Theorem osumcllem8N

Description: Lemma for osumclN . (Contributed by NM, 24-Mar-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	osumcllem.l	⊢ ≤ = ( le ‘ 𝐾 )
	osumcllem.j	⊢ ∨ = ( join ‘ 𝐾 )
	osumcllem.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	osumcllem.p	⊢ + = ( +𝑃 ‘ 𝐾 )
	osumcllem.o	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
	osumcllem.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
	osumcllem.m	⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
	osumcllem.u	⊢ 𝑈 = ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) )
Assertion	osumcllem8N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴 ) ∧ ¬ 𝑝 ∈ ( 𝑋 + 𝑌 ) ) → ( 𝑌 ∩ 𝑀 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		osumcllem.l	⊢ ≤ = ( le ‘ 𝐾 )
2		osumcllem.j	⊢ ∨ = ( join ‘ 𝐾 )
3		osumcllem.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		osumcllem.p	⊢ + = ( +𝑃 ‘ 𝐾 )
5		osumcllem.o	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
6		osumcllem.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
7		osumcllem.m	⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
8		osumcllem.u	⊢ 𝑈 = ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) )
9		n0	⊢ ( ( 𝑌 ∩ 𝑀 ) ≠ ∅ ↔ ∃ 𝑞 𝑞 ∈ ( 𝑌 ∩ 𝑀 ) )
10	1 2 3 4 5 6 7 8	osumcllem7N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑞 ∈ ( 𝑌 ∩ 𝑀 ) ) → 𝑝 ∈ ( 𝑋 + 𝑌 ) )
11	10	3expia	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴 ) ) → ( 𝑞 ∈ ( 𝑌 ∩ 𝑀 ) → 𝑝 ∈ ( 𝑋 + 𝑌 ) ) )
12	11	exlimdv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴 ) ) → ( ∃ 𝑞 𝑞 ∈ ( 𝑌 ∩ 𝑀 ) → 𝑝 ∈ ( 𝑋 + 𝑌 ) ) )
13	9 12	syl5bi	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴 ) ) → ( ( 𝑌 ∩ 𝑀 ) ≠ ∅ → 𝑝 ∈ ( 𝑋 + 𝑌 ) ) )
14	13	necon1bd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴 ) ) → ( ¬ 𝑝 ∈ ( 𝑋 + 𝑌 ) → ( 𝑌 ∩ 𝑀 ) = ∅ ) )
15	14	3impia	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴 ) ∧ ¬ 𝑝 ∈ ( 𝑋 + 𝑌 ) ) → ( 𝑌 ∩ 𝑀 ) = ∅ )