Metamath Proof Explorer

Theorem cdlemd7

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 1-Jun-2012)

Ref Expression
Hypotheses cdlemd4.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemd4.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemd4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemd4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemd4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemd7 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 cdlemd4.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemd4.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemd4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemd4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 cdlemd4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 simp1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) )
7 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
8 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
9 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝐾 ∈ HL )
10 9 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝐾 ∈ Lat )
11 simp2rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝑄 ∈ 𝐴 )
12 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
13 12 3 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
14 11 13 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
15 simp2ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝑃 ∈ 𝐴 )
16 12 3 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
17 15 16 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
18 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝐹 ∈ 𝑇 )
20 12 4 5 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
21 18 19 17 20 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
22 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
23 12 1 2 latnlej1l ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝑄 ≠ 𝑃 )
24 23 necomd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝑃 ≠ 𝑄 )
25 10 14 17 21 22 24 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → 𝑃 ≠ 𝑄 )
26 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
27 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) )
28 1 2 3 4 5 cdlemd6 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) )
29 18 27 7 8 22 26 28 syl231anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) )
30 1 2 3 4 5 cdlemd5 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )
31 6 7 8 25 26 29 30 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )