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Theorem cvrval4N 35018
 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 34875 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
5 eqid 2651 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
6 cvrval4.j . . . . . . 7 = (join‘𝐾)
7 cvrval4.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
81, 5, 6, 3, 7cvrval3 35017 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
9 simpr 476 . . . . . . 7 ((¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 𝑝) = 𝑌)
109reximi 3040 . . . . . 6 (∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌) → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)
118, 10syl6bi 243 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌))
1211imp 444 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)
134, 12jca 553 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌))
1413ex 449 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
15 simp1r 1106 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < 𝑌)
16 simp3 1083 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 𝑝) = 𝑌)
1715, 16breqtrrd 4713 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < (𝑋 𝑝))
18 simp1l1 1174 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝐾 ∈ HL)
19 simp1l2 1175 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋𝐵)
20 simp2 1082 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑝𝐴)
211, 5, 6, 3, 7cvr1 35014 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
2218, 19, 20, 21syl3anc 1366 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
231, 2, 6, 3, 7cvr2N 35015 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2418, 19, 20, 23syl3anc 1366 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2522, 24bitr4d 271 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋 < (𝑋 𝑝)))
2617, 25mpbird 247 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → ¬ 𝑝(le‘𝐾)𝑋)
2726, 16jca 553 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌))
28273exp 1283 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑝𝐴 → ((𝑋 𝑝) = 𝑌 → (¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌))))
2928reximdvai 3044 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (∃𝑝𝐴 (𝑋 𝑝) = 𝑌 → ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
3029expimpd 628 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌) → ∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌)))
3130, 8sylibrd 249 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌) → 𝑋𝐶𝑌))
3214, 31impbid 202 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∃wrex 2942   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  ltcplt 16988  joincjn 16991   ⋖ ccvr 34867  Atomscatm 34868  HLchlt 34955 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956 This theorem is referenced by: (None)
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