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Metamath Proof Explorer

Theorem dihjustlem

Description: Part of proof after Lemma N of Crawley p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-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 )
Assertion dihjustlem ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 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 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ HL )
12 11 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ Lat )
13 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ∈ 𝐴 )
14 1 5 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
15 13 14 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ∈ 𝐵 )
16 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑋 ∈ 𝐵 )
17 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐻 )
18 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
19 17 18 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐵 )
20 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
21 12 16 19 20 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
22 1 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) )
23 12 15 21 22 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) )
24 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) )
25 23 24 breqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ≤ ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) )
26 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
27 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
28 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
29 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
30 12 16 19 29 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
31 21 30 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) )
32 1 2 3 5 6 7 8 9 10 cdlemn ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) ) → ( 𝑄 ≤ ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ↔ ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
33 26 27 28 31 32 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑄 ≤ ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ↔ ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
34 25 33 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
35 6 9 26 dvhlmod ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑈 ∈ LMod )
36 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
37 36 lsssssubg ⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
38 35 37 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
39 2 5 6 9 8 36 diclss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) )
40 26 27 39 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) )
41 38 40 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) )
42 1 2 6 9 7 36 diblss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
43 26 21 30 42 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
44 38 43 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
45 10 lsmub2 ⊢ ( ( ( 𝐽 ‘ 𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
46 41 44 45 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )
47 2 5 6 9 8 36 diclss ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
48 26 28 47 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
49 38 48 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
50 36 10 lsmcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) )
51 35 40 43 50 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) )
52 38 51 sseldd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) )
53 10 lsmlub ⊢ ( ( ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
54 49 44 52 53 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) )
55 34 46 54 mpbi2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) )