34 2 3 1 4 15 16 19 31 lcvp ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( ( 𝑥𝑄 ) = { 0 } ↔ 𝑥 ( ⋖L𝑊 ) ( 𝑥 𝑄 ) ) )
35 33 34 mpbid ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥 ( ⋖L𝑊 ) ( 𝑥 𝑄 ) )
36 lmodabl ( 𝑊 ∈ LMod → 𝑊 ∈ Abel )
37 17 36 syl ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑊 ∈ Abel )
38 2 lsssssubg ( 𝑊 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) )
39 17 38 syl ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑆 ⊆ ( SubGrp ‘ 𝑊 ) )
40 39 19 sseldd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥 ∈ ( SubGrp ‘ 𝑊 ) )
41 39 21 sseldd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑄 ∈ ( SubGrp ‘ 𝑊 ) )
42 3 lsmcom ( ( 𝑊 ∈ Abel ∧ 𝑥 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ) → ( 𝑥 𝑄 ) = ( 𝑄 𝑥 ) )
43 37 40 41 42 syl3anc ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( 𝑥 𝑄 ) = ( 𝑄 𝑥 ) )
44 35 43 breqtrd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥 ( ⋖L𝑊 ) ( 𝑄 𝑥 ) )
45 simp3 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥𝑈 )
46 10 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑈 ⊊ ( 𝑄 𝑅 ) )
47 3 lsmub1 ( ( 𝑄 ∈ ( SubGrp ‘ 𝑊 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝑊 ) ) → 𝑄 ⊆ ( 𝑄 𝑥 ) )
48 41 40 47 syl2anc ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑄 ⊆ ( 𝑄 𝑥 ) )
49 8 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑅𝐴 )
50 10 pssssd ( 𝜑𝑈 ⊆ ( 𝑄 𝑅 ) )
51 50 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑈 ⊆ ( 𝑄 𝑅 ) )
52 45 51 sstrd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑥 ⊆ ( 𝑄 𝑅 ) )
53 3 4 16 18 49 31 52 30 lsatexch1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑅 ⊆ ( 𝑄 𝑥 ) )
54 2 4 13 8 lsatlssel ( 𝜑𝑅𝑆 )
55 54 3ad2ant1 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑅𝑆 )
56 39 55 sseldd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑅 ∈ ( SubGrp ‘ 𝑊 ) )
57 39 23 sseldd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( 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 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( ( 𝑄 ⊆ ( 𝑄 𝑥 ) ∧ 𝑅 ⊆ ( 𝑄 𝑥 ) ) ↔ ( 𝑄 𝑅 ) ⊆ ( 𝑄 𝑥 ) ) )
60 48 53 59 mpbi2and ( ( 𝜑𝑥𝐴𝑥𝑈 ) → ( 𝑄 𝑅 ) ⊆ ( 𝑄 𝑥 ) )
61 46 60 psssstrd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑈 ⊊ ( 𝑄 𝑥 ) )
62 2 15 16 19 23 24 44 45 61 lcvnbtwn3 ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑈 = 𝑥 )
63 62 18 eqeltrd ( ( 𝜑𝑥𝐴𝑥𝑈 ) → 𝑈𝐴 )
64 63 rexlimdv3a ( 𝜑 → ( ∃ 𝑥𝐴 𝑥𝑈𝑈𝐴 ) )
65 14 64 mpd ( 𝜑𝑈𝐴 )