Theorem osumcllem5N

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	osumcllem5N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑝 ∈ ( 𝑋 + 𝑌 ) )

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		simp11	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝐾 ∈ HL )
10	9	hllatd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝐾 ∈ Lat )
11		simp12	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑋 ⊆ 𝐴 )
12		simp13	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑌 ⊆ 𝐴 )
13		simp31	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑟 ∈ 𝑋 )
14		simp32	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑞 ∈ 𝑌 )
15		simp2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑝 ∈ 𝐴 )
16		simp33	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) )
17	1 2 3 4	elpaddri	⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑝 ∈ ( 𝑋 + 𝑌 ) )
18	10 11 12 13 14 15 16 17	syl322anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝐴 ∧ ( 𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ ( 𝑟 ∨ 𝑞 ) ) ) → 𝑝 ∈ ( 𝑋 + 𝑌 ) )