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Metamath Proof Explorer

Theorem cdlemb2

Description: Given two atoms not under the fiducial (reference) co-atom W , there is a third. Lemma B in Crawley p. 112. (Contributed by NM, 30-May-2012)

Ref Expression
Hypotheses cdlemb2.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemb2.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemb2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemb2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion cdlemb2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemb2.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemb2.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemb2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemb2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ HL )
6 simp2ll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ∈ 𝐴 )
7 simp2rl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ∈ 𝐴 )
8 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ 𝐻 )
9 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10 9 4 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
11 8 10 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
12 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ≠ 𝑄 )
13 eqid ⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
14 eqid ⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
15 13 14 4 lhp1cvr ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) )
16 15 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) )
17 simp2lr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ¬ 𝑃 ≤ 𝑊 )
18 simp2rr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ¬ 𝑄 ≤ 𝑊 )
19 9 1 2 13 14 3 cdlemb ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ∧ ¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )
20 5 6 7 11 12 16 17 18 19 syl323anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )