Metamath Proof Explorer


Theorem cdleme7

Description: Part of proof of Lemma E in Crawley p. 113. G and F represent f_s(r) and f(s) respectively. W is the fiducial co-atom (hyperplane) that they call w. Here and in cdleme7ga above, we show that f_s(r) e. W (top of p. 114), meaning it is an atom and not under w, which in our notation is expressed as G e. A /\ -. G .<_ W . (Note that we do not have a symbol for their W.) Their proof provides no details of our cdleme7aa through cdleme7 , so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012)

Ref Expression
Hypotheses cdleme4.l = ( le ‘ 𝐾 )
cdleme4.j = ( join ‘ 𝐾 )
cdleme4.m = ( meet ‘ 𝐾 )
cdleme4.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme4.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme4.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme4.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme4.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑅 𝑆 ) 𝑊 ) ) )
Assertion cdleme7 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐺 𝑊 )

Proof

Step Hyp Ref Expression
1 cdleme4.l = ( le ‘ 𝐾 )
2 cdleme4.j = ( join ‘ 𝐾 )
3 cdleme4.m = ( meet ‘ 𝐾 )
4 cdleme4.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme4.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme4.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme4.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme4.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑅 𝑆 ) 𝑊 ) ) )
9 eqid ( ( 𝑅 𝑆 ) 𝑊 ) = ( ( 𝑅 𝑆 ) 𝑊 )
10 1 2 3 4 5 6 7 8 9 cdleme7d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐺𝑈 )
11 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
12 simp2ll ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑅𝐴 )
13 1 2 3 4 5 6 7 8 cdleme7ga ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐺𝐴 )
14 1 2 4 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑅𝐴𝐺𝐴 ) → 𝐺 ( 𝑅 𝐺 ) )
15 11 12 13 14 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐺 ( 𝑅 𝐺 ) )
16 15 biantrurd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐺 𝑊 ↔ ( 𝐺 ( 𝑅 𝐺 ) ∧ 𝐺 𝑊 ) ) )
17 11 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ Lat )
18 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
19 18 4 atbase ( 𝐺𝐴𝐺 ∈ ( Base ‘ 𝐾 ) )
20 13 19 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐺 ∈ ( Base ‘ 𝐾 ) )
21 18 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑅𝐴𝐺𝐴 ) → ( 𝑅 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
22 11 12 13 21 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑅 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
23 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑊𝐻 )
24 18 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
25 23 24 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
26 18 1 3 latlem12 ( ( 𝐾 ∈ Lat ∧ ( 𝐺 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 𝐺 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐺 ( 𝑅 𝐺 ) ∧ 𝐺 𝑊 ) ↔ 𝐺 ( ( 𝑅 𝐺 ) 𝑊 ) ) )
27 17 20 22 25 26 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( ( 𝐺 ( 𝑅 𝐺 ) ∧ 𝐺 𝑊 ) ↔ 𝐺 ( ( 𝑅 𝐺 ) 𝑊 ) ) )
28 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
29 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑃𝐴 )
30 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑄𝐴 )
31 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) )
32 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
33 simp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑅 ( 𝑃 𝑄 ) )
34 1 2 3 4 5 6 7 8 cdleme6 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝑅 𝐺 ) 𝑊 ) = 𝑈 )
35 28 29 30 31 32 33 34 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( ( 𝑅 𝐺 ) 𝑊 ) = 𝑈 )
36 35 breq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐺 ( ( 𝑅 𝐺 ) 𝑊 ) ↔ 𝐺 𝑈 ) )
37 27 36 bitrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( ( 𝐺 ( 𝑅 𝐺 ) ∧ 𝐺 𝑊 ) ↔ 𝐺 𝑈 ) )
38 hlatl ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
39 11 38 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ AtLat )
40 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
41 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑃𝑄 )
42 1 2 3 4 5 6 lhpat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )
43 28 40 30 41 42 syl112anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑈𝐴 )
44 1 4 atcmp ( ( 𝐾 ∈ AtLat ∧ 𝐺𝐴𝑈𝐴 ) → ( 𝐺 𝑈𝐺 = 𝑈 ) )
45 39 13 43 44 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐺 𝑈𝐺 = 𝑈 ) )
46 16 37 45 3bitrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐺 𝑊𝐺 = 𝑈 ) )
47 46 necon3bbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( ¬ 𝐺 𝑊𝐺𝑈 ) )
48 10 47 mpbird ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐺 𝑊 )