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Metamath Proof Explorer

Theorem elpadd2at2

Description: Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012)

Ref Expression
Hypotheses paddfval.l ⊢ ≤ = ( le ‘ 𝐾 )
paddfval.j ⊢ ∨ = ( join ‘ 𝐾 )
paddfval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
paddfval.p ⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion elpadd2at2 ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑆 ∈ ( { 𝑄 } + { 𝑅 } ) ↔ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 paddfval.l ⊢ ≤ = ( le ‘ 𝐾 )
2 paddfval.j ⊢ ∨ = ( join ‘ 𝐾 )
3 paddfval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 paddfval.p ⊢ + = ( +𝑃 ‘ 𝐾 )
5 1 2 3 4 elpadd2at ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑆 ∈ ( { 𝑄 } + { 𝑅 } ) ↔ ( 𝑆 ∈ 𝐴 ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )
6 5 3adant3r3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑆 ∈ ( { 𝑄 } + { 𝑅 } ) ↔ ( 𝑆 ∈ 𝐴 ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )
7 simpr3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑆 ∈ 𝐴 )
8 7 biantrurd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑆 ∈ 𝐴 ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )
9 6 8 bitr4d ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑆 ∈ ( { 𝑄 } + { 𝑅 } ) ↔ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) )