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Metamath Proof Explorer

Theorem padd12N

Description: Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses paddass.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
paddass.p ⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion padd12N ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( 𝑋 + ( 𝑌 + 𝑍 ) ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 paddass.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2 paddass.p ⊢ + = ( +𝑃 ‘ 𝐾 )
3 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
4 3 adantr ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝐾 ∈ Lat )
5 simpr1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝑋 ⊆ 𝐴 )
6 simpr2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝑌 ⊆ 𝐴 )
7 1 2 paddcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
8 4 5 6 7 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
9 8 oveq1d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( ( 𝑌 + 𝑋 ) + 𝑍 ) )
10 1 2 paddass ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )
11 simpl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝐾 ∈ HL )
12 simpr3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝑍 ⊆ 𝐴 )
13 1 2 paddass ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑌 + 𝑋 ) + 𝑍 ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) )
14 11 6 5 12 13 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑌 + 𝑋 ) + 𝑍 ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) )
15 9 10 14 3eqtr3d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( 𝑋 + ( 𝑌 + 𝑍 ) ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) )