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Metamath Proof Explorer

Theorem latnle

Description: Equivalent expressions for "not less than" in a lattice. ( chnle analog.) (Contributed by NM, 16-Nov-2011)

Ref Expression
Hypotheses latnle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latnle.l ⊢ ≤ = ( le ‘ 𝐾 )
latnle.s ⊢ < = ( lt ‘ 𝐾 )
latnle.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latnle ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ¬ 𝑌 ≤ 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 latnle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latnle.l ⊢ ≤ = ( le ‘ 𝐾 )
3 latnle.s ⊢ < = ( lt ‘ 𝐾 )
4 latnle.j ⊢ ∨ = ( join ‘ 𝐾 )
5 1 2 4 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) )
6 5 biantrurd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≠ ( 𝑋 ∨ 𝑌 ) ↔ ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑋 ≠ ( 𝑋 ∨ 𝑌 ) ) ) )
7 1 2 4 latleeqj1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑋 ↔ ( 𝑌 ∨ 𝑋 ) = 𝑋 ) )
8 7 3com23 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑋 ↔ ( 𝑌 ∨ 𝑋 ) = 𝑋 ) )
9 eqcom ⊢ ( ( 𝑌 ∨ 𝑋 ) = 𝑋 ↔ 𝑋 = ( 𝑌 ∨ 𝑋 ) )
10 8 9 bitrdi ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑋 ↔ 𝑋 = ( 𝑌 ∨ 𝑋 ) ) )
11 1 4 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
12 11 eqeq2d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = ( 𝑋 ∨ 𝑌 ) ↔ 𝑋 = ( 𝑌 ∨ 𝑋 ) ) )
13 10 12 bitr4d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑋 ↔ 𝑋 = ( 𝑋 ∨ 𝑌 ) ) )
14 13 necon3bbid ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ¬ 𝑌 ≤ 𝑋 ↔ 𝑋 ≠ ( 𝑋 ∨ 𝑌 ) ) )
15 1 4 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )
16 2 3 pltval ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 < ( 𝑋 ∨ 𝑌 ) ↔ ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑋 ≠ ( 𝑋 ∨ 𝑌 ) ) ) )
17 15 16 syld3an3 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < ( 𝑋 ∨ 𝑌 ) ↔ ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑋 ≠ ( 𝑋 ∨ 𝑌 ) ) ) )
18 6 14 17 3bitr4d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ¬ 𝑌 ≤ 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑌 ) ) )