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Metamath Proof Explorer

Theorem dihord11c

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 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 ) )

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 simp1 ⊢ ( ( ( ( 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 simp2 ⊢ ( ( ( ( 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 ∧ 𝑌 ∈ 𝐵 ) )
20 simp31 ⊢ ( ( ( ( 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 ) ) ) )
21 simp32 ⊢ ( ( ( ( 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 )
22 simp33 ⊢ ( ( ( ( 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 ) )
23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 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 ) ) ) )
24 18 19 20 21 22 23 syl32anc ⊢ ( ( ( ( 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 ) ) ) )
25 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 ) ) ) ∧ 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
26 6 9 25 dvhlmod ⊢ ( ( ( ( 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 ) ) ) → 𝑈 ∈ LMod )
27 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
28 27 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
29 26 28 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 ) ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
30 simp13 ⊢ ( ( ( ( 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 ) )
31 2 5 6 9 8 27 diclss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑁 ) ∈ ( LSubSp ‘ 𝑈 ) )
32 25 30 31 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 ) ) ) → ( 𝐽 ‘ 𝑁 ) ∈ ( LSubSp ‘ 𝑈 ) )
33 29 32 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 ) ) ) → ( 𝐽 ‘ 𝑁 ) ∈ ( SubGrp ‘ 𝑈 ) )
34 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 ) ) ) ∧ 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ HL )
35 34 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 )
36 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 ) ) ) ∧ 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑌 ∈ 𝐵 )
37 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 ) ) ) ∧ 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐻 )
38 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
39 37 38 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 )
40 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
41 35 36 39 40 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 )
42 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 )
43 35 36 39 42 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 )
44 1 2 6 9 7 27 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
45 25 41 43 44 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 ) ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
46 29 45 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 ) ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
47 16 10 lsmelval ⊢ ( ( ( 𝐽 ‘ 𝑁 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ ∈ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) ) )
48 33 46 47 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 ) ) ) → ( ⟨ 𝑓 , 𝑂 ⟩ ∈ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ⟨ 𝑓 , 𝑂 ⟩ = ( 𝑦 + 𝑧 ) ) )
49 24 48 mpbid ⊢ ( ( ( ( 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 ) )