Metamath Proof Explorer

Theorem dihmeetlem11N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem9.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem9.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetlem9.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihmeetlem9.j ⊢ ∨ = ( join ‘ 𝐾 )
dihmeetlem9.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetlem9.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihmeetlem9.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihmeetlem9.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihmeetlem9.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihmeetlem11N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem9.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem9.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetlem9.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihmeetlem9.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihmeetlem9.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihmeetlem9.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihmeetlem9.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihmeetlem9.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihmeetlem9.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 1 2 3 4 5 6 7 8 9 dihmeetlem10N ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) )
11 10 ineq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
12 inass ⊢ ( ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
13 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝐾 ∈ HL )
14 13 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝐾 ∈ Lat )
15 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑌 ∈ 𝐵 )
16 simprll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑝 ∈ 𝐴 )
17 1 6 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
18 16 17 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑝 ∈ 𝐵 )
19 1 2 4 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) )
20 14 15 18 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) )
21 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22 1 4 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑌 ∨ 𝑝 ) ∈ 𝐵 )
23 14 15 18 22 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝑌 ∨ 𝑝 ) ∈ 𝐵 )
24 1 2 3 9 dihord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑌 ∨ 𝑝 ) ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ↔ 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) ) )
25 21 15 23 24 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ↔ 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) ) )
26 20 25 mpbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) )
27 sseqin2 ⊢ ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ↔ ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( 𝐼 ‘ 𝑌 ) )
28 26 27 sylib ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( 𝐼 ‘ 𝑌 ) )
29 28 ineq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ 𝑋 ) ∩ ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
30 12 29 eqtrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
31 11 30 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )