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 ∧ 𝑊𝐻 ) )
dihjatcclem.p ( 𝜑 → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
dihjatcclem.q ( 𝜑 → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
Assertion dihjatcclem1 ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) ( 𝐼𝑉 ) ) )

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 ∧ 𝑊𝐻 ) )
12 dihjatcclem.p ( 𝜑 → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
13 dihjatcclem.q ( 𝜑 → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
14 3 7 11 dvhlmod ( 𝜑𝑈 ∈ LMod )
15 lmodabl ( 𝑈 ∈ LMod → 𝑈 ∈ Abel )
16 14 15 syl ( 𝜑𝑈 ∈ 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 ( 𝜑𝑃𝐴 )
21 1 6 atbase ( 𝑃𝐴𝑃𝐵 )
22 20 21 syl ( 𝜑𝑃𝐵 )
23 1 3 9 7 17 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑃𝐵 ) → ( 𝐼𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
24 11 22 23 syl2anc ( 𝜑 → ( 𝐼𝑃 ) ∈ ( LSubSp ‘ 𝑈 ) )
25 19 24 sseldd ( 𝜑 → ( 𝐼𝑃 ) ∈ ( SubGrp ‘ 𝑈 ) )
26 11 simpld ( 𝜑𝐾 ∈ HL )
27 26 hllatd ( 𝜑𝐾 ∈ Lat )
28 13 simpld ( 𝜑𝑄𝐴 )
29 1 4 6 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ 𝐵 )
30 26 20 28 29 syl3anc ( 𝜑 → ( 𝑃 𝑄 ) ∈ 𝐵 )
31 11 simprd ( 𝜑𝑊𝐻 )
32 1 3 lhpbase ( 𝑊𝐻𝑊𝐵 )
33 31 32 syl ( 𝜑𝑊𝐵 )
34 1 5 latmcl ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑄 ) ∈ 𝐵𝑊𝐵 ) → ( ( 𝑃 𝑄 ) 𝑊 ) ∈ 𝐵 )
35 27 30 33 34 syl3anc ( 𝜑 → ( ( 𝑃 𝑄 ) 𝑊 ) ∈ 𝐵 )
36 10 35 eqeltrid ( 𝜑𝑉𝐵 )
37 1 3 9 7 17 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑉𝐵 ) → ( 𝐼𝑉 ) ∈ ( LSubSp ‘ 𝑈 ) )
38 11 36 37 syl2anc ( 𝜑 → ( 𝐼𝑉 ) ∈ ( LSubSp ‘ 𝑈 ) )
39 19 38 sseldd ( 𝜑 → ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) )
40 1 6 atbase ( 𝑄𝐴𝑄𝐵 )
41 28 40 syl ( 𝜑𝑄𝐵 )
42 1 3 9 7 17 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑄𝐵 ) → ( 𝐼𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
43 11 41 42 syl2anc ( 𝜑 → ( 𝐼𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
44 19 43 sseldd ( 𝜑 → ( 𝐼𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
45 8 lsm4 ( ( 𝑈 ∈ Abel ∧ ( ( 𝐼𝑃 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) ) ∧ ( ( 𝐼𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) ) ) → ( ( ( 𝐼𝑃 ) ( 𝐼𝑉 ) ) ( ( 𝐼𝑄 ) ( 𝐼𝑉 ) ) ) = ( ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) ( ( 𝐼𝑉 ) ( 𝐼𝑉 ) ) ) )
46 16 25 39 44 39 45 syl122anc ( 𝜑 → ( ( ( 𝐼𝑃 ) ( 𝐼𝑉 ) ) ( ( 𝐼𝑄 ) ( 𝐼𝑉 ) ) ) = ( ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) ( ( 𝐼𝑉 ) ( 𝐼𝑉 ) ) ) )
