Metamath Proof Explorer


Theorem latjlej12

Description: Add join to both sides of a lattice ordering. ( chlej12i analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses latlej.b 𝐵 = ( Base ‘ 𝐾 )
latlej.l = ( le ‘ 𝐾 )
latlej.j = ( join ‘ 𝐾 )
Assertion latjlej12 ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( ( 𝑋 𝑌𝑍 𝑊 ) → ( 𝑋 𝑍 ) ( 𝑌 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 latlej.b 𝐵 = ( Base ‘ 𝐾 )
2 latlej.l = ( le ‘ 𝐾 )
3 latlej.j = ( join ‘ 𝐾 )
4 simp1 ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → 𝐾 ∈ Lat )
5 simp2l ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → 𝑋𝐵 )
6 simp2r ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → 𝑌𝐵 )
7 simp3l ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → 𝑍𝐵 )
8 1 2 3 latjlej1 ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 𝑌 → ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ) )
9 4 5 6 7 8 syl13anc ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( 𝑋 𝑌 → ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ) )
10 simp3r ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → 𝑊𝐵 )
11 1 2 3 latjlej2 ( ( 𝐾 ∈ Lat ∧ ( 𝑍𝐵𝑊𝐵𝑌𝐵 ) ) → ( 𝑍 𝑊 → ( 𝑌 𝑍 ) ( 𝑌 𝑊 ) ) )
12 4 7 10 6 11 syl13anc ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( 𝑍 𝑊 → ( 𝑌 𝑍 ) ( 𝑌 𝑊 ) ) )
13 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵 ) → ( 𝑋 𝑍 ) ∈ 𝐵 )
14 4 5 7 13 syl3anc ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( 𝑋 𝑍 ) ∈ 𝐵 )
15 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵 ) → ( 𝑌 𝑍 ) ∈ 𝐵 )
16 4 6 7 15 syl3anc ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( 𝑌 𝑍 ) ∈ 𝐵 )
17 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵 ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
18 4 6 10 17 syl3anc ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
19 1 2 lattr ( ( 𝐾 ∈ Lat ∧ ( ( 𝑋 𝑍 ) ∈ 𝐵 ∧ ( 𝑌 𝑍 ) ∈ 𝐵 ∧ ( 𝑌 𝑊 ) ∈ 𝐵 ) ) → ( ( ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ∧ ( 𝑌 𝑍 ) ( 𝑌 𝑊 ) ) → ( 𝑋 𝑍 ) ( 𝑌 𝑊 ) ) )
20 4 14 16 18 19 syl13anc ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( ( ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ∧ ( 𝑌 𝑍 ) ( 𝑌 𝑊 ) ) → ( 𝑋 𝑍 ) ( 𝑌 𝑊 ) ) )
21 9 12 20 syl2and ( ( 𝐾 ∈ Lat ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑊𝐵 ) ) → ( ( 𝑋 𝑌𝑍 𝑊 ) → ( 𝑋 𝑍 ) ( 𝑌 𝑊 ) ) )