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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 = ( 0g ‘ 𝑈 )
Assertion dihmeetlem15N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑟 ) ∩ ( 𝐼 ‘ 𝑝 ) ) = { 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 = ( 0g ‘ 𝑈 )
11 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simpr1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) )
13 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) )
14 simpl3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ¬ 𝑝 ≤ 𝑊 )
15 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑟 = 𝑝 )
16 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑟 ≤ 𝑌 )
17 15 16 eqbrtrrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ≤ 𝑌 )
18 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝐾 ∈ HL )
19 18 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝐾 ∈ Lat )
20 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ∈ 𝐴 )
21 1 6 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
22 20 21 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ∈ 𝐵 )
23 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑌 ∈ 𝐵 )
24 1 2 5 latleeqm2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑌 ∧ 𝑝 ) = 𝑝 ) )
25 19 22 23 24 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑌 ∧ 𝑝 ) = 𝑝 ) )
26 17 25 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑌 ∧ 𝑝 ) = 𝑝 )
27 simp23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 )
28 26 27 eqbrtrrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ≤ 𝑊 )
29 28 3expia ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝑟 = 𝑝 → 𝑝 ≤ 𝑊 ) )
30 29 necon3bd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( ¬ 𝑝 ≤ 𝑊 → 𝑟 ≠ 𝑝 ) )
31 14 30 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → 𝑟 ≠ 𝑝 )
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 ⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
36 eqid ⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑟 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑟 )
37 eqid ⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑝 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑝 )
38 1 2 4 6 3 32 33 34 35 9 7 10 36 37 dihmeetlem13N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ 𝑟 ≠ 𝑝 ) → ( ( 𝐼 ‘ 𝑟 ) ∩ ( 𝐼 ‘ 𝑝 ) ) = { 0 } )
39 11 12 13 31 38 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑟 ) ∩ ( 𝐼 ‘ 𝑝 ) ) = { 0 } )