Metamath Proof Explorer

Theorem cdlemg4c

Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013)

Ref Expression
Hypotheses cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
Assertion cdlemg4c ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ) → ¬ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) )

Proof

Step Hyp Ref Expression
1 cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
7 cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
8 simpll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simplr2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
10 simplr3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝐺 ∈ 𝑇 )
11 1 2 3 4 5 6 7 cdlemg4b2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑉 ) = ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) )
12 8 9 10 11 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑉 ) = ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) )
13 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) )
14 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝐾 ∈ HL )
15 14 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝐾 ∈ Lat )
16 simpr1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝑃 ∈ 𝐴 )
17 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
18 17 2 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
19 16 18 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
20 simpl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 simpr3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝐺 ∈ 𝑇 )
22 17 3 4 5 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
23 20 21 22 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑅 ‘ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
24 7 23 eqeltrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝑉 ∈ ( Base ‘ 𝐾 ) )
25 17 1 6 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ) → 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) )
26 15 19 24 25 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) )
27 26 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) )
28 simpr2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝑄 ∈ 𝐴 )
29 17 2 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
30 28 29 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
31 17 3 4 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
32 20 21 30 31 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
33 17 6 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
34 15 19 24 33 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
35 17 1 6 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) ) ↔ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
36 15 32 24 34 35 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → ( ( ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) ) ↔ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
37 36 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( ( ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) ) ↔ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
38 13 27 37 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) )
39 12 38 eqbrtrrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ≤ ( 𝑃 ∨ 𝑉 ) )
40 15 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝐾 ∈ Lat )
41 30 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
42 32 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
43 19 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
44 8 10 22 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝑅 ‘ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
45 7 44 eqeltrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝑉 ∈ ( Base ‘ 𝐾 ) )
46 40 43 45 33 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
47 17 1 6 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) ↔ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
48 40 41 42 46 47 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( ( 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) ↔ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
49 39 48 mpbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
50 49 simpld ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) )
51 50 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ) )
52 51 con3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ) → ( ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) → ¬ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
53 52 3impia ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐺 ∈ 𝑇 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ) → ¬ ( 𝐺 ‘ 𝑄 ) ≤ ( 𝑃 ∨ 𝑉 ) )