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Theorem hlrelat2 35211
Description: A consequence of relative atomicity. (chrelat2i 29555 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 35172 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 hlrelat2.b . . . . 5 𝐵 = (Base‘𝐾)
3 hlrelat2.l . . . . 5 = (le‘𝐾)
4 eqid 2761 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
5 eqid 2761 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
62, 3, 4, 5latnlemlt 17306 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
71, 6syl3an1 1167 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
8 simp1 1131 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ HL)
92, 5latmcl 17274 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
101, 9syl3an1 1167 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
11 simp2 1132 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
12 eqid 2761 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
13 hlrelat2.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
142, 3, 4, 12, 13hlrelat 35210 . . . . . 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 449 . . . . 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 1477 . . . 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 1228 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
1817, 1syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ Lat)
1910adantr 472 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
202, 13atbase 35098 . . . . . . . . . 10 (𝑝𝐴𝑝𝐵)
2120adantl 473 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
22 simpl2 1230 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
232, 3, 12latjle12 17284 . . . . . . . . 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 1479 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) ↔ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋))
25 simpr 479 . . . . . . . 8 (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) → 𝑝 𝑋)
2624, 25syl6bir 244 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋𝑝 𝑋))
2726adantld 484 . . . . . 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 1232 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑌𝐵)
292, 3, 5latlem12 17300 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3018, 21, 22, 28, 29syl13anc 1479 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3130notbid 307 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ ¬ 𝑝 (𝑋(meet‘𝐾)𝑌)))
322, 3, 4, 12latnle 17307 . . . . . . . . . 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 1477 . . . . . . . . 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 268 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
3534, 24anbi12d 749 . . . . . . 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 pm3.21 463 . . . . . . . . . 10 (𝑝 𝑌 → (𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)))
37 orcom 401 . . . . . . . . . . 11 (((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
38 pm4.55 516 . . . . . . . . . . 11 (¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) ↔ ((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋))
39 imor 427 . . . . . . . . . . 11 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
4037, 38, 393bitr4ri 293 . . . . . . . . . 10 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4136, 40sylib 208 . . . . . . . . 9 (𝑝 𝑌 → ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4241con2i 134 . . . . . . . 8 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) → ¬ 𝑝 𝑌)
4342adantrl 754 . . . . . . 7 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ ((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋)) → ¬ 𝑝 𝑌)
4435, 43syl6bir 244 . . . . . 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 556 . . . . 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 3156 . . . 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 230 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 → ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
492, 3lattr 17278 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5018, 21, 22, 28, 49syl13anc 1479 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5150exp4b 633 . . . . . . 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 3107 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))
55 iman 439 . . . . . 6 ((𝑝 𝑋𝑝 𝑌) ↔ ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5655ralbii 3119 . . . . 5 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
57 ralnex 3131 . . . . 5 (∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5856, 57bitri 264 . . . 4 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5954, 58syl6ib 241 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
6059con2d 129 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌) → ¬ 𝑋 𝑌))
6148, 60impbid 202 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051  wrex 3052   class class class wbr 4805  cfv 6050  (class class class)co 6815  Basecbs 16080  lecple 16171  ltcplt 17163  joincjn 17166  meetcmee 17167  Latclat 17267  Atomscatm 35072  HLchlt 35159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-preset 17150  df-poset 17168  df-plt 17180  df-lub 17196  df-glb 17197  df-join 17198  df-meet 17199  df-p0 17261  df-lat 17268  df-clat 17330  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160
This theorem is referenced by:  lhpj1  35830
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