Metamath Proof Explorer


Theorem cdleme27cl

Description: Part of proof of Lemma E in Crawley p. 113. Closure of C . (Contributed by NM, 6-Feb-2013)

Ref Expression
Hypotheses cdleme26.b 𝐵 = ( Base ‘ 𝐾 )
cdleme26.l = ( le ‘ 𝐾 )
cdleme26.j = ( join ‘ 𝐾 )
cdleme26.m = ( meet ‘ 𝐾 )
cdleme26.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme26.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme27.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme27.f 𝐹 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
cdleme27.z 𝑍 = ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) )
cdleme27.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝑍 ( ( 𝑠 𝑧 ) 𝑊 ) ) )
cdleme27.d 𝐷 = ( 𝑢𝐵𝑧𝐴 ( ( ¬ 𝑧 𝑊 ∧ ¬ 𝑧 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )
cdleme27.c 𝐶 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐷 , 𝐹 )
Assertion cdleme27cl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝐶𝐵 )

Proof

Step Hyp Ref Expression
1 cdleme26.b 𝐵 = ( Base ‘ 𝐾 )
2 cdleme26.l = ( le ‘ 𝐾 )
3 cdleme26.j = ( join ‘ 𝐾 )
4 cdleme26.m = ( meet ‘ 𝐾 )
5 cdleme26.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme26.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme27.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme27.f 𝐹 = ( ( 𝑠 𝑈 ) ( 𝑄 ( ( 𝑃 𝑠 ) 𝑊 ) ) )
9 cdleme27.z 𝑍 = ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) )
10 cdleme27.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝑍 ( ( 𝑠 𝑧 ) 𝑊 ) ) )
11 cdleme27.d 𝐷 = ( 𝑢𝐵𝑧𝐴 ( ( ¬ 𝑧 𝑊 ∧ ¬ 𝑧 ( 𝑃 𝑄 ) ) → 𝑢 = 𝑁 ) )
12 cdleme27.c 𝐶 = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐷 , 𝐹 )
13 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ 𝑠 ( 𝑃 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 simpl2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ 𝑠 ( 𝑃 𝑄 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
15 simpl2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ 𝑠 ( 𝑃 𝑄 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
16 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ 𝑠 ( 𝑃 𝑄 ) ) → ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) )
17 simpl3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ 𝑠 ( 𝑃 𝑄 ) ) → 𝑃𝑄 )
18 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ 𝑠 ( 𝑃 𝑄 ) ) → 𝑠 ( 𝑃 𝑄 ) )
19 1 2 3 4 5 6 7 9 10 11 cdleme25cl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ ( 𝑃𝑄𝑠 ( 𝑃 𝑄 ) ) ) → 𝐷𝐵 )
20 13 14 15 16 17 18 19 syl312anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ 𝑠 ( 𝑃 𝑄 ) ) → 𝐷𝐵 )
21 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝐾 ∈ HL )
22 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝑊𝐻 )
23 simp2ll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝑃𝐴 )
24 simp2rl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝑄𝐴 )
25 simp3ll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝑠𝐴 )
26 2 3 4 5 6 7 8 1 cdleme1b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴𝑠𝐴 ) ) → 𝐹𝐵 )
27 21 22 23 24 25 26 syl23anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝐹𝐵 )
28 27 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) ∧ ¬ 𝑠 ( 𝑃 𝑄 ) ) → 𝐹𝐵 )
29 20 28 ifclda ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → if ( 𝑠 ( 𝑃 𝑄 ) , 𝐷 , 𝐹 ) ∈ 𝐵 )
30 12 29 eqeltrid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑠𝐴 ∧ ¬ 𝑠 𝑊 ) ∧ 𝑃𝑄 ) ) → 𝐶𝐵 )