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Metamath Proof Explorer

Theorem cdlemn10

Description: Part of proof of Lemma N of Crawley p. 121 line 36. (Contributed by NM, 27-Feb-2014)

Ref Expression
Hypotheses cdlemn10.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn10.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn10.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemn10.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn10.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn10.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn10.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemn10 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 cdlemn10.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn10.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn10.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemn10.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemn10.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemn10.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemn10.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝐾 ∈ HL )
9 8 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝐾 ∈ Lat )
10 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ∈ 𝐴 )
11 1 4 atbase ⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐵 )
12 10 11 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ∈ 𝐵 )
13 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 )
14 1 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 )
15 8 13 10 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 )
16 1 4 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
17 13 16 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑄 ∈ 𝐵 )
18 simp23l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 )
19 1 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 )
20 9 17 18 19 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 )
21 2 3 4 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑆 ≤ ( 𝑄 ∨ 𝑆 ) )
22 8 13 10 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ≤ ( 𝑄 ∨ 𝑆 ) )
23 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑊 ∈ 𝐻 )
24 1 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
25 23 24 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑊 ∈ 𝐵 )
26 2 3 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑄 ≤ ( 𝑄 ∨ 𝑆 ) )
27 8 13 10 26 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑄 ≤ ( 𝑄 ∨ 𝑆 ) )
28 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
29 1 2 3 28 4 atmod3i1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ∧ 𝑄 ≤ ( 𝑄 ∨ 𝑆 ) ) → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑊 ) ) )
30 8 13 15 25 27 29 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑊 ) ) )
31 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
32 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
33 eqid ⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
34 2 3 33 4 5 lhpjat2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑄 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) )
35 31 32 34 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) )
36 35 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑊 ) ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) )
37 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
38 8 37 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝐾 ∈ OL )
39 1 28 33 olm11 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑄 ∨ 𝑆 ) )
40 38 15 39 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑄 ∨ 𝑆 ) )
41 30 36 40 3eqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
42 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑔 ∈ 𝑇 )
43 2 3 28 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝑔 ) = ( ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
44 31 42 32 43 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑔 ) = ( ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
45 simp32 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑔 ‘ 𝑄 ) = 𝑆 )
46 45 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) = ( 𝑄 ∨ 𝑆 ) )
47 46 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) )
48 44 47 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑔 ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) )
49 simp33 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 )
50 48 49 eqbrtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑋 )
51 1 28 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
52 9 15 25 51 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
53 1 2 3 latjlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) ) → ( ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑋 → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ≤ ( 𝑄 ∨ 𝑋 ) ) )
54 9 52 18 17 53 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑋 → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ≤ ( 𝑄 ∨ 𝑋 ) ) )
55 50 54 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ≤ ( 𝑄 ∨ 𝑋 ) )
56 41 55 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) ≤ ( 𝑄 ∨ 𝑋 ) )
57 1 2 9 12 15 20 22 56 lattrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) )