Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrval4N Structured version   Visualization version   GIF version

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)
  Copyright terms: Public domain W3C validator