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Theorem cvrval4N 36419
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 36275 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
5 eqid 2825 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
6 cvrval4.j . . . . . . 7 = (join‘𝐾)
7 cvrval4.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
81, 5, 6, 3, 7cvrval3 36418 . . . . . 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 3247 . . . . . 6 (∃𝑝𝐴𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌) → ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)
118, 10syl6bi 254 . . . . 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 1192 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < 𝑌)
16 simp3 1132 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 𝑝) = 𝑌)
1715, 16breqtrrd 5090 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋 < (𝑋 𝑝))
18 simp1l1 1260 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝐾 ∈ HL)
19 simp1l2 1261 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑋𝐵)
20 simp2 1131 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → 𝑝𝐴)
211, 5, 6, 3, 7cvr1 36415 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
2218, 19, 20, 21syl3anc 1365 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
231, 2, 6, 3, 7cvr2N 36416 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2418, 19, 20, 23syl3anc 1365 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (𝑋 < (𝑋 𝑝) ↔ 𝑋𝐶(𝑋 𝑝)))
2522, 24bitr4d 283 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝𝐴 ∧ (𝑋 𝑝) = 𝑌) → (¬ 𝑝(le‘𝐾)𝑋𝑋 < (𝑋 𝑝)))
2617, 25mpbird 258 . . . . . . 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 1113 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑝𝐴 → ((𝑋 𝑝) = 𝑌 → (¬ 𝑝(le‘𝐾)𝑋 ∧ (𝑋 𝑝) = 𝑌))))
2928reximdvai 3276 . . . 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 260 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌) → 𝑋𝐶𝑌))
3214, 31impbid 213 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wrex 3143   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  ltcplt 17543  joincjn 17546  ccvr 36267  Atomscatm 36268  HLchlt 36355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36181  df-ol 36183  df-oml 36184  df-covers 36271  df-ats 36272  df-atl 36303  df-cvlat 36327  df-hlat 36356
This theorem is referenced by: (None)
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