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Theorem trlord 36728
 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.)
Hypotheses
Ref Expression
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
Ref Expression
trlord (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 ↔ ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌)))
Distinct variable groups:   ,𝑓   𝐵,𝑓   𝑓,𝐻   𝑓,𝐾   𝑅,𝑓   𝑇,𝑓   𝑓,𝑊   𝑓,𝑋   𝑓,𝑌
Allowed substitution hint:   𝐴(𝑓)

Proof of Theorem trlord
Dummy variables 𝑔 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trlord.b . . . . 5 𝐵 = (Base‘𝐾)
2 trlord.l . . . . 5 = (le‘𝐾)
3 simpl1l 1250 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ HL)
43hllatd 35523 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ Lat)
5 simpl1 1199 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6 simprlr 770 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑓𝑇)
7 trlord.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
8 trlord.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
9 trlord.r . . . . . . 7 𝑅 = ((trL‘𝐾)‘𝑊)
101, 7, 8, 9trlcl 36323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇) → (𝑅𝑓) ∈ 𝐵)
115, 6, 10syl2anc 579 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) ∈ 𝐵)
12 simpl2l 1254 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋𝐵)
13 simpl3l 1258 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑌𝐵)
14 simprr 763 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑋)
15 simprll 769 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋 𝑌)
161, 2, 4, 11, 12, 13, 14, 15lattrd 17448 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑌)
1716exp44 430 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 → (𝑓𝑇 → ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌))))
1817ralrimdv 3150 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 → ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌)))
19 simp11l 1340 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ HL)
2019hllatd 35523 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ Lat)
21 simp2r 1214 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢𝐴)
22 trlord.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
231, 22atbase 35448 . . . . . . . . . 10 (𝑢𝐴𝑢𝐵)
2421, 23syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢𝐵)
25 simp12l 1342 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑋𝐵)
26 simp11r 1341 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐻)
271, 7lhpbase 36157 . . . . . . . . . 10 (𝑊𝐻𝑊𝐵)
2826, 27syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐵)
29 simp3 1129 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑋)
30 simp12r 1343 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑋 𝑊)
311, 2, 20, 24, 25, 28, 29, 30lattrd 17448 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑊)
3231, 29jca 507 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → (𝑢 𝑊𝑢 𝑋))
33323expia 1111 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑋 → (𝑢 𝑊𝑢 𝑋)))
34 simp11 1217 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝐾 ∈ HL ∧ 𝑊𝐻))
35 simp2r 1214 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢𝐴)
36 simp3 1129 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢 𝑊)
372, 22, 7, 8, 9cdlemf 36722 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑢𝐴𝑢 𝑊)) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
3834, 35, 36, 37syl12anc 827 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
39 simp2l 1213 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌))
40 fveq2 6448 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑅𝑓) = (𝑅𝑔))
4140breq1d 4898 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝑔) 𝑋))
4240breq1d 4898 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑌 ↔ (𝑅𝑔) 𝑌))
4341, 42imbi12d 336 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ↔ ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
4443rspccv 3508 . . . . . . . . . . . 12 (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → (𝑔𝑇 → ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
4539, 44syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝑔𝑇 → ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
46 breq1 4891 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑋𝑢 𝑋))
47 breq1 4891 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑌𝑢 𝑌))
4846, 47imbi12d 336 . . . . . . . . . . . 12 ((𝑅𝑔) = 𝑢 → (((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌) ↔ (𝑢 𝑋𝑢 𝑌)))
4948biimpcd 241 . . . . . . . . . . 11 (((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌) → ((𝑅𝑔) = 𝑢 → (𝑢 𝑋𝑢 𝑌)))
5045, 49syl6 35 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝑔𝑇 → ((𝑅𝑔) = 𝑢 → (𝑢 𝑋𝑢 𝑌))))
5150rexlimdv 3212 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (∃𝑔𝑇 (𝑅𝑔) = 𝑢 → (𝑢 𝑋𝑢 𝑌)))
5238, 51mpd 15 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝑢 𝑋𝑢 𝑌))
53523expia 1111 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑊 → (𝑢 𝑋𝑢 𝑌)))
5453impd 400 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → ((𝑢 𝑊𝑢 𝑋) → 𝑢 𝑌))
5533, 54syld 47 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑋𝑢 𝑌))
5655exp32 413 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → (𝑢𝐴 → (𝑢 𝑋𝑢 𝑌))))
5756ralrimdv 3150 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
58 simp1l 1211 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝐾 ∈ HL)
59 simp2l 1213 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑋𝐵)
60 simp3l 1215 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑌𝐵)
611, 2, 22hlatle 35557 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6258, 59, 60, 61syl3anc 1439 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6357, 62sylibrd 251 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → 𝑋 𝑌))
6418, 63impbid 204 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 ↔ ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091   class class class wbr 4888  ‘cfv 6137  Basecbs 16259  lecple 16349  Atomscatm 35422  HLchlt 35509  LHypclh 36143  LTrncltrn 36260  trLctrl 36317 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-riotaBAD 35112 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-undef 7683  df-map 8144  df-proset 17318  df-poset 17336  df-plt 17348  df-lub 17364  df-glb 17365  df-join 17366  df-meet 17367  df-p0 17429  df-p1 17430  df-lat 17436  df-clat 17498  df-oposet 35335  df-ol 35337  df-oml 35338  df-covers 35425  df-ats 35426  df-atl 35457  df-cvlat 35481  df-hlat 35510  df-llines 35657  df-lplanes 35658  df-lvols 35659  df-lines 35660  df-psubsp 35662  df-pmap 35663  df-padd 35955  df-lhyp 36147  df-laut 36148  df-ldil 36263  df-ltrn 36264  df-trl 36318 This theorem is referenced by:  diaord  37206  dihord2pre  37384
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