Theorem pmaplubN

Description: The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pmaplub.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pmaplub.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
	pmaplub.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
Assertion	pmaplubN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ ( 𝑀 ‘ 𝑋 ) ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		pmaplub.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		pmaplub.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
3		pmaplub.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
4		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6	1 4 5 3	pmapval	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) = { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } )
7	6	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ ( 𝑀 ‘ 𝑋 ) ) = ( 𝑈 ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) )
8		hlomcmat	⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) )
9	1 4 2 5	atlatmstc	⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) = 𝑋 )
10	8 9	sylan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ { 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∣ 𝑝 ( le ‘ 𝐾 ) 𝑋 } ) = 𝑋 )
11	7 10	eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑈 ‘ ( 𝑀 ‘ 𝑋 ) ) = 𝑋 )