Metamath Proof Explorer

Theorem dihmeetbN

Description: Isomorphism H of a lattice meet when one element is under the fiducial hyperplane W . (Contributed by NM, 26-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetc.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetc.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetc.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetc.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihmeetc.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihmeetbN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 dihmeetc.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetc.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetc.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 dihmeetc.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 dihmeetc.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
6 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑊 ) → 𝑋 ∈ 𝐵 )
8 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑊 ) → 𝑋 ≤ 𝑊 )
9 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑊 ) → ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) )
10 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
11 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
12 eqid ⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
13 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
14 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
15 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
16 eqid ⊢ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) = ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 )
17 eqid ⊢ ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
18 1 3 4 5 2 10 11 12 13 14 15 16 17 dihmeetlem2N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
19 6 7 8 9 18 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
20 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ¬ 𝑋 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ¬ 𝑋 ≤ 𝑊 ) → 𝑋 ∈ 𝐵 )
22 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ¬ 𝑋 ≤ 𝑊 ) → ¬ 𝑋 ≤ 𝑊 )
23 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ¬ 𝑋 ≤ 𝑊 ) → ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) )
24 1 3 4 5 2 10 11 12 13 14 15 16 17 dihmeetlem1N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
25 20 21 22 23 24 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) ∧ ¬ 𝑋 ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )
26 19 25 pm2.61dan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) )