Metamath Proof Explorer


Theorem atbtwnexOLDN

Description: There exists a 3rd atom r on a line P .\/ Q intersecting element X at P , such that r is different from Q and not in X . (Contributed by NM, 30-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses atbtwn.b 𝐵 = ( Base ‘ 𝐾 )
atbtwn.l = ( le ‘ 𝐾 )
atbtwn.j = ( join ‘ 𝐾 )
atbtwn.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atbtwnexOLDN ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → ∃ 𝑟𝐴 ( 𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 ( 𝑃 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 atbtwn.b 𝐵 = ( Base ‘ 𝐾 )
2 atbtwn.l = ( le ‘ 𝐾 )
3 atbtwn.j = ( join ‘ 𝐾 )
4 atbtwn.a 𝐴 = ( Atoms ‘ 𝐾 )
5 simpr2 ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → 𝑃 𝑋 )
6 simpr3 ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → ¬ 𝑄 𝑋 )
7 nbrne2 ( ( 𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) → 𝑃𝑄 )
8 5 6 7 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → 𝑃𝑄 )
9 2 3 4 hlsupr ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ 𝑃𝑄 ) → ∃ 𝑟𝐴 ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) )
10 8 9 syldan ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → ∃ 𝑟𝐴 ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) )
11 simp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → 𝑟𝑄 )
12 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → 𝑟𝑃 )
13 simp1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) )
14 simp2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → 𝑟𝐴 )
15 simp1r1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → 𝑋𝐵 )
16 simp1r2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → 𝑃 𝑋 )
17 simp1r3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → ¬ 𝑄 𝑋 )
18 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → 𝑟 ( 𝑃 𝑄 ) )
19 1 2 3 4 atbtwn ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑟𝐴𝑋𝐵 ) ∧ ( 𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑟 ( 𝑃 𝑄 ) ) ) → ( 𝑟𝑃 ↔ ¬ 𝑟 𝑋 ) )
20 13 14 15 16 17 18 19 syl123anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → ( 𝑟𝑃 ↔ ¬ 𝑟 𝑋 ) )
21 12 20 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → ¬ 𝑟 𝑋 )
22 11 21 18 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) ∧ 𝑟𝐴 ∧ ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) ) → ( 𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 ( 𝑃 𝑄 ) ) )
23 22 3exp ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → ( 𝑟𝐴 → ( ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) → ( 𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 ( 𝑃 𝑄 ) ) ) ) )
24 23 reximdvai ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → ( ∃ 𝑟𝐴 ( 𝑟𝑃𝑟𝑄𝑟 ( 𝑃 𝑄 ) ) → ∃ 𝑟𝐴 ( 𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 ( 𝑃 𝑄 ) ) ) )
25 10 24 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) ∧ ( 𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋 ) ) → ∃ 𝑟𝐴 ( 𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 ( 𝑃 𝑄 ) ) )