Metamath Proof Explorer


Theorem cdleme0ex1N

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 cdleme0ex1N ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → ∃ 𝑢𝐴 ( 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 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
8 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
9 simp2r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑄𝐴 )
10 simp3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑃𝑄 )
11 1 2 3 4 5 6 lhpat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )
12 7 8 9 10 11 syl112anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑈𝐴 )
13 simp2ll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑃𝐴 )
14 1 2 3 4 5 6 cdlemeulpq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ) ) → 𝑈 ( 𝑃 𝑄 ) )
15 7 13 9 14 syl12anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑈 ( 𝑃 𝑄 ) )
16 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝐾 ∈ HL )
17 16 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝐾 ∈ Lat )
18 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
19 18 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
20 16 13 9 19 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
21 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑊𝐻 )
22 18 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
23 21 22 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
24 18 1 3 latmle2 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑄 ) 𝑊 ) 𝑊 )
25 17 20 23 24 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑃 𝑄 ) 𝑊 ) 𝑊 )
26 6 25 eqbrtrid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → 𝑈 𝑊 )
27 breq1 ( 𝑢 = 𝑈 → ( 𝑢 ( 𝑃 𝑄 ) ↔ 𝑈 ( 𝑃 𝑄 ) ) )
28 breq1 ( 𝑢 = 𝑈 → ( 𝑢 𝑊𝑈 𝑊 ) )
29 27 28 anbi12d ( 𝑢 = 𝑈 → ( ( 𝑢 ( 𝑃 𝑄 ) ∧ 𝑢 𝑊 ) ↔ ( 𝑈 ( 𝑃 𝑄 ) ∧ 𝑈 𝑊 ) ) )
30 29 rspcev ( ( 𝑈𝐴 ∧ ( 𝑈 ( 𝑃 𝑄 ) ∧ 𝑈 𝑊 ) ) → ∃ 𝑢𝐴 ( 𝑢 ( 𝑃 𝑄 ) ∧ 𝑢 𝑊 ) )
31 12 15 26 30 syl12anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ 𝑃𝑄 ) → ∃ 𝑢𝐴 ( 𝑢 ( 𝑃 𝑄 ) ∧ 𝑢 𝑊 ) )