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Metamath Proof Explorer

Theorem atcvrneN

Description: Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses atcvrne.j ⊢ ∨ = ( join ‘ 𝐾 )
atcvrne.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
atcvrne.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atcvrneN ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑄 ≠ 𝑅 )

Proof

Step Hyp Ref Expression
1 atcvrne.j ⊢ ∨ = ( join ‘ 𝐾 )
2 atcvrne.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3 atcvrne.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
5 4 3ad2ant1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝐾 ∈ AtLat )
6 simp21 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 ∈ 𝐴 )
7 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
8 7 3 atn0 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) )
9 5 6 8 syl2anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) )
10 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝐾 ∈ HL )
11 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 11 3 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
13 6 12 syl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
14 simp22 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑄 ∈ 𝐴 )
15 simp23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑅 ∈ 𝐴 )
16 simp3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) )
17 11 1 7 2 3 atcvrj0 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → ( 𝑃 = ( 0. ‘ 𝐾 ) ↔ 𝑄 = 𝑅 ) )
18 10 13 14 15 16 17 syl131anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → ( 𝑃 = ( 0. ‘ 𝐾 ) ↔ 𝑄 = 𝑅 ) )
19 18 necon3bid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → ( 𝑃 ≠ ( 0. ‘ 𝐾 ) ↔ 𝑄 ≠ 𝑅 ) )
20 9 19 mpbid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) → 𝑄 ≠ 𝑅 )