Theorem pmapmeet

Description: The projective map of a meet. (Contributed by NM, 25-Jan-2012)

	Ref	Expression
Hypotheses	pmapmeet.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pmapmeet.m	⊢ ∧ = ( meet ‘ 𝐾 )
	pmapmeet.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	pmapmeet.p	⊢ 𝑃 = ( pmap ‘ 𝐾 )
Assertion	pmapmeet	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝑃 ‘ 𝑋 ) ∩ ( 𝑃 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pmapmeet.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		pmapmeet.m	⊢ ∧ = ( meet ‘ 𝐾 )
3		pmapmeet.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		pmapmeet.p	⊢ 𝑃 = ( pmap ‘ 𝐾 )
5		eqid	⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 )
6		simp1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ HL )
7		simp2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
8		simp3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
9	5 2 6 7 8	meetval	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) = ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) )
10	9	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( 𝑃 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) )
11		prssi	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → { 𝑋 , 𝑌 } ⊆ 𝐵 )
12	11	3adant1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → { 𝑋 , 𝑌 } ⊆ 𝐵 )
13		prnzg	⊢ ( 𝑋 ∈ 𝐵 → { 𝑋 , 𝑌 } ≠ ∅ )
14	13	3ad2ant2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → { 𝑋 , 𝑌 } ≠ ∅ )
15	1 5 4	pmapglb	⊢ ( ( 𝐾 ∈ HL ∧ { 𝑋 , 𝑌 } ⊆ 𝐵 ∧ { 𝑋 , 𝑌 } ≠ ∅ ) → ( 𝑃 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) = ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) )
16	6 12 14 15	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) = ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) )
17		fveq2	⊢ ( 𝑥 = 𝑋 → ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ 𝑋 ) )
18		fveq2	⊢ ( 𝑥 = 𝑌 → ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ 𝑌 ) )
19	17 18	iinxprg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑋 ) ∩ ( 𝑃 ‘ 𝑌 ) ) )
20	19	3adant1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝑃 ‘ 𝑥 ) = ( ( 𝑃 ‘ 𝑋 ) ∩ ( 𝑃 ‘ 𝑌 ) ) )
21	10 16 20	3eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝑃 ‘ 𝑋 ) ∩ ( 𝑃 ‘ 𝑌 ) ) )