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Metamath Proof Explorer

Theorem dihmeetlem12N

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 dihmeetlem12N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 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 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
12 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
13 simpr1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) )
14 simpr2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑝 ≤ 𝑋 )
15 simpr3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 )
16 1 2 3 4 5 6 7 8 9 dihmeetlem8N ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )
17 10 11 12 13 14 15 16 syl312anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )
18 17 ineq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
19 1 2 3 4 5 6 7 8 9 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 ) ) )
20 19 3adantr3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
21 simpr1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑝 ∈ 𝐴 )
22 1 2 3 4 5 6 7 8 9 dihmeetlem9N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) )
23 10 11 12 21 22 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) )
24 18 20 23 3eqtr3rd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )