This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )