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Theorem hlrelat2 36421
Description: A consequence of relative atomicity. (chrelat2i 30070 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 36381 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 hlrelat2.b . . . . 5 𝐵 = (Base‘𝐾)
3 hlrelat2.l . . . . 5 = (le‘𝐾)
4 eqid 2821 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
5 eqid 2821 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
62, 3, 4, 5latnlemlt 17684 . . . 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 17652 . . . . . 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 2821 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
13 hlrelat2.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
142, 3, 4, 12, 13hlrelat 36420 . . . . . 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 36382 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ Lat)
1910adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
202, 13atbase 36307 . . . . . . . . . 10 (𝑝𝐴𝑝𝐵)
2120adantl 482 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
22 simpl2 1184 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
232, 3, 12latjle12 17662 . . . . . . . . 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 17678 . . . . . . . . . . 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 17685 . . . . . . . . . 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 3274 . . . 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 17656 . . . . . . . . 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 3188 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))
55 iman 402 . . . . . 6 ((𝑝 𝑋𝑝 𝑌) ↔ ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5655ralbii 3165 . . . . 5 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
57 ralnex 3236 . . . . 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 3138  wrex 3139   class class class wbr 5058  cfv 6349  (class class class)co 7145  Basecbs 16473  lecple 16562  ltcplt 17541  joincjn 17544  meetcmee 17545  Latclat 17645  Atomscatm 36281  HLchlt 36368
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 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-clat 17708  df-oposet 36194  df-ol 36196  df-oml 36197  df-covers 36284  df-ats 36285  df-atl 36316  df-cvlat 36340  df-hlat 36369
This theorem is referenced by:  lhpj1  37040
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