Metamath Proof Explorer

Theorem osumcllem2N

Description: Lemma for osumclN . (Contributed by NM, 25-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 osumcllem2N ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ ( 𝑈 ∩ 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 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝐾 ∈ HL )
10 simpl2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ 𝐴 )
11 simpr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑝 ∈ 𝑈 )
12 11 snssd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → { 𝑝 } ⊆ 𝑈 )
13 3 4 paddssat ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) ⊆ 𝐴 )
14 13 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑋 + 𝑌 ) ⊆ 𝐴 )
15 3 5 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 + 𝑌 ) ⊆ 𝐴 ) → ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ⊆ 𝐴 )
16 9 14 15 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ⊆ 𝐴 )
17 3 5 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) ⊆ 𝐴 )
18 9 16 17 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) ⊆ 𝐴 )
19 8 18 eqsstrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑈 ⊆ 𝐴 )
20 12 19 sstrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → { 𝑝 } ⊆ 𝐴 )
21 3 4 sspadd1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ { 𝑝 } ⊆ 𝐴 ) → 𝑋 ⊆ ( 𝑋 + { 𝑝 } ) )
22 9 10 20 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ ( 𝑋 + { 𝑝 } ) )
23 22 7 sseqtrrdi ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ 𝑀 )
24 1 2 3 4 5 6 7 8 osumcllem1N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑈 ∩ 𝑀 ) = 𝑀 )
25 23 24 sseqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ ( 𝑈 ∩ 𝑀 ) )