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Metamath Proof Explorer

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 ) )