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Theorem lhprelat3N 39377
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 38749. (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 484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2 simpll1 1211 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3 lhprelat3.b . . . . . . . 8 𝐵 = (Base‘𝐾)
4 eqid 2731 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 38625 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
65adantl 481 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
7 eqid 2731 . . . . . . 7 (oc‘𝐾) = (oc‘𝐾)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
93, 7, 4, 8lhpoc2N 39352 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
102, 6, 9syl2anc 583 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
111, 10mpbid 231 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
1211adantr 480 . . 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 38698 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
142, 13syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
152hllatd 38700 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
16 simpll3 1213 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
173, 7opoccl 38530 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
1814, 6, 17syl2anc 583 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
19 lhprelat3.m . . . . . . . . . 10 = (meet‘𝐾)
203, 19latmcl 18403 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
2115, 16, 18, 20syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑌 ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
22 lhprelat3.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
233, 7, 22cvrcon3b 38613 . . . . . . . 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 1370 . . . . . . 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 38697 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ OL)
262, 25syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OL)
27 eqid 2731 . . . . . . . . . 10 (join‘𝐾) = (join‘𝐾)
283, 27, 19, 7oldmm3N 38555 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑝𝐵) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
2926, 16, 6, 28syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
3029breq2d 5160 . . . . . . 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 280 . . . . . 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 1212 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
33 lhprelat3.l . . . . . . . . 9 = (le‘𝐾)
343, 33, 7oplecon3b 38536 . . . . . . . 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 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ((oc‘𝐾)‘𝑝))) ((oc‘𝐾)‘𝑋)))
3629breq1d 5158 . . . . . . 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 280 . . . . . 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 630 . . . . 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 476 . . . 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 461 . . 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 7420 . . . . . 6 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 𝑤) = (𝑌 ((oc‘𝐾)‘𝑝)))
4241breq2d 5160 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 (𝑌 𝑤) ↔ 𝑋 (𝑌 ((oc‘𝐾)‘𝑝))))
4341breq1d 5158 . . . . 5 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 𝑤)𝐶𝑌 ↔ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌))
4442, 43anbi12d 630 . . . 4 (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌) ↔ (𝑋 (𝑌 ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ((oc‘𝐾)‘𝑝))𝐶𝑌)))
4544rspcev 3612 . . 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 583 . 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 1190 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ HL)
4847, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ OP)
49 simpl3 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
503, 7opoccl 38530 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
5148, 49, 50syl2anc 583 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
52 simpl2 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
533, 7opoccl 38530 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
5448, 52, 53syl2anc 583 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
55 simpr 484 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
56 lhprelat3.s . . . . . 6 < = (lt‘𝐾)
573, 56, 7opltcon3b 38540 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5848, 52, 49, 57syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
5955, 58mpbid 231 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋))
603, 33, 56, 27, 22, 4hlrelat3 38749 . . 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 1372 . 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 3161 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤𝐻 (𝑋 (𝑌 𝑤) ∧ (𝑌 𝑤)𝐶𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  occoc 17212  ltcplt 18271  joincjn 18274  meetcmee 18275  Latclat 18394  OPcops 38508  OLcol 38510  ccvr 38598  Atomscatm 38599  HLchlt 38686  LHypclh 39321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-oposet 38512  df-ol 38514  df-oml 38515  df-covers 38602  df-ats 38603  df-atl 38634  df-cvlat 38658  df-hlat 38687  df-lhyp 39325
This theorem is referenced by: (None)
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