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Theorem cvrval4N 37165
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 37021 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
5 eqid 2737 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
6 cvrval4.j . . . . . . 7 = (join‘𝐾)
7 cvrval4.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
81, 5, 6, 3, 7cvrval3 37164 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
9 simpr 488 . . . . . . 7 ((¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 𝑝) = 𝑌)
109reximi 3166 . . . . . 6 (∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌) → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)
118, 10syl6bi 256 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌))
1211imp 410 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)
134, 12jca 515 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌))
1413ex 416 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
15 simp1r 1200 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < 𝑌)
16 simp3 1140 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 𝑝) = 𝑌)
1715, 16breqtrrd 5081 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < (𝑋 𝑝))
18 simp1l1 1268 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝐾 ∈ HL)
19 simp1l2 1269 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋𝐵)
20 simp2 1139 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑝𝐴)
211, 5, 6, 3, 7cvr1 37161 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
2218, 19, 20, 21syl3anc 1373 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
231, 2, 6, 3, 7cvr2N 37162 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2418, 19, 20, 23syl3anc 1373 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2522, 24bitr4d 285 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋 < (𝑋 𝑝)))
2617, 25mpbird 260 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → ¬ 𝑝(le‘𝐾)𝑋)
2726, 16jca 515 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌))
28273exp 1121 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑝𝐴 → ((𝑋 𝑝) = 𝑌 → (¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌))))
2928reximdvai 3191 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (∃𝑝𝐴 (𝑋 𝑝) = 𝑌 → ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
3029expimpd 457 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌) → ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
3130, 8sylibrd 262 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌) → 𝑋𝐶𝑌))
3214, 31impbid 215 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wrex 3062   class class class wbr 5053  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  ltcplt 17815  joincjn 17818  ccvr 37013  Atomscatm 37014  HLchlt 37101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-lat 17938  df-clat 18005  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102
This theorem is referenced by: (None)
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