Metamath Proof Explorer

Theorem cdleme01N

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 5-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme0.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme0.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme0.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
Assertion cdleme01N ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑈 ≠ 𝑃 ∧ 𝑈 ≠ 𝑄 ∧ 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdleme0.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme0.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme0.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ HL )
8 7 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ Lat )
9 simp2ll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ∈ 𝐴 )
10 simp2rl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ∈ 𝐴 )
11 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 11 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
13 7 9 10 12 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
14 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ 𝐻 )
15 11 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
16 14 15 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
17 11 1 3 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
18 8 13 16 17 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
19 6 18 eqbrtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ≤ 𝑊 )
20 simp2lr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ¬ 𝑃 ≤ 𝑊 )
21 nbrne2 ⊢ ( ( 𝑈 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊 ) → 𝑈 ≠ 𝑃 )
22 19 20 21 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ≠ 𝑃 )
23 simp2rr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ¬ 𝑄 ≤ 𝑊 )
24 nbrne2 ⊢ ( ( 𝑈 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → 𝑈 ≠ 𝑄 )
25 19 23 24 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ≠ 𝑄 )
26 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
27 1 2 3 4 5 6 cdlemeulpq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) )
28 26 9 10 27 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) )
29 22 25 28 3jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑈 ≠ 𝑃 ∧ 𝑈 ≠ 𝑄 ∧ 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) )
30 29 19 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑈 ≠ 𝑃 ∧ 𝑈 ≠ 𝑄 ∧ 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ 𝑊 ) )