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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝐾 ∈ HL )
12 11 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝐾 ∈ Lat )
13 simp21l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝑄𝐴 )
14 1 5 atbase ( 𝑄𝐴𝑄𝐵 )
15 13 14 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝑄𝐵 )
16 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝑋𝐵 )
17 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝑊𝐻 )
18 1 6 lhpbase ( 𝑊𝐻𝑊𝐵 )
19 17 18 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝑊𝐵 )
20 1 4 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
21 12 16 19 20 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
22 1 2 3 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ ( 𝑋 𝑊 ) ∈ 𝐵 ) → 𝑄 ( 𝑄 ( 𝑋 𝑊 ) ) )
23 12 15 21 22 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝑄 ( 𝑄 ( 𝑋 𝑊 ) ) )
24 simp3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) )
25 23 24 breqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → 𝑄 ( 𝑅 ( 𝑋 𝑊 ) ) )
26 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
27 simp22 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) )
28 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
29 1 2 4 latmle2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 𝑊 ) 𝑊 )
30 12 16 19 29 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝑋 𝑊 ) 𝑊 )
31 21 30 jca ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( ( 𝑋 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 𝑊 ) 𝑊 ) ) ) → ( 𝑄 ( 𝑅 ( 𝑋 𝑊 ) ) ↔ ( 𝐽𝑄 ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ) )
33 26 27 28 31 32 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝑄 ( 𝑅 ( 𝑋 𝑊 ) ) ↔ ( 𝐽𝑄 ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ) )
34 25 33 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐽𝑄 ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) )
35 6 9 26 dvhlmod ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
39 2 5 6 9 8 36 diclss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) → ( 𝐽𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) )
40 26 27 39 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐽𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) )
41 38 40 sseldd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐽𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) )
42 1 2 6 9 7 36 diblss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑋 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 𝑊 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
43 26 21 30 42 syl12anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
44 38 43 sseldd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
45 10 lsmub2 ( ( ( 𝐽𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) )
46 41 44 45 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) )
47 2 5 6 9 8 36 diclss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐽𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
48 26 28 47 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐽𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
49 38 48 sseldd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( 𝐽𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
50 36 10 lsmcl ( ( 𝑈 ∈ LMod ∧ ( 𝐽𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) )
51 35 40 43 50 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) )
52 38 51 sseldd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) )
53 10 lsmlub ( ( ( 𝐽𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( 𝐽𝑄 ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ) ↔ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ) )
54 49 44 52 53 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( ( ( 𝐽𝑄 ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ) ↔ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ) )
55 34 46 54 mpbi2and ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ 𝑋𝐵 ) ∧ ( 𝑄 ( 𝑋 𝑊 ) ) = ( 𝑅 ( 𝑋 𝑊 ) ) ) → ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) )