Metamath Proof Explorer


Theorem dihord2b

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 )
Assertion dihord2b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑅 ) ( 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 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
12 6 9 11 dvhlmod ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → 𝑈 ∈ LMod )
13 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
14 13 lsssssubg ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
15 12 14 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) )
16 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
17 2 5 6 9 8 13 diclss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐽𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
18 11 16 17 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐽𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) )
19 15 18 sseldd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐽𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) )
20 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → 𝐾 ∈ HL )
21 20 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → 𝐾 ∈ Lat )
22 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → 𝑋𝐵 )
23 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → 𝑊𝐻 )
24 1 6 lhpbase ( 𝑊𝐻𝑊𝐵 )
25 23 24 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → 𝑊𝐵 )
26 1 4 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
27 21 22 25 26 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
28 1 2 4 latmle2 ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 𝑊 ) 𝑊 )
29 21 22 25 28 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝑋 𝑊 ) 𝑊 )
30 1 2 6 9 7 13 diblss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑋 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 𝑊 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
31 11 27 29 30 syl12anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) )
32 15 31 sseldd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) )
33 10 lsmub2 ( ( ( 𝐽𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) )
34 19 32 33 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) )
35 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) )
36 34 35 sstrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ⊆ ( ( 𝐽𝑅 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) )