Metamath Proof Explorer


Theorem trlord

Description: The ordering of two Hilbert lattice elements (under the fiducial hyperplane W ) is determined by the translations whose traces are under them. (Contributed by NM, 3-Mar-2014)

Ref Expression
Hypotheses trlord.b 𝐵 = ( Base ‘ 𝐾 )
trlord.l = ( le ‘ 𝐾 )
trlord.a 𝐴 = ( Atoms ‘ 𝐾 )
trlord.h 𝐻 = ( LHyp ‘ 𝐾 )
trlord.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlord.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝑋 𝑌 ↔ ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 trlord.b 𝐵 = ( Base ‘ 𝐾 )
2 trlord.l = ( le ‘ 𝐾 )
3 trlord.a 𝐴 = ( Atoms ‘ 𝐾 )
4 trlord.h 𝐻 = ( LHyp ‘ 𝐾 )
5 trlord.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 trlord.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7 simpl1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → 𝐾 ∈ HL )
8 7 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → 𝐾 ∈ Lat )
9 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simprlr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → 𝑓𝑇 )
11 1 4 5 6 trlcl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑓𝑇 ) → ( 𝑅𝑓 ) ∈ 𝐵 )
12 9 10 11 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → ( 𝑅𝑓 ) ∈ 𝐵 )
13 simpl2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → 𝑋𝐵 )
14 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → 𝑌𝐵 )
15 simprr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → ( 𝑅𝑓 ) 𝑋 )
16 simprll ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → 𝑋 𝑌 )
17 1 2 8 12 13 14 15 16 lattrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ( 𝑋 𝑌𝑓𝑇 ) ∧ ( 𝑅𝑓 ) 𝑋 ) ) → ( 𝑅𝑓 ) 𝑌 )
18 17 exp44 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝑋 𝑌 → ( 𝑓𝑇 → ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ) ) )
19 18 ralrimdv ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝑋 𝑌 → ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ) )
20 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝐾 ∈ HL )
21 20 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝐾 ∈ Lat )
22 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑢𝐴 )
23 1 3 atbase ( 𝑢𝐴𝑢𝐵 )
24 22 23 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑢𝐵 )
25 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑋𝐵 )
26 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑊𝐻 )
27 1 4 lhpbase ( 𝑊𝐻𝑊𝐵 )
28 26 27 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑊𝐵 )
29 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑢 𝑋 )
30 simp12r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑋 𝑊 )
31 1 2 21 24 25 28 29 30 lattrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → 𝑢 𝑊 )
32 31 29 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑋 ) → ( 𝑢 𝑊𝑢 𝑋 ) )
33 32 3expia ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ) → ( 𝑢 𝑋 → ( 𝑢 𝑊𝑢 𝑋 ) ) )
34 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
35 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → 𝑢𝐴 )
36 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → 𝑢 𝑊 )
37 2 3 4 5 6 cdlemf ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑢𝐴𝑢 𝑊 ) ) → ∃ 𝑔𝑇 ( 𝑅𝑔 ) = 𝑢 )
38 34 35 36 37 syl12anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → ∃ 𝑔𝑇 ( 𝑅𝑔 ) = 𝑢 )
39 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) )
40 fveq2 ( 𝑓 = 𝑔 → ( 𝑅𝑓 ) = ( 𝑅𝑔 ) )
41 40 breq1d ( 𝑓 = 𝑔 → ( ( 𝑅𝑓 ) 𝑋 ↔ ( 𝑅𝑔 ) 𝑋 ) )
42 40 breq1d ( 𝑓 = 𝑔 → ( ( 𝑅𝑓 ) 𝑌 ↔ ( 𝑅𝑔 ) 𝑌 ) )
43 41 42 imbi12d ( 𝑓 = 𝑔 → ( ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ↔ ( ( 𝑅𝑔 ) 𝑋 → ( 𝑅𝑔 ) 𝑌 ) ) )
44 43 rspccv ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) → ( 𝑔𝑇 → ( ( 𝑅𝑔 ) 𝑋 → ( 𝑅𝑔 ) 𝑌 ) ) )
45 39 44 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → ( 𝑔𝑇 → ( ( 𝑅𝑔 ) 𝑋 → ( 𝑅𝑔 ) 𝑌 ) ) )
46 breq1 ( ( 𝑅𝑔 ) = 𝑢 → ( ( 𝑅𝑔 ) 𝑋𝑢 𝑋 ) )
47 breq1 ( ( 𝑅𝑔 ) = 𝑢 → ( ( 𝑅𝑔 ) 𝑌𝑢 𝑌 ) )
48 46 47 imbi12d ( ( 𝑅𝑔 ) = 𝑢 → ( ( ( 𝑅𝑔 ) 𝑋 → ( 𝑅𝑔 ) 𝑌 ) ↔ ( 𝑢 𝑋𝑢 𝑌 ) ) )
49 48 biimpcd ( ( ( 𝑅𝑔 ) 𝑋 → ( 𝑅𝑔 ) 𝑌 ) → ( ( 𝑅𝑔 ) = 𝑢 → ( 𝑢 𝑋𝑢 𝑌 ) ) )
50 45 49 syl6 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → ( 𝑔𝑇 → ( ( 𝑅𝑔 ) = 𝑢 → ( 𝑢 𝑋𝑢 𝑌 ) ) ) )
51 50 rexlimdv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → ( ∃ 𝑔𝑇 ( 𝑅𝑔 ) = 𝑢 → ( 𝑢 𝑋𝑢 𝑌 ) ) )
52 38 51 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ∧ 𝑢 𝑊 ) → ( 𝑢 𝑋𝑢 𝑌 ) )
53 52 3expia ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ) → ( 𝑢 𝑊 → ( 𝑢 𝑋𝑢 𝑌 ) ) )
54 53 impd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ) → ( ( 𝑢 𝑊𝑢 𝑋 ) → 𝑢 𝑌 ) )
55 33 54 syld ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) ∧ ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ∧ 𝑢𝐴 ) ) → ( 𝑢 𝑋𝑢 𝑌 ) )
56 55 exp32 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) → ( 𝑢𝐴 → ( 𝑢 𝑋𝑢 𝑌 ) ) ) )
57 56 ralrimdv ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) → ∀ 𝑢𝐴 ( 𝑢 𝑋𝑢 𝑌 ) ) )
58 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → 𝐾 ∈ HL )
59 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → 𝑋𝐵 )
60 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → 𝑌𝐵 )
61 1 2 3 hlatle ( ( 𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ↔ ∀ 𝑢𝐴 ( 𝑢 𝑋𝑢 𝑌 ) ) )
62 58 59 60 61 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝑋 𝑌 ↔ ∀ 𝑢𝐴 ( 𝑢 𝑋𝑢 𝑌 ) ) )
63 57 62 sylibrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) → 𝑋 𝑌 ) )
64 19 63 impbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ∧ ( 𝑌𝐵𝑌 𝑊 ) ) → ( 𝑋 𝑌 ↔ ∀ 𝑓𝑇 ( ( 𝑅𝑓 ) 𝑋 → ( 𝑅𝑓 ) 𝑌 ) ) )