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Metamath Proof Explorer

Theorem cdlemg2fv2

Description: Value of a translation in terms of an associated atom. TODO: FIX COMMENT. TODO: Is this useful elsewhere e.g. around cdlemeg46fjv that use more complex proofs? TODO: Use ltrnj to vastly simplify. (Contributed by NM, 23-Apr-2013)

Ref Expression
Hypotheses cdlemg2inv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg2inv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg2j.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg2j.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg2j.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg2j.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg2j.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
Assertion cdlemg2fv2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( 𝑅 ∨ 𝑈 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑈 ) )

Proof

Step Hyp Ref Expression
1 cdlemg2inv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 cdlemg2inv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 cdlemg2j.l ⊢ ≤ = ( le ‘ 𝐾 )
4 cdlemg2j.j ⊢ ∨ = ( join ‘ 𝐾 )
5 cdlemg2j.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemg2j.m ⊢ ∧ = ( meet ‘ 𝐾 )
7 cdlemg2j.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
10 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ HL )
11 10 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ Lat )
12 simp23l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑅 ∈ 𝐴 )
13 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 5 atbase ⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
15 12 14 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
16 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑊 ∈ 𝐻 )
17 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑃 ∈ 𝐴 )
18 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑄 ∈ 𝐴 )
19 3 4 6 5 1 7 13 cdleme0aa ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
20 10 16 17 18 19 syl211anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
21 13 4 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
22 11 15 20 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
23 simp23r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ¬ 𝑅 ≤ 𝑊 )
24 13 3 4 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → 𝑅 ≤ ( 𝑅 ∨ 𝑈 ) )
25 11 15 20 24 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑅 ≤ ( 𝑅 ∨ 𝑈 ) )
26 13 1 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
27 16 26 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
28 13 3 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑅 ≤ ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑈 ) ≤ 𝑊 ) → 𝑅 ≤ 𝑊 ) )
29 11 15 22 27 28 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ≤ ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑈 ) ≤ 𝑊 ) → 𝑅 ≤ 𝑊 ) )
30 25 29 mpand ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ∨ 𝑈 ) ≤ 𝑊 → 𝑅 ≤ 𝑊 ) )
31 23 30 mtod ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ¬ ( 𝑅 ∨ 𝑈 ) ≤ 𝑊 )
32 22 31 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ∧ ¬ ( 𝑅 ∨ 𝑈 ) ≤ 𝑊 ) )
33 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 )
34 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
35 3 6 34 5 1 lhpmat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝑅 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) )
36 8 9 35 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) )
37 36 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ∧ 𝑊 ) ∨ 𝑈 ) = ( ( 0. ‘ 𝐾 ) ∨ 𝑈 ) )
38 13 4 5 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
39 10 17 18 38 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
40 13 3 6 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
41 11 39 27 40 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
42 7 41 eqbrtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝑈 ≤ 𝑊 )
43 13 3 4 6 5 atmod4i2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑈 ≤ 𝑊 ) → ( ( 𝑅 ∧ 𝑊 ) ∨ 𝑈 ) = ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) )
44 10 12 20 27 42 43 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ∧ 𝑊 ) ∨ 𝑈 ) = ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) )
45 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
46 10 45 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ OL )
47 13 4 34 olj02 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → ( ( 0. ‘ 𝐾 ) ∨ 𝑈 ) = 𝑈 )
48 46 20 47 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 0. ‘ 𝐾 ) ∨ 𝑈 ) = 𝑈 )
49 37 44 48 3eqtr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) = 𝑈 )
50 49 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ∨ ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) ) = ( 𝑅 ∨ 𝑈 ) )
51 1 2 3 4 5 6 13 cdlemg2fv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ∧ ¬ ( 𝑅 ∨ 𝑈 ) ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ∨ ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) ) = ( 𝑅 ∨ 𝑈 ) ) ) → ( 𝐹 ‘ ( 𝑅 ∨ 𝑈 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) ) )
52 8 9 32 33 50 51 syl122anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( 𝑅 ∨ 𝑈 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) ) )
53 49 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( ( 𝑅 ∨ 𝑈 ) ∧ 𝑊 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑈 ) )
54 52 53 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( 𝑅 ∨ 𝑈 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑈 ) )