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Metamath Proof Explorer

Theorem cdlemg11b

Description: TODO: FIX COMMENT. (Contributed by NM, 5-May-2013)

Ref Expression
Hypotheses cdlemg8.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg8.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg8.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg10.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg11b ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) ≠ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg8.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg8.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg8.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemg8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg10.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp33 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) )
9 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simpl31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝐺 ∈ 𝑇 )
11 simpl2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
12 1 2 3 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) )
13 9 10 11 12 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) )
14 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
15 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝐾 ∈ HL )
16 15 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝐾 ∈ Lat )
17 simp2ll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
18 17 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
19 14 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
20 18 19 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
21 14 5 6 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
22 9 10 20 21 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
23 14 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
24 16 20 22 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
25 simpl1r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑊 ∈ 𝐻 )
26 14 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
27 25 26 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
28 14 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
29 16 24 27 28 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
30 simpl2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
31 14 4 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
32 30 31 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
33 14 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
34 16 20 32 33 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
35 14 1 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
36 16 24 27 35 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
37 14 1 2 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) )
38 16 20 32 37 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) )
39 14 5 6 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
40 9 10 32 39 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
41 14 1 2 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑃 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
42 16 22 40 41 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
43 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
44 42 43 breqtrrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) )
45 14 1 2 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
46 16 20 22 34 45 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
47 38 44 46 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )
48 14 1 16 29 24 34 36 47 lattrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
49 13 48 eqbrtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) )
50 49 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
51 50 necon3bd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ≠ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
52 8 51 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) ≠ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )