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	syl6bi	⊢ ( ( ( 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 ) )