atmod3i2

Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012) (Revised by Mario Carneiro, 10-May-2013)

	Ref	Expression
Hypotheses	atmod.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	atmod.l	⊢ ≤ = ( le ‘ 𝐾 )
	atmod.j	⊢ ∨ = ( join ‘ 𝐾 )
	atmod.m	⊢ ∧ = ( meet ‘ 𝐾 )
	atmod.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	atmod3i2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( 𝑌 ∧ 𝑃 ) ) = ( 𝑌 ∧ ( 𝑋 ∨ 𝑃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		atmod.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		atmod.l	⊢ ≤ = ( le ‘ 𝐾 )
3		atmod.j	⊢ ∨ = ( join ‘ 𝐾 )
4		atmod.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		atmod.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
7	6	3ad2ant1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝐾 ∈ Lat )
8		simp23	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑌 ∈ 𝐵 )
9		simp22	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑋 ∈ 𝐵 )
10		simp21	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑃 ∈ 𝐴 )
11	1 5	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 )
12	10 11	syl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑃 ∈ 𝐵 )
13	1 3	latjcl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 )
14	7 9 12 13	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 )
15	1 4	latmcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 ) → ( 𝑌 ∧ ( 𝑋 ∨ 𝑃 ) ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) )
16	7 8 14 15	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑌 ∧ ( 𝑋 ∨ 𝑃 ) ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) )
17	1 2 3 4 5	atmod1i2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( 𝑃 ∧ 𝑌 ) ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) )
18	1 4	latmcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ∧ 𝑌 ) = ( 𝑌 ∧ 𝑃 ) )
19	7 12 8 18	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑃 ∧ 𝑌 ) = ( 𝑌 ∧ 𝑃 ) )
20	19	oveq2d	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( 𝑃 ∧ 𝑌 ) ) = ( 𝑋 ∨ ( 𝑌 ∧ 𝑃 ) ) )
21	16 17 20	3eqtr2rd	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( 𝑌 ∧ 𝑃 ) ) = ( 𝑌 ∧ ( 𝑋 ∨ 𝑃 ) ) )

Metamath Proof Explorer

Theorem atmod3i2

Proof