Metamath Proof Explorer


Theorem dihmeetlem15N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem14.b 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem14.l = ( le ‘ 𝐾 )
dihmeetlem14.h 𝐻 = ( LHyp ‘ 𝐾 )
dihmeetlem14.j = ( join ‘ 𝐾 )
dihmeetlem14.m = ( meet ‘ 𝐾 )
dihmeetlem14.a 𝐴 = ( Atoms ‘ 𝐾 )
dihmeetlem14.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihmeetlem14.s = ( LSSum ‘ 𝑈 )
dihmeetlem14.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihmeetlem15.z 0 = ( 0g𝑈 )
Assertion dihmeetlem15N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( ( 𝐼𝑟 ) ∩ ( 𝐼𝑝 ) ) = { 0 } )

Proof

Step Hyp Ref Expression
1 dihmeetlem14.b 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem14.l = ( le ‘ 𝐾 )
3 dihmeetlem14.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dihmeetlem14.j = ( join ‘ 𝐾 )
5 dihmeetlem14.m = ( meet ‘ 𝐾 )
6 dihmeetlem14.a 𝐴 = ( Atoms ‘ 𝐾 )
7 dihmeetlem14.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihmeetlem14.s = ( LSSum ‘ 𝑈 )
9 dihmeetlem14.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 dihmeetlem15.z 0 = ( 0g𝑈 )
11 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
12 simpr1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) )
13 simpl3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) )
14 simpl3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ¬ 𝑝 𝑊 )
15 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑟 = 𝑝 )
16 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑟 𝑌 )
17 15 16 eqbrtrrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 𝑌 )
18 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝐾 ∈ HL )
19 18 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝐾 ∈ Lat )
20 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝𝐴 )
21 1 6 atbase ( 𝑝𝐴𝑝𝐵 )
22 20 21 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝𝐵 )
23 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑌𝐵 )
24 1 2 5 latleeqm2 ( ( 𝐾 ∈ Lat ∧ 𝑝𝐵𝑌𝐵 ) → ( 𝑝 𝑌 ↔ ( 𝑌 𝑝 ) = 𝑝 ) )
25 19 22 23 24 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑝 𝑌 ↔ ( 𝑌 𝑝 ) = 𝑝 ) )
26 17 25 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑌 𝑝 ) = 𝑝 )
27 simp23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑌 𝑝 ) 𝑊 )
28 26 27 eqbrtrrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 𝑊 )
29 28 3expia ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( 𝑟 = 𝑝𝑝 𝑊 ) )
30 29 necon3bd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( ¬ 𝑝 𝑊𝑟𝑝 ) )
31 14 30 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → 𝑟𝑝 )
32 eqid ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
33 eqid ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
34 eqid ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
35 eqid ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
36 eqid ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑟 ) = ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑟 )
37 eqid ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑝 ) = ( ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑝 )
38 1 2 4 6 3 32 33 34 35 9 7 10 36 37 dihmeetlem13N ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ 𝑟𝑝 ) → ( ( 𝐼𝑟 ) ∩ ( 𝐼𝑝 ) ) = { 0 } )
39 11 12 13 31 38 syl121anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑌𝐵 ∧ ( 𝑝𝐴 ∧ ¬ 𝑝 𝑊 ) ) ∧ ( ( 𝑟𝐴 ∧ ¬ 𝑟 𝑊 ) ∧ 𝑟 𝑌 ∧ ( 𝑌 𝑝 ) 𝑊 ) ) → ( ( 𝐼𝑟 ) ∩ ( 𝐼𝑝 ) ) = { 0 } )