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Metamath Proof Explorer

Theorem paddasslem11

Description: Lemma for paddass . The case when p = z . (Contributed by NM, 11-Jan-2012)

Ref Expression
Hypotheses paddasslem.l ⊢ ≤ = ( le ‘ 𝐾 )
paddasslem.j ⊢ ∨ = ( join ‘ 𝐾 )
paddasslem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
paddasslem.p ⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion paddasslem11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑝 ∈ ( ( 𝑋 + 𝑌 ) + 𝑍 ) )

Proof

Step Hyp Ref Expression
1 paddasslem.l ⊢ ≤ = ( le ‘ 𝐾 )
2 paddasslem.j ⊢ ∨ = ( join ‘ 𝐾 )
3 paddasslem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 paddasslem.p ⊢ + = ( +𝑃 ‘ 𝐾 )
5 simplll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝐾 ∈ HL )
6 simplr3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑍 ⊆ 𝐴 )
7 simplr1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑋 ⊆ 𝐴 )
8 simplr2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑌 ⊆ 𝐴 )
9 3 4 paddssat ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) ⊆ 𝐴 )
10 5 7 8 9 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → ( 𝑋 + 𝑌 ) ⊆ 𝐴 )
11 3 4 sspadd2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑍 ⊆ 𝐴 ∧ ( 𝑋 + 𝑌 ) ⊆ 𝐴 ) → 𝑍 ⊆ ( ( 𝑋 + 𝑌 ) + 𝑍 ) )
12 5 6 10 11 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑍 ⊆ ( ( 𝑋 + 𝑌 ) + 𝑍 ) )
13 simpllr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑝 = 𝑧 )
14 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑧 ∈ 𝑍 )
15 13 14 eqeltrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑝 ∈ 𝑍 )
16 12 15 sseldd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑝 = 𝑧 ) ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) ∧ 𝑧 ∈ 𝑍 ) → 𝑝 ∈ ( ( 𝑋 + 𝑌 ) + 𝑍 ) )