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 + = ( +𝑃𝐾 )
Assertion padd12N ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐴𝑌𝐴𝑍𝐴 ) ) → ( 𝑋 + ( 𝑌 + 𝑍 ) ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) )

Proof

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