Metamath Proof Explorer


Theorem dihjatc1

Description: Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change .\/ order of ( X ./\ Y ) .\/ Q here and down? (Contributed by NM, 6-Apr-2014)

Ref Expression
Hypotheses dihjatc1.b 𝐵 = ( Base ‘ 𝐾 )
dihjatc1.l = ( le ‘ 𝐾 )
dihjatc1.h 𝐻 = ( LHyp ‘ 𝐾 )
dihjatc1.j = ( join ‘ 𝐾 )
dihjatc1.m = ( meet ‘ 𝐾 )
dihjatc1.a 𝐴 = ( Atoms ‘ 𝐾 )
dihjatc1.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjatc1.s = ( LSSum ‘ 𝑈 )
dihjatc1.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihjatc1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑄 ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjatc1.b 𝐵 = ( Base ‘ 𝐾 )
2 dihjatc1.l = ( le ‘ 𝐾 )
3 dihjatc1.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dihjatc1.j = ( join ‘ 𝐾 )
5 dihjatc1.m = ( meet ‘ 𝐾 )
6 dihjatc1.a 𝐴 = ( Atoms ‘ 𝐾 )
7 dihjatc1.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihjatc1.s = ( LSSum ‘ 𝑈 )
9 dihjatc1.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
11 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝐾 ∈ HL )
12 11 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝐾 ∈ Lat )
13 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑋𝐵 )
14 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑌𝐵 )
15 1 5 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) ∈ 𝐵 )
16 12 13 14 15 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝑋 𝑌 ) ∈ 𝐵 )
17 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑄𝐴 )
18 1 6 atbase ( 𝑄𝐴𝑄𝐵 )
19 17 18 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑄𝐵 )
20 1 4 latjcl ( ( 𝐾 ∈ Lat ∧ ( 𝑋 𝑌 ) ∈ 𝐵𝑄𝐵 ) → ( ( 𝑋 𝑌 ) 𝑄 ) ∈ 𝐵 )
21 12 16 19 20 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( ( 𝑋 𝑌 ) 𝑄 ) ∈ 𝐵 )
22 simp2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
23 simp3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑄 𝑋 )
24 1 2 3 4 5 6 dihmeetlem6 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑄 𝑋 ) ) → ¬ ( 𝑋 ( 𝑌 𝑄 ) ) 𝑊 )
25 10 13 14 22 23 24 syl32anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ¬ ( 𝑋 ( 𝑌 𝑄 ) ) 𝑊 )
26 1 2 4 5 6 dihmeetlem5 ( ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴𝑄 𝑋 ) ) → ( 𝑋 ( 𝑌 𝑄 ) ) = ( ( 𝑋 𝑌 ) 𝑄 ) )
27 11 13 14 17 23 26 syl32anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝑋 ( 𝑌 𝑄 ) ) = ( ( 𝑋 𝑌 ) 𝑄 ) )
28 27 breq1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( ( 𝑋 ( 𝑌 𝑄 ) ) 𝑊 ↔ ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) )
29 25 28 mtbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ¬ ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 )
30 1 2 4 latlej2 ( ( 𝐾 ∈ Lat ∧ ( 𝑋 𝑌 ) ∈ 𝐵𝑄𝐵 ) → 𝑄 ( ( 𝑋 𝑌 ) 𝑄 ) )
31 12 16 19 30 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑄 ( ( 𝑋 𝑌 ) 𝑄 ) )
32 1 2 4 5 6 3 9 7 8 dihvalcq2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( ( 𝑋 𝑌 ) 𝑄 ) ∈ 𝐵 ∧ ¬ ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑄 ( ( 𝑋 𝑌 ) 𝑄 ) ) ) → ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑄 ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) ) ) )
33 10 21 29 22 31 32 syl122anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑄 ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) ) ) )
34 eqid ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
35 2 5 34 6 3 lhpmat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝑄 𝑊 ) = ( 0. ‘ 𝐾 ) )
36 10 22 35 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝑄 𝑊 ) = ( 0. ‘ 𝐾 ) )
37 36 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( ( 𝑋 𝑌 ) ( 𝑄 𝑊 ) ) = ( ( 𝑋 𝑌 ) ( 0. ‘ 𝐾 ) ) )
38 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑊𝐻 )
39 1 3 lhpbase ( 𝑊𝐻𝑊𝐵 )
40 38 39 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝑊𝐵 )
41 simp3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝑋 𝑌 ) 𝑊 )
42 1 2 4 5 6 atmod1i2 ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴 ∧ ( 𝑋 𝑌 ) ∈ 𝐵𝑊𝐵 ) ∧ ( 𝑋 𝑌 ) 𝑊 ) → ( ( 𝑋 𝑌 ) ( 𝑄 𝑊 ) ) = ( ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) )
43 11 17 16 40 41 42 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( ( 𝑋 𝑌 ) ( 𝑄 𝑊 ) ) = ( ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) )
44 hlol ( 𝐾 ∈ HL → 𝐾 ∈ OL )
45 11 44 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → 𝐾 ∈ OL )
46 1 4 34 olj01 ( ( 𝐾 ∈ OL ∧ ( 𝑋 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 𝑌 ) ( 0. ‘ 𝐾 ) ) = ( 𝑋 𝑌 ) )
47 45 16 46 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( ( 𝑋 𝑌 ) ( 0. ‘ 𝐾 ) ) = ( 𝑋 𝑌 ) )
48 37 43 47 3eqtr3d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) = ( 𝑋 𝑌 ) )
49 48 fveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝐼 ‘ ( ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) ) = ( 𝐼 ‘ ( 𝑋 𝑌 ) ) )
50 49 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( ( ( 𝑋 𝑌 ) 𝑄 ) 𝑊 ) ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) )
51 33 50 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑄 𝑋 ∧ ( 𝑋 𝑌 ) 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 𝑌 ) 𝑄 ) ) = ( ( 𝐼𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑌 ) ) ) )