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Metamath Proof Explorer

Theorem 2pmaplubN

Description: Double projective map of an LUB. (Contributed by NM, 6-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses sspmaplub.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
sspmaplub.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
sspmaplub.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
Assertion 2pmaplubN ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( 𝑀 ‘ ( 𝑈 ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) = ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) )

Proof

Step Hyp Ref Expression
1 sspmaplub.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
2 sspmaplub.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 sspmaplub.m ⊢ 𝑀 = ( pmap ‘ 𝐾 )
4 eqid ⊢ ( ⊥𝑃 ‘ 𝐾 ) = ( ⊥𝑃 ‘ 𝐾 )
5 1 2 3 4 2polvalN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) = ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) )
6 5 fveq2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) ) = ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) )
7 6 fveq2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) ) ) = ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) )
8 2 4 polssatN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ⊆ 𝐴 )
9 2 4 3polN ⊢ ( ( 𝐾 ∈ HL ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) ) ) = ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) )
10 8 9 syldan ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) ) ) = ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) )
11 7 10 eqtr3d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) = ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) )
12 hlclat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat )
13 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 2 atssbase ⊢ 𝐴 ⊆ ( Base ‘ 𝐾 )
15 sstr ⊢ ( ( 𝑆 ⊆ 𝐴 ∧ 𝐴 ⊆ ( Base ‘ 𝐾 ) ) → 𝑆 ⊆ ( Base ‘ 𝐾 ) )
16 14 15 mpan2 ⊢ ( 𝑆 ⊆ 𝐴 → 𝑆 ⊆ ( Base ‘ 𝐾 ) )
17 13 1 clatlubcl ⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ ( Base ‘ 𝐾 ) ) → ( 𝑈 ‘ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
18 12 16 17 syl2an ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( 𝑈 ‘ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
19 13 2 3 pmapssat ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑈 ‘ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ⊆ 𝐴 )
20 18 19 syldan ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ⊆ 𝐴 )
21 1 2 3 4 2polvalN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) = ( 𝑀 ‘ ( 𝑈 ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) )
22 20 21 syldan ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) = ( 𝑀 ‘ ( 𝑈 ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) )
23 11 22 eqtr3d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑆 ) ) = ( 𝑀 ‘ ( 𝑈 ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) )
24 23 5 eqtr3d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ) → ( 𝑀 ‘ ( 𝑈 ‘ ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) ) ) = ( 𝑀 ‘ ( 𝑈 ‘ 𝑆 ) ) )