Metamath Proof Explorer


Theorem cdlemc4

Description: Part of proof of Lemma C in Crawley p. 113. (Contributed by NM, 26-May-2012)

Ref Expression
Hypotheses cdlemc3.l = ( le ‘ 𝐾 )
cdlemc3.j = ( join ‘ 𝐾 )
cdlemc3.m = ( meet ‘ 𝐾 )
cdlemc3.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemc3.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemc3.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemc3.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemc4 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ¬ 𝑄 ( 𝑃 ( 𝐹𝑃 ) ) ) → ( 𝑄 ( 𝑅𝐹 ) ) ≠ ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemc3.l = ( le ‘ 𝐾 )
2 cdlemc3.j = ( join ‘ 𝐾 )
3 cdlemc3.m = ( meet ‘ 𝐾 )
4 cdlemc3.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemc3.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemc3.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemc3.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simpll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → 𝐾 ∈ HL )
9 8 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → 𝐾 ∈ Lat )
10 simpl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
11 simpr1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → 𝐹𝑇 )
12 simpr2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → 𝑃𝐴 )
13 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 4 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
15 12 14 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
16 13 5 6 ltrncl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) )
17 10 11 15 16 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) )
18 simpr3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → 𝑄𝐴 )
19 13 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
20 8 12 18 19 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
21 13 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
22 21 ad2antlr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
23 13 3 latmcl ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑄 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
24 9 20 22 23 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( ( 𝑃 𝑄 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
25 13 1 2 latlej1 ( ( 𝐾 ∈ Lat ∧ ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 𝑄 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹𝑃 ) ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
26 9 17 24 25 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( 𝐹𝑃 ) ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
27 breq2 ( ( 𝑄 ( 𝑅𝐹 ) ) = ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) → ( ( 𝐹𝑃 ) ( 𝑄 ( 𝑅𝐹 ) ) ↔ ( 𝐹𝑃 ) ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) ) )
28 26 27 syl5ibrcom ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( ( 𝑄 ( 𝑅𝐹 ) ) = ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) → ( 𝐹𝑃 ) ( 𝑄 ( 𝑅𝐹 ) ) ) )
29 1 2 3 4 5 6 7 cdlemc3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( ( 𝐹𝑃 ) ( 𝑄 ( 𝑅𝐹 ) ) → 𝑄 ( 𝑃 ( 𝐹𝑃 ) ) ) )
30 28 29 syld ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( ( 𝑄 ( 𝑅𝐹 ) ) = ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) → 𝑄 ( 𝑃 ( 𝐹𝑃 ) ) ) )
31 30 necon3bd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( ¬ 𝑄 ( 𝑃 ( 𝐹𝑃 ) ) → ( 𝑄 ( 𝑅𝐹 ) ) ≠ ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) ) )
32 31 3impia ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ¬ 𝑄 ( 𝑃 ( 𝐹𝑃 ) ) ) → ( 𝑄 ( 𝑅𝐹 ) ) ≠ ( ( 𝐹𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )