Metamath Proof Explorer

Theorem cdlemb3

Description: Given two atoms not under the fiducial co-atom W , there is a third. Lemma B in Crawley p. 112. TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle ? Then replace cdlemb2 with it. This is a more general version of cdlemb2 without P =/= Q condition. (Contributed by NM, 27-Apr-2013)

Ref Expression
Hypotheses cdlemg5.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg5.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion cdlemb3 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg5.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg5.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemg5.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
7 1 2 3 4 cdlemg5 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( 𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊 ) )
8 5 6 7 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( 𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊 ) )
9 ancom ⊢ ( ( 𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊 ) ↔ ( ¬ 𝑟 ≤ 𝑊 ∧ 𝑃 ≠ 𝑟 ) )
10 eqcom ⊢ ( 𝑃 = 𝑟 ↔ 𝑟 = 𝑃 )
11 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → 𝑃 = 𝑄 )
12 11 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑃 ) = ( 𝑃 ∨ 𝑄 ) )
13 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → 𝐾 ∈ HL )
14 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → 𝑃 ∈ 𝐴 )
15 2 3 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑃 ) = 𝑃 )
16 13 14 15 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑃 ) = 𝑃 )
17 12 16 eqtr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = 𝑃 )
18 17 breq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑟 ≤ 𝑃 ) )
19 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
20 13 19 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → 𝐾 ∈ AtLat )
21 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → 𝑟 ∈ 𝐴 )
22 1 3 atcmp ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑟 ≤ 𝑃 ↔ 𝑟 = 𝑃 ) )
23 20 21 14 22 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑟 ≤ 𝑃 ↔ 𝑟 = 𝑃 ) )
24 18 23 bitr2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑟 = 𝑃 ↔ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )
25 10 24 bitrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑃 = 𝑟 ↔ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )
26 25 necon3abid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑟 ↔ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )
27 26 anbi2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( ( ¬ 𝑟 ≤ 𝑊 ∧ 𝑃 ≠ 𝑟 ) ↔ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
28 9 27 bitrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ∧ 𝑟 ∈ 𝐴 ) → ( ( 𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊 ) ↔ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
29 28 3expa ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ) ∧ 𝑟 ∈ 𝐴 ) → ( ( 𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊 ) ↔ ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
30 29 rexbidva ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ) → ( ∃ 𝑟 ∈ 𝐴 ( 𝑃 ≠ 𝑟 ∧ ¬ 𝑟 ≤ 𝑊 ) ↔ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
31 8 30 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 = 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )
32 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
33 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
34 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
35 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ≠ 𝑄 )
36 1 2 3 4 cdlemb2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )
37 32 33 34 35 36 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )
38 31 37 pm2.61dane ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) )