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Metamath Proof Explorer

Theorem latabs1

Description: Lattice absorption law. From definition of lattice in Kalmbach p. 14. ( chabs1 analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses latabs1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latabs1.j ⊢ ∨ = ( join ‘ 𝐾 )
latabs1.m ⊢ ∧ = ( meet ‘ 𝐾 )
Assertion latabs1 ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ ( 𝑋 ∧ 𝑌 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 latabs1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latabs1.j ⊢ ∨ = ( join ‘ 𝐾 )
3 latabs1.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5 1 4 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 )
6 1 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
7 1 4 2 latleeqj2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝑋 ∨ ( 𝑋 ∧ 𝑌 ) ) = 𝑋 ) )
8 7 3com23 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝑋 ∨ ( 𝑋 ∧ 𝑌 ) ) = 𝑋 ) )
9 6 8 syld3an3 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝑋 ∨ ( 𝑋 ∧ 𝑌 ) ) = 𝑋 ) )
10 5 9 mpbid ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ ( 𝑋 ∧ 𝑌 ) ) = 𝑋 )