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Metamath Proof Explorer

Theorem dihjatcclem1

Description: Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014)

Ref Expression
Hypotheses dihjatcclem.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihjatcclem.l ⊢ ≤ = ( le ‘ 𝐾 )
dihjatcclem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjatcclem.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjatcclem.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihjatcclem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjatcclem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjatcclem.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjatcclem.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihjatcclem.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
dihjatcclem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
dihjatcclem.p ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
dihjatcclem.q ⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
Assertion dihjatcclem1 ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( 𝐼 ‘ 𝑉 ) ) )

Proof

Step Hyp Ref Expression
1 dihjatcclem.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihjatcclem.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihjatcclem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihjatcclem.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihjatcclem.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihjatcclem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihjatcclem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihjatcclem.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihjatcclem.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 dihjatcclem.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
11 dihjatcclem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 dihjatcclem.p ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
13 dihjatcclem.q ⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
14 3 7 11 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
15 lmodabl ⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Abel )
16 14 15 syl ⊢ ( 𝜑 → 𝑈 ∈ Abel )
17 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
18 17 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
19 14 18 syl ⊢ ( 𝜑 → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
20 12 simpld ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
21 1 6 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
22 20 21 syl ⊢ ( 𝜑 → 𝑃 ∈ 𝐵 )
23 1 3 9 7 17 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
24 11 22 23 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
25 19 24 sseldd ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ ( SubGrp ‘ 𝑈 ) )
26 11 simpld ⊢ ( 𝜑 → 𝐾 ∈ HL )
27 26 hllatd ⊢ ( 𝜑 → 𝐾 ∈ Lat )
28 13 simpld ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
29 1 4 6 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
30 26 20 28 29 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 )
31 11 simprd ⊢ ( 𝜑 → 𝑊 ∈ 𝐻 )
32 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
33 31 32 syl ⊢ ( 𝜑 → 𝑊 ∈ 𝐵 )
34 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐵 )
35 27 30 33 34 syl3anc ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐵 )
36 10 35 eqeltrid ⊢ ( 𝜑 → 𝑉 ∈ 𝐵 )
37 1 3 9 7 17 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑉 ) ∈ ( LSubSp ‘ 𝑈 ) )
38 11 36 37 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ ( LSubSp ‘ 𝑈 ) )
39 19 38 sseldd ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) )
40 1 6 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
41 28 40 syl ⊢ ( 𝜑 → 𝑄 ∈ 𝐵 )
42 1 3 9 7 17 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑄 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
43 11 41 42 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
44 19 43 sseldd ⊢ ( 𝜑 → ( 𝐼 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
45 8 lsm4 ⊢ ( ( 𝑈 ∈ Abel ∧ ( ( 𝐼 ‘ 𝑃 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) ) ∧ ( ( 𝐼 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) ) ) → ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ⊕ ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) = ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) )
46 16 25 39 44 39 45 syl122anc ⊢ ( 𝜑 → ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ⊕ ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) = ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) )
47 13 simprd ⊢ ( 𝜑 → ¬ 𝑄 ≤ 𝑊 )
48 47 intnand ⊢ ( 𝜑 → ¬ ( 𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) )
49 1 2 4 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) )
50 27 22 41 33 49 syl13anc ⊢ ( 𝜑 → ( ( 𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) )
51 48 50 mtbid ⊢ ( 𝜑 → ¬ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 )
52 2 4 6 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) )
53 26 20 28 52 syl3anc ⊢ ( 𝜑 → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) )
54 1 2 4 5 6 3 9 7 8 dihvalcq2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ ¬ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
55 11 30 51 12 53 54 syl122anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
56 10 fveq2i ⊢ ( 𝐼 ‘ 𝑉 ) = ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) )
57 56 oveq2i ⊢ ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) )
58 55 57 eqtr4di ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) )
59 2 4 6 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) )
60 26 20 28 59 syl3anc ⊢ ( 𝜑 → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) )
61 1 2 4 5 6 3 9 7 8 dihvalcq2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ ¬ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
62 11 30 51 13 60 61 syl122anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
63 56 oveq2i ⊢ ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) )
64 62 63 eqtr4di ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) )
65 58 64 oveq12d ⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ⊕ ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ) = ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ⊕ ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) )
66 1 3 9 7 17 dihlss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
67 11 30 66 syl2anc ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
68 19 67 sseldd ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
69 8 lsmidm ⊢ ( ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ∈ ( SubGrp ‘ 𝑈 ) → ( ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ⊕ ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ) = ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) )
70 68 69 syl ⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ⊕ ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ) = ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) )
71 65 70 eqtr3d ⊢ ( 𝜑 → ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ⊕ ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) = ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) )
72 8 lsmidm ⊢ ( ( 𝐼 ‘ 𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) → ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) = ( 𝐼 ‘ 𝑉 ) )
73 39 72 syl ⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) = ( 𝐼 ‘ 𝑉 ) )
74 73 oveq2d ⊢ ( 𝜑 → ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( ( 𝐼 ‘ 𝑉 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) = ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( 𝐼 ‘ 𝑉 ) ) )
75 46 71 74 3eqtr3d ⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ⊕ ( 𝐼 ‘ 𝑉 ) ) )