Metamath Proof Explorer

Theorem dihjatc2N

Description: Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihjatc1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihjatc1.l ⊢ ≤ = ( le ‘ 𝐾 )
dihjatc1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjatc1.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjatc1.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihjatc1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjatc1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjatc1.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihjatc1.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion dihjatc2N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑄 ∨ ( 𝑋 ∧ 𝑌 ) ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjatc1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihjatc1.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihjatc1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihjatc1.j ⊢ ∨ = ( join ‘ 𝐾 )
5 dihjatc1.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dihjatc1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 dihjatc1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8 dihjatc1.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
9 dihjatc1.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
10 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ HL )
11 10 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
12 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ∈ 𝐴 )
13 1 6 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
14 12 13 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ∈ 𝐵 )
15 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
16 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 )
17 1 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
18 11 15 16 17 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
19 1 4 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )
20 11 14 18 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) )
21 20 fveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑄 ∨ ( 𝑋 ∧ 𝑌 ) ) ) = ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) )
22 1 2 3 4 5 6 7 8 9 dihjatc1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )
23 21 22 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑄 ∨ ( 𝑋 ∧ 𝑌 ) ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) )