Metamath Proof Explorer


Theorem cdleme02N

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 9-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 cdleme02N ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → ( ( 𝑃 𝑈 ) = ( 𝑄 𝑈 ) ∧ 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 1 2 3 4 5 6 cdleme01N ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → ( ( 𝑈𝑃𝑈𝑄𝑈 ( 𝑃 𝑄 ) ) ∧ 𝑈 𝑊 ) )
8 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → 𝐾 ∈ HL )
9 hlcvl ( 𝐾 ∈ HL → 𝐾 ∈ CvLat )
10 8 9 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → 𝐾 ∈ CvLat )
11 simp2ll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → 𝑃𝐴 )
12 simp2rl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → 𝑄𝐴 )
13 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
15 simp3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → 𝑃𝑄 )
16 1 2 3 4 5 6 lhpat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )
17 13 14 12 15 16 syl112anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → 𝑈𝐴 )
18 4 1 2 cvlsupr2 ( ( 𝐾 ∈ CvLat ∧ ( 𝑃𝐴𝑄𝐴𝑈𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑃 𝑈 ) = ( 𝑄 𝑈 ) ↔ ( 𝑈𝑃𝑈𝑄𝑈 ( 𝑃 𝑄 ) ) ) )
19 10 11 12 17 15 18 syl131anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → ( ( 𝑃 𝑈 ) = ( 𝑄 𝑈 ) ↔ ( 𝑈𝑃𝑈𝑄𝑈 ( 𝑃 𝑄 ) ) ) )
20 19 anbi1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → ( ( ( 𝑃 𝑈 ) = ( 𝑄 𝑈 ) ∧ 𝑈 𝑊 ) ↔ ( ( 𝑈𝑃𝑈𝑄𝑈 ( 𝑃 𝑄 ) ) ∧ 𝑈 𝑊 ) ) )
21 7 20 mpbird ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝑃𝑄 ) → ( ( 𝑃 𝑈 ) = ( 𝑄 𝑈 ) ∧ 𝑈 𝑊 ) )