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Theorem lhprelat3N 35996
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 35368. (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 477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2 simpll1 1269 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3 lhprelat3.b . . . . . . . 8 𝐵 = (Base‘𝐾)
4 eqid 2765 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 35245 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 473 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 eqid 2765 . . . . . . 7 (oc‘𝐾) = (oc‘𝐾)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
93, 7, 4, 8lhpoc2N 35971 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
102, 6, 9syl2anc 579 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
111, 10mpbid 223 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
1211adantr 472 . . 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 35318 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
142, 13syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
152hllatd 35320 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
16 simpll3 1273 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
173, 7opoccl 35150 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
1814, 6, 17syl2anc 579 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
19 lhprelat3.m . . . . . . . . . 10 = (meet‘𝐾)
203, 19latmcl 17318 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
2115, 16, 18, 20syl3anc 1490 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
22 lhprelat3.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
233, 7, 22cvrcon3b 35233 . . . . . . . 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 1490 . . . . . . 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 35317 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ OL)
262, 25syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OL)
27 eqid 2765 . . . . . . . . . 10 (join‘𝐾) = (join‘𝐾)
283, 27, 19, 7oldmm3N 35175 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑝𝐵) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
2926, 16, 6, 28syl3anc 1490 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
3029breq2d 4821 . . . . . . 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 271 . . . . . 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 1271 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
33 lhprelat3.l . . . . . . . . 9 = (le‘𝐾)
343, 33, 7oplecon3b 35156 . . . . . . . 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 1490 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋)))
3629breq1d 4819 . . . . . . 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 271 . . . . . 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 624 . . . . 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 468 . . . 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 453 . . 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 6850 . . . . . 6 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 𝑤) = (𝑌 ((oc‘𝐾)‘𝑝)))
4241breq2d 4821 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 (𝑌 𝑤) ↔ 𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
4341breq1d 4819 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 𝑤)𝐶𝑌 ↔ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
4442, 43anbi12d 624 . . . 4 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌) ↔ (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌)))
4544rspcev 3461 . . 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 579 . 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 1242 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ HL)
4847, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ OP)
49 simpl3 1246 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
503, 7opoccl 35150 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
5148, 49, 50syl2anc 579 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
52 simpl2 1244 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
533, 7opoccl 35150 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
5448, 52, 53syl2anc 579 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
55 simpr 477 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
56 lhprelat3.s . . . . . 6 < = (lt‘𝐾)
573, 56, 7opltcon3b 35160 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5848, 52, 49, 57syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5955, 58mpbid 223 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋))
603, 33, 56, 27, 22, 4hlrelat3 35368 . . 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 1492 . 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 3225 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wrex 3056   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  occoc 16222  ltcplt 17207  joincjn 17210  meetcmee 17211  Latclat 17311  OPcops 35128  OLcol 35130  ccvr 35218  Atomscatm 35219  HLchlt 35306  LHypclh 35940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-p1 17306  df-lat 17312  df-clat 17374  df-oposet 35132  df-ol 35134  df-oml 35135  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278  df-hlat 35307  df-lhyp 35944
This theorem is referenced by: (None)
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