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Metamath Proof Explorer

Theorem latnlemlt

Description: Negation of "less than or equal to" expressed in terms of meet and less-than. ( nssinpss analog.) (Contributed by NM, 5-Feb-2012)

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

Proof

Step Hyp Ref Expression
1 latnlemlt.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latnlemlt.l ⊢ ≤ = ( le ‘ 𝐾 )
3 latnlemlt.s ⊢ < = ( lt ‘ 𝐾 )
4 latnlemlt.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 1 2 4 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 )
6 5 biantrurd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ≠ 𝑋 ↔ ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≠ 𝑋 ) ) )
7 1 2 4 latleeqm1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∧ 𝑌 ) = 𝑋 ) )
8 7 necon3bbid ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ¬ 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∧ 𝑌 ) ≠ 𝑋 ) )
9 simp1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat )
10 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
11 simp2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
12 2 3 pltval ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) < 𝑋 ↔ ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≠ 𝑋 ) ) )
13 9 10 11 12 syl3anc ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) < 𝑋 ↔ ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≠ 𝑋 ) ) )
14 6 8 13 3bitr4d ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ¬ 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∧ 𝑌 ) < 𝑋 ) )