Metamath Proof Explorer

Theorem dihmeetlem18N

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 )
dihmeetlem18.z ⊢ 0 = ( 0g ‘ 𝑈 )
Assertion dihmeetlem18N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( ( 𝐼 ‘ 𝑌 ) ∩ ( 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 dihmeetlem18.z ⊢ 0 = ( 0g ‘ 𝑈 )
11 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
13 simpr1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) )
14 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → 𝑌 ∈ 𝐵 )
15 simpr33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 )
16 simpr31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → 𝑝 ≤ 𝑋 )
17 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
18 1 2 3 4 5 6 7 8 9 17 dihmeetlem17N ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝑌 ∧ 𝑝 ) = ( 0. ‘ 𝐾 ) )
19 11 12 13 14 15 16 18 syl33anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝑌 ∧ 𝑝 ) = ( 0. ‘ 𝐾 ) )
20 19 fveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑝 ) ) = ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) )
21 simpr2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) )
22 simpr32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → 𝑟 ≤ 𝑌 )
23 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → 𝐾 ∈ HL )
24 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
25 23 24 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → 𝐾 ∈ OP )
26 simpl1r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → 𝑊 ∈ 𝐻 )
27 1 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
28 26 27 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → 𝑊 ∈ 𝐵 )
29 1 2 17 op0le ⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵 ) → ( 0. ‘ 𝐾 ) ≤ 𝑊 )
30 25 28 29 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 0. ‘ 𝐾 ) ≤ 𝑊 )
31 19 30 eqbrtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 )
32 1 2 3 4 5 6 7 8 9 dihmeetlem16N ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑌 ) ∩ ( 𝐼 ‘ 𝑝 ) ) )
33 11 14 13 21 22 31 32 syl33anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑌 ) ∩ ( 𝐼 ‘ 𝑝 ) ) )
34 17 3 9 7 10 dih0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) = { 0 } )
35 11 34 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) = { 0 } )
36 20 33 35 3eqtr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ) → ( ( 𝐼 ‘ 𝑌 ) ∩ ( 𝐼 ‘ 𝑝 ) ) = { 0 } )