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Metamath Proof Explorer

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 biimtrid ⊢ ( ( ( 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 ) = ∅ )