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Metamath Proof Explorer

Theorem cdlemn2

Description: Part of proof of Lemma N of Crawley p. 121 line 30. (Contributed by NM, 21-Feb-2014)

Ref Expression
Hypotheses cdlemn2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemn2.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemn2.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemn2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemn2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemn2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemn2.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemn2.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑄 ) = 𝑆 )
Assertion cdlemn2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 )

Proof

Step Hyp Ref Expression
1 cdlemn2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemn2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemn2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemn2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemn2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemn2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemn2.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemn2.f ⊢ 𝐹 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑄 ) = 𝑆 )
9 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
11 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
12 2 4 5 6 8 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
13 9 10 11 12 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝐹 ∈ 𝑇 )
14 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
15 2 3 14 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
16 9 13 10 15 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
17 2 4 5 6 8 ltrniotaval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑄 ) = 𝑆 )
18 9 10 11 17 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐹 ‘ 𝑄 ) = 𝑆 )
19 18 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ 𝑆 ) )
20 19 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) )
21 16 20 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) )
22 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝐾 ∈ HL )
23 22 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝐾 ∈ Lat )
24 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑄 ∈ 𝐴 )
25 1 4 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
26 24 25 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑄 ∈ 𝐵 )
27 simp23l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑋 ∈ 𝐵 )
28 1 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑄 ≤ ( 𝑄 ∨ 𝑋 ) )
29 23 26 27 28 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑄 ≤ ( 𝑄 ∨ 𝑋 ) )
30 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) )
31 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑆 ∈ 𝐴 )
32 1 4 atbase ⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐵 )
33 31 32 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑆 ∈ 𝐵 )
34 1 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 )
35 23 26 27 34 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 )
36 1 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵 ∧ ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 ) ) → ( ( 𝑄 ≤ ( 𝑄 ∨ 𝑋 ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) ↔ ( 𝑄 ∨ 𝑆 ) ≤ ( 𝑄 ∨ 𝑋 ) ) )
37 23 26 33 35 36 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝑄 ≤ ( 𝑄 ∨ 𝑋 ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) ↔ ( 𝑄 ∨ 𝑆 ) ≤ ( 𝑄 ∨ 𝑋 ) ) )
38 29 30 37 mpbi2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) ≤ ( 𝑄 ∨ 𝑋 ) )
39 1 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 )
40 22 24 31 39 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 )
41 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑊 ∈ 𝐻 )
42 1 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
43 41 42 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → 𝑊 ∈ 𝐵 )
44 1 2 14 latmlem1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 ∧ ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑄 ∨ 𝑆 ) ≤ ( 𝑄 ∨ 𝑋 ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ ( ( 𝑄 ∨ 𝑋 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
45 23 40 35 43 44 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ≤ ( 𝑄 ∨ 𝑋 ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ ( ( 𝑄 ∨ 𝑋 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
46 38 45 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ ( ( 𝑄 ∨ 𝑋 ) ( meet ‘ 𝐾 ) 𝑊 ) )
47 21 46 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝑄 ∨ 𝑋 ) ( meet ‘ 𝐾 ) 𝑊 ) )
48 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) )
49 1 2 3 14 4 5 lhple ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑄 ∨ 𝑋 ) ( meet ‘ 𝐾 ) 𝑊 ) = 𝑋 )
50 9 10 48 49 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑋 ) ( meet ‘ 𝐾 ) 𝑊 ) = 𝑋 )
51 47 50 breqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 )