Metamath Proof Explorer

Theorem dihord11b

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014)

Ref Expression
Hypotheses dihjust.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihjust.l ⊢ ≤ = ( le ‘ 𝐾 )
dihjust.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjust.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihjust.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjust.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjust.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
dihjust.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
dihjust.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjust.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihord2c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dihord2c.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
dihord2c.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
dihord2.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
dihord2.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dihord2.d ⊢ + = ( +g ‘ 𝑈 )
dihord2.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
Assertion dihord11b ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ⟨ 𝑓 , 𝑂 ⟩ ∈ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjust.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihjust.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihjust.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihjust.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihjust.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 dihjust.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 dihjust.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8 dihjust.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
9 dihjust.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
10 dihjust.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
11 dihord2c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
12 dihord2c.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
13 dihord2c.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
14 dihord2.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
15 dihord2.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
16 dihord2.d ⊢ + = ( +g ‘ 𝑈 )
17 dihord2.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
18 1 2 3 4 5 6 7 8 9 10 dihord2b ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) )
19 18 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) )
20 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) )
21 eqidd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑂 = 𝑂 )
22 simpl11 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
23 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝐾 ∈ HL )
24 23 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ HL )
25 24 hllatd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ Lat )
26 simpl2l ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑋 ∈ 𝐵 )
27 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝑊 ∈ 𝐻 )
28 27 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐻 )
29 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
30 28 29 syl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐵 )
31 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
32 25 26 30 31 syl3anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
33 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
34 25 26 30 33 syl3anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
35 1 2 6 11 12 13 7 dibopelval3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ ∈ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ↔ ( ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑂 = 𝑂 ) ) )
36 22 32 34 35 syl12anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ ∈ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ↔ ( ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑂 = 𝑂 ) ) )
37 20 21 36 mpbir2and ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ⟨ 𝑓 , 𝑂 ⟩ ∈ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) )
38 19 37 sseldd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ⟨ 𝑓 , 𝑂 ⟩ ∈ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) )