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Metamath Proof Explorer

Theorem elpaddatiN

Description: Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses paddfval.l ⊢ ≤ = ( le ‘ 𝐾 )
paddfval.j ⊢ ∨ = ( join ‘ 𝐾 )
paddfval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
paddfval.p ⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion elpaddatiN ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ 𝑅 ∈ ( 𝑋 + { 𝑄 } ) ) ) → ∃ 𝑝 ∈ 𝑋 𝑅 ≤ ( 𝑝 ∨ 𝑄 ) )

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 elpaddat ⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 ≠ ∅ ) → ( 𝑅 ∈ ( 𝑋 + { 𝑄 } ) ↔ ( 𝑅 ∈ 𝐴 ∧ ∃ 𝑝 ∈ 𝑋 𝑅 ≤ ( 𝑝 ∨ 𝑄 ) ) ) )
6 simpr ⊢ ( ( 𝑅 ∈ 𝐴 ∧ ∃ 𝑝 ∈ 𝑋 𝑅 ≤ ( 𝑝 ∨ 𝑄 ) ) → ∃ 𝑝 ∈ 𝑋 𝑅 ≤ ( 𝑝 ∨ 𝑄 ) )
7 5 6 biimtrdi ⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 ≠ ∅ ) → ( 𝑅 ∈ ( 𝑋 + { 𝑄 } ) → ∃ 𝑝 ∈ 𝑋 𝑅 ≤ ( 𝑝 ∨ 𝑄 ) ) )
8 7 impr ⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ 𝑅 ∈ ( 𝑋 + { 𝑄 } ) ) ) → ∃ 𝑝 ∈ 𝑋 𝑅 ≤ ( 𝑝 ∨ 𝑄 ) )