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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 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 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 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝐾 ∈ HL )
4 simp21 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑃 ∈ 𝐴 )
5 simp22 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ∈ 𝐴 )
6 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7 6 1 2 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) )
8 3 4 5 7 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) )
9 simp23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ∈ 𝐴 )
10 6 1 2 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) )
11 3 4 9 10 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) )
12 simp3r ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) )
13 11 12 breqtrrd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) )
14 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
15 14 3ad2ant1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝐾 ∈ Lat )
16 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
17 16 2 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
18 5 17 syl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
19 16 2 atbase ⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
20 9 19 syl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
21 16 1 2 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
22 3 4 5 21 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 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 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 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 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) )
26 simp3l ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ≠ 𝑅 )
27 6 1 2 ps-1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ≠ 𝑅 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ↔ ( 𝑄 ∨ 𝑅 ) = ( 𝑃 ∨ 𝑄 ) ) )
28 3 5 9 26 4 5 27 syl132anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ↔ ( 𝑄 ∨ 𝑅 ) = ( 𝑃 ∨ 𝑄 ) ) )
29 25 28 mpbid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑃 ∨ 𝑄 ) )
30 29 eqcomd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑅 ) )