Metamath Proof Explorer

Theorem dihord2pre

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 dihord2pre ( ( ( ( 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 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 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 simpl1 ⊢ ( ( ( ( ( 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 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) )
19 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 )
20 simpl2r ⊢ ( ( ( ( ( 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 ) ) ) → 𝑌 ∈ 𝐵 )
21 simpl3 ⊢ ( ( ( ( ( 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 ) ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) )
22 simprl ⊢ ( ( ( ( ( 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 )
23 simprr ⊢ ( ( ( ( ( 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 ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) )
24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 dihord11c ⊢ ( ( ( ( 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 ) ) ) → ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) )
25 18 19 20 21 22 23 24 syl123anc ⊢ ( ( ( ( ( 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 ) ) ) → ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) )
26 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 ) )
27 simpl13 ⊢ ( ( ( ( ( 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 ) ) ) → ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) )
28 2 5 6 14 11 15 8 17 dicelval3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) → ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ↔ ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ) )
29 26 27 28 syl2anc ⊢ ( ( ( ( ( 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 ) ) ) → ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ↔ ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ) )
30 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 )
31 30 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 )
32 31 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 )
33 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 )
34 33 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 )
35 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
36 34 35 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 )
37 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
38 32 20 36 37 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 ) ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
39 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 )
40 32 20 36 39 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 ) ) ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 )
41 1 2 6 11 12 13 7 dibelval3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ↔ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
42 26 38 40 41 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 ) ) ) → ( 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ↔ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
43 29 42 anbi12d ⊢ ( ( ( ( ( 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 ) ) ) → ( ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∧ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) )
44 reeanv ⊢ ( ∃ 𝑠 ∈ 𝐸 ∃ 𝑔 ∈ 𝑇 ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
45 simpll1 ⊢ ( ( ( ( ( ( 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 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) )
46 simplr ⊢ ( ( ( ( ( ( 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 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) )
47 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 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) )
48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 dihord10 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) )
49 45 46 47 48 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 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) )
50 49 3exp2 ⊢ ( ( ( ( ( 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 ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) )
51 oveq12 ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑦 + 𝑧 ) = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) )
52 51 eqeq2d ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) ↔ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) )
53 52 imbi1d ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ↔ ( ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
54 53 imbi2d ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) )
55 54 biimprd ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) )
56 55 com23 ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ) → ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) )
57 56 impr ⊢ ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
58 57 com12 ⊢ ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
59 50 58 syl6 ⊢ ( ( ( ( ( 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 ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) )
60 59 rexlimdvv ⊢ ( ( ( ( ( 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 ) ) ) → ( ∃ 𝑠 ∈ 𝐸 ∃ 𝑔 ∈ 𝑇 ( 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
61 44 60 biimtrrid ⊢ ( ( ( ( ( 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 ) ) ) → ( ( ∃ 𝑠 ∈ 𝐸 𝑦 = ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = ⟨ 𝑔 , 𝑂 ⟩ ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
62 43 61 sylbid ⊢ ( ( ( ( ( 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 ) ) ) → ( ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∧ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
63 62 rexlimdvv ⊢ ( ( ( ( ( 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 ) ) ) → ( ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) )
64 25 63 mpd ⊢ ( ( ( ( ( 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 ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) )
65 64 exp32 ⊢ ( ( ( ( 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 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
66 65 ralrimiv ⊢ ( ( ( ( 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 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) )
67 simp11 ⊢ ( ( ( ( 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 ∧ 𝑊 ∈ 𝐻 ) )
68 30 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 ) ) ) ) → 𝐾 ∈ Lat )
69 simp2l ⊢ ( ( ( ( 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 ) ) ) ) → 𝑋 ∈ 𝐵 )
70 33 35 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 ) ) ) ) → 𝑊 ∈ 𝐵 )
71 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
72 68 69 70 71 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 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
73 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
74 68 69 70 73 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 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
75 simp2r ⊢ ( ( ( ( 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 ) ) ) ) → 𝑌 ∈ 𝐵 )
76 68 75 70 37 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 ) ) ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
77 68 75 70 39 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 ) ) ) ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 )
78 1 2 5 6 11 12 trlord ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ↔ ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
79 67 72 74 76 77 78 syl122anc ⊢ ( ( ( ( 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 ) ) ) ) → ( ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ↔ ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) )
80 66 79 mpbird ⊢ ( ( ( ( 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 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) )