Metamath Proof Explorer


Theorem atmod1i2

Description: Version of modular law pmod1i that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-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 atmod1i2 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑋 𝑌 ) → ( 𝑋 ( 𝑃 𝑌 ) ) = ( ( 𝑋 𝑃 ) 𝑌 ) )

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 simpl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → 𝐾 ∈ HL )
7 simpr2 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → 𝑋𝐵 )
8 simpr1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → 𝑃𝐴 )
9 eqid ( pmap ‘ 𝐾 ) = ( pmap ‘ 𝐾 )
10 eqid ( +𝑃𝐾 ) = ( +𝑃𝐾 )
11 1 3 5 9 10 pmapjat1 ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 𝑃 ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ 𝑃 ) ) )
12 6 7 8 11 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 𝑃 ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ 𝑃 ) ) )
13 1 5 atbase ( 𝑃𝐴𝑃𝐵 )
14 8 13 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → 𝑃𝐵 )
15 simpr3 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → 𝑌𝐵 )
16 1 2 3 4 9 10 hlmod1i ( ( 𝐾 ∈ HL ∧ ( 𝑋𝐵𝑃𝐵𝑌𝐵 ) ) → ( ( 𝑋 𝑌 ∧ ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 𝑃 ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ 𝑃 ) ) ) → ( ( 𝑋 𝑃 ) 𝑌 ) = ( 𝑋 ( 𝑃 𝑌 ) ) ) )
17 6 7 14 15 16 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → ( ( 𝑋 𝑌 ∧ ( ( pmap ‘ 𝐾 ) ‘ ( 𝑋 𝑃 ) ) = ( ( ( pmap ‘ 𝐾 ) ‘ 𝑋 ) ( +𝑃𝐾 ) ( ( pmap ‘ 𝐾 ) ‘ 𝑃 ) ) ) → ( ( 𝑋 𝑃 ) 𝑌 ) = ( 𝑋 ( 𝑃 𝑌 ) ) ) )
18 12 17 mpan2d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝑌 → ( ( 𝑋 𝑃 ) 𝑌 ) = ( 𝑋 ( 𝑃 𝑌 ) ) ) )
19 18 3impia ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑋 𝑌 ) → ( ( 𝑋 𝑃 ) 𝑌 ) = ( 𝑋 ( 𝑃 𝑌 ) ) )
20 19 eqcomd ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑋𝐵𝑌𝐵 ) ∧ 𝑋 𝑌 ) → ( 𝑋 ( 𝑃 𝑌 ) ) = ( ( 𝑋 𝑃 ) 𝑌 ) )