Metamath Proof Explorer

Theorem dihjat

Description: Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014)

Ref Expression
Hypotheses dihjat.h ⊢ H = LHyp ⁡ K
dihjat.j ⊢ ∨ ˙ = join ⁡ K
dihjat.a ⊢ A = Atoms ⁡ K
dihjat.u ⊢ U = DVecH ⁡ K ⁡ W
dihjat.s ⊢ ⊕ ˙ = LSSum ⁡ U
dihjat.i ⊢ I = DIsoH ⁡ K ⁡ W
dihjat.k ⊢ φ → K ∈ HL ∧ W ∈ H
dihjat.p ⊢ φ → P ∈ A
dihjat.q ⊢ φ → Q ∈ A
Assertion dihjat ⊢ φ → I ⁡ P ∨ ˙ Q = I ⁡ P ⊕ ˙ I ⁡ Q

Proof

Step Hyp Ref Expression
1 dihjat.h ⊢ H = LHyp ⁡ K
2 dihjat.j ⊢ ∨ ˙ = join ⁡ K
3 dihjat.a ⊢ A = Atoms ⁡ K
4 dihjat.u ⊢ U = DVecH ⁡ K ⁡ W
5 dihjat.s ⊢ ⊕ ˙ = LSSum ⁡ U
6 dihjat.i ⊢ I = DIsoH ⁡ K ⁡ W
7 dihjat.k ⊢ φ → K ∈ HL ∧ W ∈ H
8 dihjat.p ⊢ φ → P ∈ A
9 dihjat.q ⊢ φ → Q ∈ A
10 eqid ⊢ ≤ K = ≤ K
11 7 adantr ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → K ∈ HL ∧ W ∈ H
12 8 adantr ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → P ∈ A
13 simprl ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → P ≤ K W
14 12 13 jca ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → P ∈ A ∧ P ≤ K W
15 9 adantr ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → Q ∈ A
16 simprr ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → Q ≤ K W
17 15 16 jca ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → Q ∈ A ∧ Q ≤ K W
18 10 1 2 3 4 5 6 11 14 17 dihjatb ⊢ φ ∧ P ≤ K W ∧ Q ≤ K W → I ⁡ P ∨ ˙ Q = I ⁡ P ⊕ ˙ I ⁡ Q
19 eqid ⊢ Base K = Base K
20 7 adantr ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → K ∈ HL ∧ W ∈ H
21 19 3 atbase ⊢ P ∈ A → P ∈ Base K
22 8 21 syl ⊢ φ → P ∈ Base K
23 22 adantr ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → P ∈ Base K
24 simprl ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → P ≤ K W
25 23 24 jca ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → P ∈ Base K ∧ P ≤ K W
26 9 adantr ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → Q ∈ A
27 simprr ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → ¬ Q ≤ K W
28 26 27 jca ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → Q ∈ A ∧ ¬ Q ≤ K W
29 19 10 1 2 3 4 5 6 20 25 28 dihjatc ⊢ φ ∧ P ≤ K W ∧ ¬ Q ≤ K W → I ⁡ P ∨ ˙ Q = I ⁡ P ⊕ ˙ I ⁡ Q
30 7 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → K ∈ HL ∧ W ∈ H
31 19 3 atbase ⊢ Q ∈ A → Q ∈ Base K
32 9 31 syl ⊢ φ → Q ∈ Base K
33 32 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → Q ∈ Base K
34 simprr ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → Q ≤ K W
35 33 34 jca ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → Q ∈ Base K ∧ Q ≤ K W
36 8 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → P ∈ A
37 simprl ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → ¬ P ≤ K W
38 36 37 jca ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → P ∈ A ∧ ¬ P ≤ K W
39 19 10 1 2 3 4 5 6 30 35 38 dihjatc ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → I ⁡ Q ∨ ˙ P = I ⁡ Q ⊕ ˙ I ⁡ P
40 7 simpld ⊢ φ → K ∈ HL
41 2 3 hlatjcom ⊢ K ∈ HL ∧ P ∈ A ∧ Q ∈ A → P ∨ ˙ Q = Q ∨ ˙ P
42 40 8 9 41 syl3anc ⊢ φ → P ∨ ˙ Q = Q ∨ ˙ P
43 42 fveq2d ⊢ φ → I ⁡ P ∨ ˙ Q = I ⁡ Q ∨ ˙ P
44 43 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → I ⁡ P ∨ ˙ Q = I ⁡ Q ∨ ˙ P
45 1 4 7 dvhlmod ⊢ φ → U ∈ LMod
46 lmodabl ⊢ U ∈ LMod → U ∈ Abel
47 45 46 syl ⊢ φ → U ∈ Abel
48 eqid ⊢ LSubSp ⁡ U = LSubSp ⁡ U
49 48 lsssssubg ⊢ U ∈ LMod → LSubSp ⁡ U ⊆ SubGrp ⁡ U
50 45 49 syl ⊢ φ → LSubSp ⁡ U ⊆ SubGrp ⁡ U
51 19 1 6 4 48 dihlss ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ Base K → I ⁡ P ∈ LSubSp ⁡ U
52 7 22 51 syl2anc ⊢ φ → I ⁡ P ∈ LSubSp ⁡ U
53 50 52 sseldd ⊢ φ → I ⁡ P ∈ SubGrp ⁡ U
54 19 1 6 4 48 dihlss ⊢ K ∈ HL ∧ W ∈ H ∧ Q ∈ Base K → I ⁡ Q ∈ LSubSp ⁡ U
55 7 32 54 syl2anc ⊢ φ → I ⁡ Q ∈ LSubSp ⁡ U
56 50 55 sseldd ⊢ φ → I ⁡ Q ∈ SubGrp ⁡ U
57 5 lsmcom ⊢ U ∈ Abel ∧ I ⁡ P ∈ SubGrp ⁡ U ∧ I ⁡ Q ∈ SubGrp ⁡ U → I ⁡ P ⊕ ˙ I ⁡ Q = I ⁡ Q ⊕ ˙ I ⁡ P
58 47 53 56 57 syl3anc ⊢ φ → I ⁡ P ⊕ ˙ I ⁡ Q = I ⁡ Q ⊕ ˙ I ⁡ P
59 58 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → I ⁡ P ⊕ ˙ I ⁡ Q = I ⁡ Q ⊕ ˙ I ⁡ P
60 39 44 59 3eqtr4d ⊢ φ ∧ ¬ P ≤ K W ∧ Q ≤ K W → I ⁡ P ∨ ˙ Q = I ⁡ P ⊕ ˙ I ⁡ Q
61 7 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → K ∈ HL ∧ W ∈ H
62 8 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → P ∈ A
63 simprl ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → ¬ P ≤ K W
64 62 63 jca ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → P ∈ A ∧ ¬ P ≤ K W
65 9 adantr ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → Q ∈ A
66 simprr ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → ¬ Q ≤ K W
67 65 66 jca ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → Q ∈ A ∧ ¬ Q ≤ K W
68 10 1 2 3 4 5 6 61 64 67 dihjatcc ⊢ φ ∧ ¬ P ≤ K W ∧ ¬ Q ≤ K W → I ⁡ P ∨ ˙ Q = I ⁡ P ⊕ ˙ I ⁡ Q
69 18 29 60 68 4casesdan ⊢ φ → I ⁡ P ∨ ˙ Q = I ⁡ P ⊕ ˙ I ⁡ Q