Metamath Proof Explorer


Theorem hlatexch3N

Description: Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013) (New usage is discouraged.)

Ref Expression
Hypotheses hlatexch4.j = ( join ‘ 𝐾 )
hlatexch4.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlatexch3N ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑅 ) )

Proof

Step Hyp Ref Expression
1 hlatexch4.j = ( join ‘ 𝐾 )
2 hlatexch4.a 𝐴 = ( Atoms ‘ 𝐾 )
3 simp1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝐾 ∈ HL )
4 simp21 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑃𝐴 )
5 simp22 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑄𝐴 )
6 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7 6 1 2 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) )
8 3 4 5 7 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) )
9 simp23 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑅𝐴 )
10 6 1 2 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴 ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 𝑅 ) )
11 3 4 9 10 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 𝑅 ) )
12 simp3r ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) )
13 11 12 breqtrrd ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) )
14 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
15 14 3ad2ant1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝐾 ∈ Lat )
16 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
17 16 2 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
18 5 17 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
19 16 2 atbase ( 𝑅𝐴𝑅 ∈ ( Base ‘ 𝐾 ) )
20 9 19 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
21 16 1 2 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
22 3 4 5 21 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
23 16 6 1 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ∧ 𝑅 ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ) ↔ ( 𝑄 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ) )
24 15 18 20 22 23 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( ( 𝑄 ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ∧ 𝑅 ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ) ↔ ( 𝑄 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ) )
25 8 13 24 mpbi2and ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( 𝑄 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) )
26 simp3l ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → 𝑄𝑅 )
27 6 1 2 ps-1 ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑄𝑅 ) ∧ ( 𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑄 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ↔ ( 𝑄 𝑅 ) = ( 𝑃 𝑄 ) ) )
28 3 5 9 26 4 5 27 syl132anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( ( 𝑄 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 𝑄 ) ↔ ( 𝑄 𝑅 ) = ( 𝑃 𝑄 ) ) )
29 25 28 mpbid ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( 𝑄 𝑅 ) = ( 𝑃 𝑄 ) )
30 29 eqcomd ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑄𝑅 ∧ ( 𝑃 𝑄 ) = ( 𝑃 𝑅 ) ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑅 ) )