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Theorem lhprelat3N 37333
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 36705. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhprelat3.b 𝐵 = (Base‘𝐾)
lhprelat3.l = (le‘𝐾)
lhprelat3.s < = (lt‘𝐾)
lhprelat3.m = (meet‘𝐾)
lhprelat3.c 𝐶 = ( ⋖ ‘𝐾)
lhprelat3.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
lhprelat3N (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
Distinct variable groups:   𝑤,𝐶   𝑤,𝐻   𝑤,𝐾   𝑤,   𝑤,   𝑤,𝑋   𝑤,𝑌
Allowed substitution hints:   𝐵(𝑤)   < (𝑤)

Proof of Theorem lhprelat3N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpr 488 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2 simpll1 1209 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3 lhprelat3.b . . . . . . . 8 𝐵 = (Base‘𝐾)
4 eqid 2798 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 36582 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 485 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 eqid 2798 . . . . . . 7 (oc‘𝐾) = (oc‘𝐾)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
93, 7, 4, 8lhpoc2N 37308 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
102, 6, 9syl2anc 587 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
111, 10mpbid 235 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
1211adantr 484 . . 3 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
13 hlop 36655 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
142, 13syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
152hllatd 36657 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
16 simpll3 1211 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
173, 7opoccl 36487 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
1814, 6, 17syl2anc 587 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
19 lhprelat3.m . . . . . . . . . 10 = (meet‘𝐾)
203, 19latmcl 17654 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
2115, 16, 18, 20syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
22 lhprelat3.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
233, 7, 22cvrcon3b 36570 . . . . . . . 8 ((𝐾 ∈ OP ∧ (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵𝑌𝐵) → ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌 ↔ ((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝)))))
2414, 21, 16, 23syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌 ↔ ((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝)))))
25 hlol 36654 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ OL)
262, 25syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OL)
27 eqid 2798 . . . . . . . . . 10 (join‘𝐾) = (join‘𝐾)
283, 27, 19, 7oldmm3N 36512 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑝𝐵) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
2926, 16, 6, 28syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
3029breq2d 5042 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ↔ ((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝)))
3124, 30bitr2d 283 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ↔ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
32 simpll2 1210 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
33 lhprelat3.l . . . . . . . . 9 = (le‘𝐾)
343, 33, 7oplecon3b 36493 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋𝐵 ∧ (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋)))
3514, 32, 21, 34syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋)))
3629breq1d 5040 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋) ↔ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)))
3735, 36bitr2d 283 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋) ↔ 𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
3831, 37anbi12d 633 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)) ↔ ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌𝑋 (𝑌 ((oc‘𝐾)‘𝑝)))))
3938biimpa 480 . . . 4 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → ((𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
4039ancomd 465 . . 3 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
41 oveq2 7143 . . . . . 6 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 𝑤) = (𝑌 ((oc‘𝐾)‘𝑝)))
4241breq2d 5042 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 (𝑌 𝑤) ↔ 𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
4341breq1d 5040 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 𝑤)𝐶𝑌 ↔ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
4442, 43anbi12d 633 . . . 4 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌) ↔ (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌)))
4544rspcev 3571 . . 3 ((((oc‘𝐾)‘𝑝) ∈ 𝐻 ∧ (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌)) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
4612, 40, 45syl2anc 587 . 2 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋))) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
47 simpl1 1188 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ HL)
4847, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ OP)
49 simpl3 1190 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
503, 7opoccl 36487 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
5148, 49, 50syl2anc 587 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
52 simpl2 1189 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
533, 7opoccl 36487 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
5448, 52, 53syl2anc 587 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
55 simpr 488 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
56 lhprelat3.s . . . . . 6 < = (lt‘𝐾)
573, 56, 7opltcon3b 36497 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5848, 52, 49, 57syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5955, 58mpbid 235 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋))
603, 33, 56, 27, 22, 4hlrelat3 36705 . . 3 (((𝐾 ∈ HL ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵) ∧ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)) → ∃𝑝 ∈ (Atoms‘𝐾)(((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)))
6147, 51, 54, 59, 60syl31anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)(((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ((oc‘𝐾)‘𝑋)))
6246, 61r19.29a 3248 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wrex 3107   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  occoc 16565  ltcplt 17543  joincjn 17546  meetcmee 17547  Latclat 17647  OPcops 36465  OLcol 36467  ccvr 36555  Atomscatm 36556  HLchlt 36643  LHypclh 37277
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36469  df-ol 36471  df-oml 36472  df-covers 36559  df-ats 36560  df-atl 36591  df-cvlat 36615  df-hlat 36644  df-lhyp 37281
This theorem is referenced by: (None)
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