47 13 simprd ( 𝜑 → ¬ 𝑄 𝑊 )
48 47 intnand ( 𝜑 → ¬ ( 𝑃 𝑊𝑄 𝑊 ) )
49 1 2 4 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑃𝐵𝑄𝐵𝑊𝐵 ) ) → ( ( 𝑃 𝑊𝑄 𝑊 ) ↔ ( 𝑃 𝑄 ) 𝑊 ) )
50 27 22 41 33 49 syl13anc ( 𝜑 → ( ( 𝑃 𝑊𝑄 𝑊 ) ↔ ( 𝑃 𝑄 ) 𝑊 ) )
51 48 50 mtbid ( 𝜑 → ¬ ( 𝑃 𝑄 ) 𝑊 )
52 2 4 6 hlatlej1 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑃 ( 𝑃 𝑄 ) )
53 26 20 28 52 syl3anc ( 𝜑𝑃 ( 𝑃 𝑄 ) )
54 1 2 4 5 6 3 9 7 8 dihvalcq2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃 𝑄 ) ∈ 𝐵 ∧ ¬ ( 𝑃 𝑄 ) 𝑊 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑃 ( 𝑃 𝑄 ) ) ) → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑃 ) ( 𝐼 ‘ ( ( 𝑃 𝑄 ) 𝑊 ) ) ) )
55 11 30 51 12 53 54 syl122anc ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑃 ) ( 𝐼 ‘ ( ( 𝑃 𝑄 ) 𝑊 ) ) ) )
56 10 fveq2i ( 𝐼𝑉 ) = ( 𝐼 ‘ ( ( 𝑃 𝑄 ) 𝑊 ) )
57 56 oveq2i ( ( 𝐼𝑃 ) ( 𝐼𝑉 ) ) = ( ( 𝐼𝑃 ) ( 𝐼 ‘ ( ( 𝑃 𝑄 ) 𝑊 ) ) )
58 55 57 eqtr4di ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑃 ) ( 𝐼𝑉 ) ) )
59 2 4 6 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑄 ( 𝑃 𝑄 ) )
60 26 20 28 59 syl3anc ( 𝜑𝑄 ( 𝑃 𝑄 ) )
61 1 2 4 5 6 3 9 7 8 dihvalcq2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃 𝑄 ) ∈ 𝐵 ∧ ¬ ( 𝑃 𝑄 ) 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑄 ( 𝑃 𝑄 ) ) ) → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( ( 𝑃 𝑄 ) 𝑊 ) ) ) )
62 11 30 51 13 60 61 syl122anc ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( ( 𝑃 𝑄 ) 𝑊 ) ) ) )
63 56 oveq2i ( ( 𝐼𝑄 ) ( 𝐼𝑉 ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( ( 𝑃 𝑄 ) 𝑊 ) ) )
64 62 63 eqtr4di ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐼𝑄 ) ( 𝐼𝑉 ) ) )
65 58 64 oveq12d ( 𝜑 → ( ( 𝐼 ‘ ( 𝑃 𝑄 ) ) ( 𝐼 ‘ ( 𝑃 𝑄 ) ) ) = ( ( ( 𝐼𝑃 ) ( 𝐼𝑉 ) ) ( ( 𝐼𝑄 ) ( 𝐼𝑉 ) ) ) )
66 1 3 9 7 17 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 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 → ( ( ( 𝐼𝑃 ) ( 𝐼𝑉 ) ) ( ( 𝐼𝑄 ) ( 𝐼𝑉 ) ) ) = ( 𝐼 ‘ ( 𝑃 𝑄 ) ) )
72 8 lsmidm ( ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑈 ) → ( ( 𝐼𝑉 ) ( 𝐼𝑉 ) ) = ( 𝐼𝑉 ) )
73 39 72 syl ( 𝜑 → ( ( 𝐼𝑉 ) ( 𝐼𝑉 ) ) = ( 𝐼𝑉 ) )
74 73 oveq2d ( 𝜑 → ( ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) ( ( 𝐼𝑉 ) ( 𝐼𝑉 ) ) ) = ( ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) ( 𝐼𝑉 ) ) )
75 46 71 74 3eqtr3d ( 𝜑 → ( 𝐼 ‘ ( 𝑃 𝑄 ) ) = ( ( ( 𝐼𝑃 ) ( 𝐼𝑄 ) ) ( 𝐼𝑉 ) ) )