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Theorem hlrelat2 36699
Description: A consequence of relative atomicity. (chrelat2i 30148 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 36659 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 hlrelat2.b . . . . 5 𝐵 = (Base‘𝐾)
3 hlrelat2.l . . . . 5 = (le‘𝐾)
4 eqid 2798 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
5 eqid 2798 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
62, 3, 4, 5latnlemlt 17686 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
71, 6syl3an1 1160 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
8 simp1 1133 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ HL)
92, 5latmcl 17654 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
101, 9syl3an1 1160 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
11 simp2 1134 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
12 eqid 2798 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
13 hlrelat2.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
142, 3, 4, 12, 13hlrelat 36698 . . . . . 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 416 . . . . 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 1368 . . . 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 1188 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
1817hllatd 36660 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ Lat)
1910adantr 484 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
202, 13atbase 36585 . . . . . . . . . 10 (𝑝𝐴𝑝𝐵)
2120adantl 485 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
22 simpl2 1189 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
232, 3, 12latjle12 17664 . . . . . . . . 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 1369 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) ↔ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋))
25 simpr 488 . . . . . . . 8 (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) → 𝑝 𝑋)
2624, 25syl6bir 257 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋𝑝 𝑋))
2726adantld 494 . . . . . 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 1190 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑌𝐵)
292, 3, 5latlem12 17680 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3018, 21, 22, 28, 29syl13anc 1369 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3130notbid 321 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ ¬ 𝑝 (𝑋(meet‘𝐾)𝑌)))
322, 3, 4, 12latnle 17687 . . . . . . . . . 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 1368 . . . . . . . . 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 282 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
3534, 24anbi12d 633 . . . . . . 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 pm3.21 475 . . . . . . . . . 10 (𝑝 𝑌 → (𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)))
37 orcom 867 . . . . . . . . . . 11 (((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
38 pm4.55 985 . . . . . . . . . . 11 (¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) ↔ ((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋))
39 imor 850 . . . . . . . . . . 11 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
4037, 38, 393bitr4ri 307 . . . . . . . . . 10 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4136, 40sylib 221 . . . . . . . . 9 (𝑝 𝑌 → ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4241con2i 141 . . . . . . . 8 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) → ¬ 𝑝 𝑌)
4342adantrl 715 . . . . . . 7 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ ((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋)) → ¬ 𝑝 𝑌)
4435, 43syl6bir 257 . . . . . 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 516 . . . . 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 3233 . . . 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 243 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 → ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
492, 3lattr 17658 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5018, 21, 22, 28, 49syl13anc 1369 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5150exp4b 434 . . . . . . 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 3153 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))
55 iman 405 . . . . . 6 ((𝑝 𝑋𝑝 𝑌) ↔ ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5655ralbii 3133 . . . . 5 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
57 ralnex 3199 . . . . 5 (∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5856, 57bitri 278 . . . 4 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5954, 58syl6ib 254 . . 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 215 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  ltcplt 17543  joincjn 17546  meetcmee 17547  Latclat 17647  Atomscatm 36559  HLchlt 36646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647
This theorem is referenced by:  lhpj1  37318
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