Metamath Proof Explorer


Theorem lsatcvat

Description: A nonzero subspace less than the sum of two atoms is an atom. ( atcvati analog.) (Contributed by NM, 10-Jan-2015)

Ref Expression
Hypotheses lsatcvat.o 0 ˙ = 0 W
lsatcvat.s S = LSubSp W
lsatcvat.p ˙ = LSSum W
lsatcvat.a A = LSAtoms W
lsatcvat.w φ W LVec
lsatcvat.u φ U S
lsatcvat.q φ Q A
lsatcvat.r φ R A
lsatcvat.n φ U 0 ˙
lsatcvat.l φ U Q ˙ R
Assertion lsatcvat φ U A

Proof

Step Hyp Ref Expression
1 lsatcvat.o 0 ˙ = 0 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