Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlord Structured version   Visualization version   GIF version

Theorem trlord 35358
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 1110 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ HL)
4 hllat 34151 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
53, 4syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝐾 ∈ Lat)
6 simpl1 1062 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simprlr 802 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑓𝑇)
8 trlord.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
9 trlord.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 trlord.r . . . . . . 7 𝑅 = ((trL‘𝐾)‘𝑊)
111, 8, 9, 10trlcl 34952 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇) → (𝑅𝑓) ∈ 𝐵)
126, 7, 11syl2anc 692 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) ∈ 𝐵)
13 simpl2l 1112 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋𝐵)
14 simpl3l 1114 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑌𝐵)
15 simprr 795 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑋)
16 simprll 801 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → 𝑋 𝑌)
171, 2, 5, 12, 13, 14, 15, 16lattrd 16982 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ ((𝑋 𝑌𝑓𝑇) ∧ (𝑅𝑓) 𝑋)) → (𝑅𝑓) 𝑌)
1817exp44 640 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 → (𝑓𝑇 → ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌))))
1918ralrimdv 2962 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 → ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌)))
20 simp11l 1170 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ HL)
2120, 4syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝐾 ∈ Lat)
22 simp2r 1086 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢𝐴)
23 trlord.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
241, 23atbase 34077 . . . . . . . . . 10 (𝑢𝐴𝑢𝐵)
2522, 24syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢𝐵)
26 simp12l 1172 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑋𝐵)
27 simp11r 1171 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐻)
281, 8lhpbase 34785 . . . . . . . . . 10 (𝑊𝐻𝑊𝐵)
2927, 28syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑊𝐵)
30 simp3 1061 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑋)
31 simp12r 1173 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑋 𝑊)
321, 2, 21, 25, 26, 29, 30, 31lattrd 16982 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → 𝑢 𝑊)
3332, 30jca 554 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑋) → (𝑢 𝑊𝑢 𝑋))
34333expia 1264 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑋 → (𝑢 𝑊𝑢 𝑋)))
35 simp11 1089 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝐾 ∈ HL ∧ 𝑊𝐻))
36 simp2r 1086 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢𝐴)
37 simp3 1061 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → 𝑢 𝑊)
382, 23, 8, 9, 10cdlemf 35352 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑢𝐴𝑢 𝑊)) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
3935, 36, 37, 38syl12anc 1321 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∃𝑔𝑇 (𝑅𝑔) = 𝑢)
40 simp2l 1085 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → ∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌))
41 fveq2 6150 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑅𝑓) = (𝑅𝑔))
4241breq1d 4625 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝑔) 𝑋))
4341breq1d 4625 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → ((𝑅𝑓) 𝑌 ↔ (𝑅𝑔) 𝑌))
4442, 43imbi12d 334 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ↔ ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
4544rspccv 3292 . . . . . . . . . . . 12 (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → (𝑔𝑇 → ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
4640, 45syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝑔𝑇 → ((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌)))
47 breq1 4618 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑋𝑢 𝑋))
48 breq1 4618 . . . . . . . . . . . . 13 ((𝑅𝑔) = 𝑢 → ((𝑅𝑔) 𝑌𝑢 𝑌))
4947, 48imbi12d 334 . . . . . . . . . . . 12 ((𝑅𝑔) = 𝑢 → (((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌) ↔ (𝑢 𝑋𝑢 𝑌)))
5049biimpcd 239 . . . . . . . . . . 11 (((𝑅𝑔) 𝑋 → (𝑅𝑔) 𝑌) → ((𝑅𝑔) = 𝑢 → (𝑢 𝑋𝑢 𝑌)))
5146, 50syl6 35 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝑔𝑇 → ((𝑅𝑔) = 𝑢 → (𝑢 𝑋𝑢 𝑌))))
5251rexlimdv 3023 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (∃𝑔𝑇 (𝑅𝑔) = 𝑢 → (𝑢 𝑋𝑢 𝑌)))
5339, 52mpd 15 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴) ∧ 𝑢 𝑊) → (𝑢 𝑋𝑢 𝑌))
54533expia 1264 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑊 → (𝑢 𝑋𝑢 𝑌)))
5554impd 447 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → ((𝑢 𝑊𝑢 𝑋) → 𝑢 𝑌))
5634, 55syld 47 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) ∧ 𝑢𝐴)) → (𝑢 𝑋𝑢 𝑌))
5756exp32 630 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → (𝑢𝐴 → (𝑢 𝑋𝑢 𝑌))))
5857ralrimdv 2962 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
59 simp1l 1083 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝐾 ∈ HL)
60 simp2l 1085 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑋𝐵)
61 simp3l 1087 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → 𝑌𝐵)
621, 2, 23hlatle 34185 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6359, 60, 61, 62syl3anc 1323 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝑋 𝑌 ↔ ∀𝑢𝐴 (𝑢 𝑋𝑢 𝑌)))
6458, 63sylibrd 249 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (∀𝑓𝑇 ((𝑅𝑓) 𝑋 → (𝑅𝑓) 𝑌) → 𝑋 𝑌))
6519, 64impbid 202 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908   class class class wbr 4615  cfv 5849  Basecbs 15784  lecple 15872  Latclat 16969  Atomscatm 34051  HLchlt 34138  LHypclh 34771  LTrncltrn 34888  trLctrl 34946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-riotaBAD 33740
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-undef 7347  df-map 7807  df-preset 16852  df-poset 16870  df-plt 16882  df-lub 16898  df-glb 16899  df-join 16900  df-meet 16901  df-p0 16963  df-p1 16964  df-lat 16970  df-clat 17032  df-oposet 33964  df-ol 33966  df-oml 33967  df-covers 34054  df-ats 34055  df-atl 34086  df-cvlat 34110  df-hlat 34139  df-llines 34285  df-lplanes 34286  df-lvols 34287  df-lines 34288  df-psubsp 34290  df-pmap 34291  df-padd 34583  df-lhyp 34775  df-laut 34776  df-ldil 34891  df-ltrn 34892  df-trl 34947
This theorem is referenced by:  diaord  35837  dihord2pre  36015
  Copyright terms: Public domain W3C validator