Theorem odudlatb

Description: The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015)

	Ref	Expression
Hypothesis	odudlat.d	⊢ 𝐷 = ( ODual ‘ 𝐾 )
Assertion	odudlatb	⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ DLat ↔ 𝐷 ∈ DLat ) )

Proof

Step	Hyp	Ref	Expression
1		odudlat.d	⊢ 𝐷 = ( ODual ‘ 𝐾 )
2		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
3		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
4		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
5	2 3 4	latdisd	⊢ ( 𝐾 ∈ Lat → ( ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( join ‘ 𝐾 ) 𝑦 ) ( meet ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) 𝑧 ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) ( 𝑦 ( join ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( meet ‘ 𝐾 ) 𝑦 ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑧 ) ) ) )
6	5	bicomd	⊢ ( 𝐾 ∈ Lat → ( ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) ( 𝑦 ( join ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( meet ‘ 𝐾 ) 𝑦 ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑧 ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( join ‘ 𝐾 ) 𝑦 ) ( meet ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) 𝑧 ) ) ) )
7	6	pm5.32i	⊢ ( ( 𝐾 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) ( 𝑦 ( join ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( meet ‘ 𝐾 ) 𝑦 ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑧 ) ) ) ↔ ( 𝐾 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( join ‘ 𝐾 ) 𝑦 ) ( meet ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) 𝑧 ) ) ) )
8	1	odulatb	⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ Lat ↔ 𝐷 ∈ Lat ) )
9	8	anbi1d	⊢ ( 𝐾 ∈ 𝑉 → ( ( 𝐾 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( join ‘ 𝐾 ) 𝑦 ) ( meet ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) 𝑧 ) ) ) ↔ ( 𝐷 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( join ‘ 𝐾 ) 𝑦 ) ( meet ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) 𝑧 ) ) ) ) )
10	7 9	syl5bb	⊢ ( 𝐾 ∈ 𝑉 → ( ( 𝐾 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) ( 𝑦 ( join ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( meet ‘ 𝐾 ) 𝑦 ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑧 ) ) ) ↔ ( 𝐷 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( join ‘ 𝐾 ) 𝑦 ) ( meet ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) 𝑧 ) ) ) ) )
11	2 3 4	isdlat	⊢ ( 𝐾 ∈ DLat ↔ ( 𝐾 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) ( 𝑦 ( join ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( meet ‘ 𝐾 ) 𝑦 ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑧 ) ) ) )
12	1 2	odubas	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐷 )
13	1 4	odujoin	⊢ ( meet ‘ 𝐾 ) = ( join ‘ 𝐷 )
14	1 3	odumeet	⊢ ( join ‘ 𝐾 ) = ( meet ‘ 𝐷 )
15	12 13 14	isdlat	⊢ ( 𝐷 ∈ DLat ↔ ( 𝐷 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) ( 𝑦 ( meet ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( join ‘ 𝐾 ) 𝑦 ) ( meet ‘ 𝐾 ) ( 𝑥 ( join ‘ 𝐾 ) 𝑧 ) ) ) )
16	10 11 15	3bitr4g	⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ DLat ↔ 𝐷 ∈ DLat ) )