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Theorem trlord 37823
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 1221 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ HL)
43hllatd 36618 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ Lat)
5 simpl1 1188 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6 simprlr 779 . . . . . 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 37418 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇) → (𝑅𝑓) ∈ 𝐵)
115, 6, 10syl2anc 587 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) ∈ 𝐵)
12 simpl2l 1223 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋𝐵)
13 simpl3l 1225 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑌𝐵)
14 simprr 772 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑋)
15 simprll 778 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋 𝑌)
161, 2, 4, 11, 12, 13, 14, 15lattrd 17659 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑌)
1716exp44 441 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 → (𝑓𝑇 → ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌))))
1817ralrimdv 3178 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 → ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌)))
19 simp11l 1281 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ HL)
2019hllatd 36618 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ Lat)
21 simp2r 1197 . . . . . . . . . 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 36543 . . . . . . . . . 10 (𝑢𝐴𝑢𝐵)
2421, 23syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢𝐵)
25 simp12l 1283 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑋𝐵)
26 simp11r 1282 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐻)
271, 7lhpbase 37252 . . . . . . . . . 10 (𝑊𝐻𝑊𝐵)
2826, 27syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐵)
29 simp3 1135 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑋)
30 simp12r 1284 . . . . . . . . 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 17659 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑊)
3231, 29jca 515 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → (𝑢 𝑊𝑢 𝑋))
33323expia 1118 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑋 → (𝑢 𝑊𝑢 𝑋)))
34 simp11 1200 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝐾 ∈ HL ∧ 𝑊𝐻))
35 simp2r 1197 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢𝐴)
36 simp3 1135 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢 𝑊)
372, 22, 7, 8, 9cdlemf 37817 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑢𝐴𝑢 𝑊)) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
3834, 35, 36, 37syl12anc 835 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
39 simp2l 1196 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌))
40 fveq2 6652 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑅𝑓) = (𝑅𝑔))
4140breq1d 5052 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝑔) 𝑋))
4240breq1d 5052 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑌 ↔ (𝑅𝑔) 𝑌))
4341, 42imbi12d 348 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ↔ ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
4443rspccv 3595 . . . . . . . . . . . 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 5045 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑋𝑢 𝑋))
47 breq1 5045 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑌𝑢 𝑌))
4846, 47imbi12d 348 . . . . . . . . . . . 12 ((𝑅𝑔) = 𝑢 → (((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌) ↔ (𝑢 𝑋𝑢 𝑌)))
4948biimpcd 252 . . . . . . . . . . 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 3269 . . . . . . . . 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 1118 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑊 → (𝑢 𝑋𝑢 𝑌)))
5453impd 414 . . . . . 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 424 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → (𝑢𝐴 → (𝑢 𝑋𝑢 𝑌))))
5756ralrimdv 3178 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
58 simp1l 1194 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝐾 ∈ HL)
59 simp2l 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑋𝐵)
60 simp3l 1198 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑌𝐵)
611, 2, 22hlatle 36652 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6258, 59, 60, 61syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6357, 62sylibrd 262 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → 𝑋 𝑌))
6418, 63impbid 215 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131   class class class wbr 5042  cfv 6334  Basecbs 16474  lecple 16563  Atomscatm 36517  HLchlt 36604  LHypclh 37238  LTrncltrn 37355  trLctrl 37412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-riotaBAD 36207
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-undef 7926  df-map 8395  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-oposet 36430  df-ol 36432  df-oml 36433  df-covers 36520  df-ats 36521  df-atl 36552  df-cvlat 36576  df-hlat 36605  df-llines 36752  df-lplanes 36753  df-lvols 36754  df-lines 36755  df-psubsp 36757  df-pmap 36758  df-padd 37050  df-lhyp 37242  df-laut 37243  df-ldil 37358  df-ltrn 37359  df-trl 37413
This theorem is referenced by:  diaord  38301  dihord2pre  38479
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