Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlrelat2 Structured version   Visualization version   GIF version

Theorem hlrelat2 36419
Description: A consequence of relative atomicity. (chrelat2i 30069 analog.) (Contributed by NM, 5-Feb-2012.)
Hypotheses
Ref Expression
hlrelat2.b 𝐵 = (Base‘𝐾)
hlrelat2.l = (le‘𝐾)
hlrelat2.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlrelat2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
Distinct variable groups:   𝐴,𝑝   𝐵,𝑝   𝐾,𝑝   ,𝑝   𝑋,𝑝   𝑌,𝑝

Proof of Theorem hlrelat2
StepHypRef Expression
1 hllat 36379 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 hlrelat2.b . . . . 5 𝐵 = (Base‘𝐾)
3 hlrelat2.l . . . . 5 = (le‘𝐾)
4 eqid 2818 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
5 eqid 2818 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
62, 3, 4, 5latnlemlt 17682 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
71, 6syl3an1 1155 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
8 simp1 1128 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ HL)
92, 5latmcl 17650 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
101, 9syl3an1 1155 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
11 simp2 1129 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
12 eqid 2818 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
13 hlrelat2.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
142, 3, 4, 12, 13hlrelat 36418 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵𝑋𝐵) ∧ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋) → ∃𝑝𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋))
1514ex 413 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵𝑋𝐵) → ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋 → ∃𝑝𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋)))
168, 10, 11, 15syl3anc 1363 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋 → ∃𝑝𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋)))
17 simpl1 1183 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
1817hllatd 36380 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ Lat)
1910adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
202, 13atbase 36305 . . . . . . . . . 10 (𝑝𝐴𝑝𝐵)
2120adantl 482 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
22 simpl2 1184 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
232, 3, 12latjle12 17660 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)𝑌) ∈ 𝐵𝑝𝐵𝑋𝐵)) → (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) ↔ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋))
2418, 19, 21, 22, 23syl13anc 1364 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) ↔ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋))
25 simpr 485 . . . . . . . 8 (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) → 𝑝 𝑋)
2624, 25syl6bir 255 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋𝑝 𝑋))
2726adantld 491 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋) → 𝑝 𝑋))
28 simpl3 1185 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑌𝐵)
292, 3, 5latlem12 17676 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3018, 21, 22, 28, 29syl13anc 1364 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3130notbid 319 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ ¬ 𝑝 (𝑋(meet‘𝐾)𝑌)))
322, 3, 4, 12latnle 17683 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵𝑝𝐵) → (¬ 𝑝 (𝑋(meet‘𝐾)𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
3318, 19, 21, 32syl3anc 1363 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ 𝑝 (𝑋(meet‘𝐾)𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
3431, 33bitrd 280 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
3534, 24anbi12d 630 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((¬ (𝑝 𝑋𝑝 𝑌) ∧ ((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋)) ↔ ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋)))
36 pm3.21 472 . . . . . . . . . 10 (𝑝 𝑌 → (𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)))
37 orcom 864 . . . . . . . . . . 11 (((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
38 pm4.55 981 . . . . . . . . . . 11 (¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) ↔ ((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋))
39 imor 847 . . . . . . . . . . 11 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
4037, 38, 393bitr4ri 305 . . . . . . . . . 10 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4136, 40sylib 219 . . . . . . . . 9 (𝑝 𝑌 → ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4241con2i 141 . . . . . . . 8 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) → ¬ 𝑝 𝑌)
4342adantrl 712 . . . . . . 7 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ ((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋)) → ¬ 𝑝 𝑌)
4435, 43syl6bir 255 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋) → ¬ 𝑝 𝑌))
4527, 44jcad 513 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋) → (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
4645reximdva 3271 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (∃𝑝𝐴 ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) ∧ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋) → ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
4716, 46syld 47 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋 → ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
487, 47sylbid 241 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 → ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
492, 3lattr 17654 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5018, 21, 22, 28, 49syl13anc 1364 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5150exp4b 431 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑝𝐴 → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
5251com34 91 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑝𝐴 → (𝑋 𝑌 → (𝑝 𝑋𝑝 𝑌))))
5352com23 86 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑝𝐴 → (𝑝 𝑋𝑝 𝑌))))
5453ralrimdv 3185 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))
55 iman 402 . . . . . 6 ((𝑝 𝑋𝑝 𝑌) ↔ ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5655ralbii 3162 . . . . 5 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
57 ralnex 3233 . . . . 5 (∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5856, 57bitri 276 . . . 4 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5954, 58syl6ib 252 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
6059con2d 136 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌) → ¬ 𝑋 𝑌))
6148, 60impbid 213 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  ltcplt 17539  joincjn 17542  meetcmee 17543  Latclat 17643  Atomscatm 36279  HLchlt 36366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367
This theorem is referenced by:  lhpj1  37038
  Copyright terms: Public domain W3C validator