This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem osumcllem1N

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 osumcllem1N ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑈 ∩ 𝑀 ) = 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 3 4 sspadd1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → 𝑋 ⊆ ( 𝑋 + 𝑌 ) )
10 9 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ ( 𝑋 + 𝑌 ) )
11 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝐾 ∈ HL )
12 3 4 paddssat ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) ⊆ 𝐴 )
13 12 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑋 + 𝑌 ) ⊆ 𝐴 )
14 3 5 2polssN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 + 𝑌 ) ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) ⊆ ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) )
15 11 13 14 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑋 + 𝑌 ) ⊆ ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) )
16 15 8 sseqtrrdi ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑋 + 𝑌 ) ⊆ 𝑈 )
17 10 16 sstrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ 𝑈 )
18 simpr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑝 ∈ 𝑈 )
19 18 snssd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → { 𝑝 } ⊆ 𝑈 )
20 simpl2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑋 ⊆ 𝐴 )
21 3 5 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 + 𝑌 ) ⊆ 𝐴 ) → ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ⊆ 𝐴 )
22 11 13 21 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ⊆ 𝐴 )
23 3 5 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) ⊆ 𝐴 )
24 11 22 23 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) ⊆ 𝐴 )
25 8 24 eqsstrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑈 ⊆ 𝐴 )
26 19 25 sstrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → { 𝑝 } ⊆ 𝐴 )
27 eqid ⊢ ( PSubSp ‘ 𝐾 ) = ( PSubSp ‘ 𝐾 )
28 3 27 5 polsubN ⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ⊆ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) ∈ ( PSubSp ‘ 𝐾 ) )
29 11 22 28 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑋 + 𝑌 ) ) ) ∈ ( PSubSp ‘ 𝐾 ) )
30 8 29 eqeltrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑈 ∈ ( PSubSp ‘ 𝐾 ) )
31 3 27 4 paddss ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ { 𝑝 } ⊆ 𝐴 ∧ 𝑈 ∈ ( PSubSp ‘ 𝐾 ) ) ) → ( ( 𝑋 ⊆ 𝑈 ∧ { 𝑝 } ⊆ 𝑈 ) ↔ ( 𝑋 + { 𝑝 } ) ⊆ 𝑈 ) )
32 11 20 26 30 31 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( ( 𝑋 ⊆ 𝑈 ∧ { 𝑝 } ⊆ 𝑈 ) ↔ ( 𝑋 + { 𝑝 } ) ⊆ 𝑈 ) )
33 17 19 32 mpbi2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑋 + { 𝑝 } ) ⊆ 𝑈 )
34 7 33 eqsstrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → 𝑀 ⊆ 𝑈 )
35 sseqin2 ⊢ ( 𝑀 ⊆ 𝑈 ↔ ( 𝑈 ∩ 𝑀 ) = 𝑀 )
36 34 35 sylib ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑝 ∈ 𝑈 ) → ( 𝑈 ∩ 𝑀 ) = 𝑀 )