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Theorem trlord 40552
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 1223 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ HL)
43hllatd 39346 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ Lat)
5 simpl1 1190 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6 simprlr 780 . . . . . 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 40147 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇) → (𝑅𝑓) ∈ 𝐵)
115, 6, 10syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) ∈ 𝐵)
12 simpl2l 1225 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋𝐵)
13 simpl3l 1227 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑌𝐵)
14 simprr 773 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑋)
15 simprll 779 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋 𝑌)
161, 2, 4, 11, 12, 13, 14, 15lattrd 18504 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑌)
1716exp44 437 . . 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 1283 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ HL)
2019hllatd 39346 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ Lat)
21 simp2r 1199 . . . . . . . . . 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 39271 . . . . . . . . . 10 (𝑢𝐴𝑢𝐵)
2421, 23syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢𝐵)
25 simp12l 1285 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑋𝐵)
26 simp11r 1284 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐻)
271, 7lhpbase 39981 . . . . . . . . . 10 (𝑊𝐻𝑊𝐵)
2826, 27syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐵)
29 simp3 1137 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑋)
30 simp12r 1286 . . . . . . . . 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 18504 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑊)
3231, 29jca 511 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → (𝑢 𝑊𝑢 𝑋))
33323expia 1120 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑋 → (𝑢 𝑊𝑢 𝑋)))
34 simp11 1202 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝐾 ∈ HL ∧ 𝑊𝐻))
35 simp2r 1199 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢𝐴)
36 simp3 1137 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢 𝑊)
372, 22, 7, 8, 9cdlemf 40546 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑢𝐴𝑢 𝑊)) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
3834, 35, 36, 37syl12anc 837 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
39 simp2l 1198 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌))
40 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑅𝑓) = (𝑅𝑔))
4140breq1d 5158 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝑔) 𝑋))
4240breq1d 5158 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑌 ↔ (𝑅𝑔) 𝑌))
4341, 42imbi12d 344 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ↔ ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
4443rspccv 3619 . . . . . . . . . . . 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 5151 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑋𝑢 𝑋))
47 breq1 5151 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑌𝑢 𝑌))
4846, 47imbi12d 344 . . . . . . . . . . . 12 ((𝑅𝑔) = 𝑢 → (((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌) ↔ (𝑢 𝑋𝑢 𝑌)))
4948biimpcd 249 . . . . . . . . . . 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 3151 . . . . . . . . 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 1120 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑊 → (𝑢 𝑋𝑢 𝑌)))
5453impd 410 . . . . . 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 420 . . . 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 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝐾 ∈ HL)
59 simp2l 1198 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑋𝐵)
60 simp3l 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑌𝐵)
611, 2, 22hlatle 39381 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6258, 59, 60, 61syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6357, 62sylibrd 259 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → 𝑋 𝑌))
6418, 63impbid 212 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  trLctrl 40141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-undef 8297  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142
This theorem is referenced by:  diaord  41030  dihord2pre  41208
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