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Theorem cvrval4N 37844
Description: Binary relation expressing 𝑌 covers 𝑋. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvrval4.b 𝐵 = (Base‘𝐾)
cvrval4.s < = (lt‘𝐾)
cvrval4.j = (join‘𝐾)
cvrval4.c 𝐶 = ( ⋖ ‘𝐾)
cvrval4.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cvrval4N ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
Distinct variable groups:   < ,𝑝   𝐴,𝑝   𝐵,𝑝   𝐶,𝑝   𝐾,𝑝   𝑋,𝑝   𝑌,𝑝
Allowed substitution hint:   (𝑝)

Proof of Theorem cvrval4N
StepHypRef Expression
1 cvrval4.b . . . . 5 𝐵 = (Base‘𝐾)
2 cvrval4.s . . . . 5 < = (lt‘𝐾)
3 cvrval4.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrlt 37699 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
5 eqid 2736 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
6 cvrval4.j . . . . . . 7 = (join‘𝐾)
7 cvrval4.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
81, 5, 6, 3, 7cvrval3 37843 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
9 simpr 485 . . . . . . 7 ((¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 𝑝) = 𝑌)
109reximi 3085 . . . . . 6 (∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌) → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)
118, 10syl6bi 252 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌))
1211imp 407 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)
134, 12jca 512 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌))
1413ex 413 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
15 simp1r 1198 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < 𝑌)
16 simp3 1138 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 𝑝) = 𝑌)
1715, 16breqtrrd 5131 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < (𝑋 𝑝))
18 simp1l1 1266 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝐾 ∈ HL)
19 simp1l2 1267 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋𝐵)
20 simp2 1137 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑝𝐴)
211, 5, 6, 3, 7cvr1 37840 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
2218, 19, 20, 21syl3anc 1371 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
231, 2, 6, 3, 7cvr2N 37841 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2418, 19, 20, 23syl3anc 1371 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2522, 24bitr4d 281 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋 < (𝑋 𝑝)))
2617, 25mpbird 256 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → ¬ 𝑝(le‘𝐾)𝑋)
2726, 16jca 512 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌))
28273exp 1119 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑝𝐴 → ((𝑋 𝑝) = 𝑌 → (¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌))))
2928reximdvai 3160 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (∃𝑝𝐴 (𝑋 𝑝) = 𝑌 → ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
3029expimpd 454 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌) → ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
3130, 8sylibrd 258 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌) → 𝑋𝐶𝑌))
3214, 31impbid 211 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3071   class class class wbr 5103  cfv 6493  (class class class)co 7353  Basecbs 17075  lecple 17132  ltcplt 18189  joincjn 18192  ccvr 37691  Atomscatm 37692  HLchlt 37779
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-proset 18176  df-poset 18194  df-plt 18211  df-lub 18227  df-glb 18228  df-join 18229  df-meet 18230  df-p0 18306  df-lat 18313  df-clat 18380  df-oposet 37605  df-ol 37607  df-oml 37608  df-covers 37695  df-ats 37696  df-atl 37727  df-cvlat 37751  df-hlat 37780
This theorem is referenced by: (None)
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