Theorem latj12

Description: Swap 1st and 2nd members of lattice join. ( chj12 analog.) (Contributed by NM, 4-Jun-2012)

	Ref	Expression
Hypotheses	latjass.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latjass.j	⊢ ∨ = ( join ‘ 𝐾 )
Assertion	latj12	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) = ( 𝑌 ∨ ( 𝑋 ∨ 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		latjass.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latjass.j	⊢ ∨ = ( join ‘ 𝐾 )
3	1 2	latjcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
4	3	3adant3r3	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
5	4	oveq1d	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ 𝑍 ) = ( ( 𝑌 ∨ 𝑋 ) ∨ 𝑍 ) )
6	1 2	latjass	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ 𝑍 ) = ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) )
7		simpl	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
8		simpr2	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
9		simpr1	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
10		simpr3	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
11	1 2	latjass	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑌 ∨ 𝑋 ) ∨ 𝑍 ) = ( 𝑌 ∨ ( 𝑋 ∨ 𝑍 ) ) )
12	7 8 9 10 11	syl13anc	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑌 ∨ 𝑋 ) ∨ 𝑍 ) = ( 𝑌 ∨ ( 𝑋 ∨ 𝑍 ) ) )
13	5 6 12	3eqtr3d	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) = ( 𝑌 ∨ ( 𝑋 ∨ 𝑍 ) ) )