Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme41.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdleme41.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdleme41.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdleme41.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdleme41.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdleme41.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdleme41.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdleme41.d |
⊢ 𝐷 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
9 |
|
cdleme41.e |
⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
10 |
|
cdleme41.g |
⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
11 |
|
cdleme41.i |
⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) ) |
12 |
|
cdleme41.n |
⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐷 ) |
13 |
|
cdleme41.o |
⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) |
14 |
|
cdleme41.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) ) |
15 |
|
cdleme34e.v |
⊢ 𝑉 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) |
16 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → 𝐾 ∈ HL ) |
17 |
|
simpr2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) |
18 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
cdleme32fvaw |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑅 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑅 ) ≤ 𝑊 ) ) |
19 |
17 18
|
syldan |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑅 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑅 ) ≤ 𝑊 ) ) |
20 |
19
|
simpld |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( 𝐹 ‘ 𝑅 ) ∈ 𝐴 ) |
21 |
3 5
|
hlatjidm |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑅 ) ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑅 ) ) = ( 𝐹 ‘ 𝑅 ) ) |
22 |
16 20 21
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑅 ) ) = ( 𝐹 ‘ 𝑅 ) ) |
23 |
|
fveq2 |
⊢ ( 𝑅 = 𝑆 → ( 𝐹 ‘ 𝑅 ) = ( 𝐹 ‘ 𝑆 ) ) |
24 |
23
|
oveq2d |
⊢ ( 𝑅 = 𝑆 → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑅 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑆 ) ) ) |
25 |
22 24
|
sylan9req |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) ∧ 𝑅 = 𝑆 ) → ( 𝐹 ‘ 𝑅 ) = ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑆 ) ) ) |
26 |
|
simpr2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → 𝑅 ∈ 𝐴 ) |
27 |
3 5
|
hlatjidm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ) → ( 𝑅 ∨ 𝑅 ) = 𝑅 ) |
28 |
16 26 27
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( 𝑅 ∨ 𝑅 ) = 𝑅 ) |
29 |
28
|
oveq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝑅 ∨ 𝑅 ) ∧ 𝑊 ) = ( 𝑅 ∧ 𝑊 ) ) |
30 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
31 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
32 |
2 4 31 5 6
|
lhpmat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝑅 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
33 |
30 17 32
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( 𝑅 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
34 |
29 33
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝑅 ∨ 𝑅 ) ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
35 |
34
|
oveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( ( 𝑅 ∨ 𝑅 ) ∧ 𝑊 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ ( 0. ‘ 𝐾 ) ) ) |
36 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
37 |
16 36
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → 𝐾 ∈ OL ) |
38 |
1 5
|
atbase |
⊢ ( ( 𝐹 ‘ 𝑅 ) ∈ 𝐴 → ( 𝐹 ‘ 𝑅 ) ∈ 𝐵 ) |
39 |
20 38
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( 𝐹 ‘ 𝑅 ) ∈ 𝐵 ) |
40 |
1 3 31
|
olj01 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝐹 ‘ 𝑅 ) ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝐹 ‘ 𝑅 ) ) |
41 |
37 39 40
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝐹 ‘ 𝑅 ) ) |
42 |
35 41
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( ( 𝑅 ∨ 𝑅 ) ∧ 𝑊 ) ) = ( 𝐹 ‘ 𝑅 ) ) |
43 |
|
oveq2 |
⊢ ( 𝑅 = 𝑆 → ( 𝑅 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑆 ) ) |
44 |
43
|
oveq1d |
⊢ ( 𝑅 = 𝑆 → ( ( 𝑅 ∨ 𝑅 ) ∧ 𝑊 ) = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) ) |
45 |
44 15
|
eqtr4di |
⊢ ( 𝑅 = 𝑆 → ( ( 𝑅 ∨ 𝑅 ) ∧ 𝑊 ) = 𝑉 ) |
46 |
45
|
oveq2d |
⊢ ( 𝑅 = 𝑆 → ( ( 𝐹 ‘ 𝑅 ) ∨ ( ( 𝑅 ∨ 𝑅 ) ∧ 𝑊 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑉 ) ) |
47 |
42 46
|
sylan9req |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) ∧ 𝑅 = 𝑆 ) → ( 𝐹 ‘ 𝑅 ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑉 ) ) |
48 |
25 47
|
eqtr3d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) ∧ 𝑅 = 𝑆 ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑆 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑉 ) ) |
49 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
cdleme42k |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ 𝑅 ≠ 𝑆 ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑆 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑉 ) ) |
50 |
49
|
3expa |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) ∧ 𝑅 ≠ 𝑆 ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑆 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑉 ) ) |
51 |
48 50
|
pm2.61dane |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑅 ) ∨ ( 𝐹 ‘ 𝑆 ) ) = ( ( 𝐹 ‘ 𝑅 ) ∨ 𝑉 ) ) |