Metamath Proof Explorer


Theorem dihord2pre2

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. (Contributed by NM, 4-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 = ( 𝑇 ↦ ( I ↾ 𝐵 ) )
dihord2.p 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
dihord2.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dihord2.d + = ( +g𝑈 )
dihord2.g 𝐺 = ( 𝑇 ( 𝑃 ) = 𝑁 )
Assertion dihord2pre2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑄 ( 𝑋 𝑊 ) ) ( 𝑁 ( 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 = ( 𝑇 ↦ ( I ↾ 𝐵 ) )
14 dihord2.p 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
15 dihord2.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
16 dihord2.d + = ( +g𝑈 )
17 dihord2.g 𝐺 = ( 𝑇 ( 𝑃 ) = 𝑁 )
18 1 2 3 4 5 6 7 8 9 10 dihord2a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑄 ( 𝑁 ( 𝑌 𝑊 ) ) )
19 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝐾 ∈ HL )
20 19 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝐾 ∈ Lat )
21 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑋𝐵 )
22 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑊𝐻 )
23 1 6 lhpbase ( 𝑊𝐻𝑊𝐵 )
24 22 23 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑊𝐵 )
25 1 4 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵 ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
26 20 21 24 25 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑋 𝑊 ) ∈ 𝐵 )
27 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑌𝐵 )
28 1 4 latmcl ( ( 𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵 ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
29 20 27 24 28 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑌 𝑊 ) ∈ 𝐵 )
30 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑁𝐴 )
31 1 5 atbase ( 𝑁𝐴𝑁𝐵 )
32 30 31 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑁𝐵 )
33 1 3 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑁𝐵 ∧ ( 𝑌 𝑊 ) ∈ 𝐵 ) → ( 𝑁 ( 𝑌 𝑊 ) ) ∈ 𝐵 )
34 20 32 29 33 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑁 ( 𝑌 𝑊 ) ) ∈ 𝐵 )
35 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 dihord2pre ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) → ( 𝑋 𝑊 ) ( 𝑌 𝑊 ) )
37 35 36 syld3an3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑋 𝑊 ) ( 𝑌 𝑊 ) )
38 1 2 3 latlej2 ( ( 𝐾 ∈ Lat ∧ 𝑁𝐵 ∧ ( 𝑌 𝑊 ) ∈ 𝐵 ) → ( 𝑌 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) )
39 20 32 29 38 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑌 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) )
40 1 2 20 26 29 34 37 39 lattrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑋 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) )
41 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑄𝐴 )
42 1 5 atbase ( 𝑄𝐴𝑄𝐵 )
43 41 42 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → 𝑄𝐵 )
44 1 2 3 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑄𝐵 ∧ ( 𝑋 𝑊 ) ∈ 𝐵 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) ∈ 𝐵 ) ) → ( ( 𝑄 ( 𝑁 ( 𝑌 𝑊 ) ) ∧ ( 𝑋 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) ) ↔ ( 𝑄 ( 𝑋 𝑊 ) ) ( 𝑁 ( 𝑌 𝑊 ) ) ) )
45 20 43 26 34 44 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( ( 𝑄 ( 𝑁 ( 𝑌 𝑊 ) ) ∧ ( 𝑋 𝑊 ) ( 𝑁 ( 𝑌 𝑊 ) ) ) ↔ ( 𝑄 ( 𝑋 𝑊 ) ) ( 𝑁 ( 𝑌 𝑊 ) ) ) )
46 18 40 45 mpbi2and ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑁𝐴 ∧ ¬ 𝑁 𝑊 ) ) ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝑄 ( 𝑋 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ( 𝑌 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽𝑄 ) ( 𝐼 ‘ ( 𝑋 𝑊 ) ) ) ⊆ ( ( 𝐽𝑁 ) ( 𝐼 ‘ ( 𝑌 𝑊 ) ) ) ) ) → ( 𝑄 ( 𝑋 𝑊 ) ) ( 𝑁 ( 𝑌 𝑊 ) ) )