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Theorem hlrelat2 33503
Description: A consequence of relative atomicity. (chrelat2i 28414 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 33464 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 hlrelat2.b . . . . 5 𝐵 = (Base‘𝐾)
3 hlrelat2.l . . . . 5 = (le‘𝐾)
4 eqid 2609 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
5 eqid 2609 . . . . 5 (meet‘𝐾) = (meet‘𝐾)
62, 3, 4, 5latnlemlt 16853 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
71, 6syl3an1 1350 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋))
8 simp1 1053 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ HL)
92, 5latmcl 16821 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
101, 9syl3an1 1350 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
11 simp2 1054 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
12 eqid 2609 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
13 hlrelat2.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
142, 3, 4, 12, 13hlrelat 33502 . . . . . 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 448 . . . . 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 1317 . . . 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 1056 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
1817, 1syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ Lat)
1910adantr 479 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
202, 13atbase 33390 . . . . . . . . . 10 (𝑝𝐴𝑝𝐵)
2120adantl 480 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
22 simpl2 1057 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
232, 3, 12latjle12 16831 . . . . . . . . 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 1319 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) ↔ ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋))
25 simpr 475 . . . . . . . 8 (((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋) → 𝑝 𝑋)
2624, 25syl6bir 242 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝) 𝑋𝑝 𝑋))
2726adantld 481 . . . . . 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 1058 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → 𝑌𝐵)
292, 3, 5latlem12 16847 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3018, 21, 22, 28, 29syl13anc 1319 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑝 𝑌) ↔ 𝑝 (𝑋(meet‘𝐾)𝑌)))
3130notbid 306 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ ¬ 𝑝 (𝑋(meet‘𝐾)𝑌)))
322, 3, 4, 12latnle 16854 . . . . . . . . . 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 1317 . . . . . . . . 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 266 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → (¬ (𝑝 𝑋𝑝 𝑌) ↔ (𝑋(meet‘𝐾)𝑌)(lt‘𝐾)((𝑋(meet‘𝐾)𝑌)(join‘𝐾)𝑝)))
3534, 24anbi12d 742 . . . . . . 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 pm3.21 462 . . . . . . . . . 10 (𝑝 𝑌 → (𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)))
37 orcom 400 . . . . . . . . . . 11 (((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
38 pm4.55 513 . . . . . . . . . . 11 (¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) ↔ ((𝑝 𝑋𝑝 𝑌) ∨ ¬ 𝑝 𝑋))
39 imor 426 . . . . . . . . . . 11 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ (¬ 𝑝 𝑋 ∨ (𝑝 𝑋𝑝 𝑌)))
4037, 38, 393bitr4ri 291 . . . . . . . . . 10 ((𝑝 𝑋 → (𝑝 𝑋𝑝 𝑌)) ↔ ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4136, 40sylib 206 . . . . . . . . 9 (𝑝 𝑌 → ¬ (¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋))
4241con2i 132 . . . . . . . 8 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ 𝑝 𝑋) → ¬ 𝑝 𝑌)
4342adantrl 747 . . . . . . 7 ((¬ (𝑝 𝑋𝑝 𝑌) ∧ ((𝑋(meet‘𝐾)𝑌) 𝑋𝑝 𝑋)) → ¬ 𝑝 𝑌)
4435, 43syl6bir 242 . . . . . 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 553 . . . . 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 2999 . . . 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 45 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(lt‘𝐾)𝑋 → ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
487, 47sylbid 228 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 → ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
492, 3lattr 16825 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5018, 21, 22, 28, 49syl13anc 1319 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐴) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
5150exp4b 629 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑝𝐴 → (𝑝 𝑋 → (𝑋 𝑌𝑝 𝑌))))
5251com34 88 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑝𝐴 → (𝑋 𝑌 → (𝑝 𝑋𝑝 𝑌))))
5352com23 83 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑝𝐴 → (𝑝 𝑋𝑝 𝑌))))
5453ralrimdv 2950 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))
55 iman 438 . . . . . 6 ((𝑝 𝑋𝑝 𝑌) ↔ ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5655ralbii 2962 . . . . 5 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
57 ralnex 2974 . . . . 5 (∀𝑝𝐴 ¬ (𝑝 𝑋 ∧ ¬ 𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5856, 57bitri 262 . . . 4 (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌))
5954, 58syl6ib 239 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ¬ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
6059con2d 127 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌) → ¬ 𝑋 𝑌))
6148, 60impbid 200 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  ltcplt 16710  joincjn 16713  meetcmee 16714  Latclat 16814  Atomscatm 33364  HLchlt 33451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33277  df-ol 33279  df-oml 33280  df-covers 33367  df-ats 33368  df-atl 33399  df-cvlat 33423  df-hlat 33452
This theorem is referenced by:  lhpj1  34122
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