Metamath Proof Explorer

Theorem olj01

Description: An ortholattice element joined with zero equals itself. ( chj0 analog.) (Contributed by NM, 19-Oct-2011)

Ref Expression
Hypotheses olj0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
olj0.j ⊢ ∨ = ( join ‘ 𝐾 )
olj0.z ⊢ 0 = ( 0. ‘ 𝐾 )
Assertion olj01 ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 0 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 olj0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 olj0.j ⊢ ∨ = ( join ‘ 𝐾 )
3 olj0.z ⊢ 0 = ( 0. ‘ 𝐾 )
4 olop ⊢ ( 𝐾 ∈ OL → 𝐾 ∈ OP )
5 1 3 op0cl ⊢ ( 𝐾 ∈ OP → 0 ∈ 𝐵 )
6 4 5 syl ⊢ ( 𝐾 ∈ OL → 0 ∈ 𝐵 )
7 6 adantr ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 0 ∈ 𝐵 )
8 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
9 ollat ⊢ ( 𝐾 ∈ OL → 𝐾 ∈ Lat )
10 9 3ad2ant1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → 𝐾 ∈ Lat )
11 1 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 ∨ 0 ) ∈ 𝐵 )
12 9 11 syl3an1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 ∨ 0 ) ∈ 𝐵 )
13 simp2 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
14 1 8 latref ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
15 9 14 sylan ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
16 15 3adant3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
17 1 8 3 op0le ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 𝑋 )
18 4 17 sylan ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 𝑋 )
19 18 3adant3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 𝑋 )
20 simp3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → 0 ∈ 𝐵 )
21 1 8 2 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑋 ∧ 0 ( le ‘ 𝐾 ) 𝑋 ) ↔ ( 𝑋 ∨ 0 ) ( le ‘ 𝐾 ) 𝑋 ) )
22 10 13 20 13 21 syl13anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑋 ∧ 0 ( le ‘ 𝐾 ) 𝑋 ) ↔ ( 𝑋 ∨ 0 ) ( le ‘ 𝐾 ) 𝑋 ) )
23 16 19 22 mpbi2and ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 ∨ 0 ) ( le ‘ 𝐾 ) 𝑋 )
24 1 8 2 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 0 ) )
25 9 24 syl3an1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 0 ) )
26 1 8 10 12 13 23 25 latasymd ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 ∨ 0 ) = 𝑋 )
27 7 26 mpd3an3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 0 ) = 𝑋 )