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Theorem cvrval4N 30211
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  =  (  <o  `  K )
cvrval4.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvrval4N  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  E. p  e.  A  ( 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  =  (  <o  `  K )
41, 2, 3cvrlt 30068 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
5 eqid 2436 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
6 cvrval4.j . . . . . . 7  |-  .\/  =  ( join `  K )
7 cvrval4.a . . . . . . 7  |-  A  =  ( Atoms `  K )
81, 5, 6, 3, 7cvrval3 30210 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  E. p  e.  A  ( -.  p ( le `  K ) X  /\  ( X  .\/  p )  =  Y ) ) )
9 simpr 448 . . . . . . 7  |-  ( ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
109reximi 2813 . . . . . 6  |-  ( E. p  e.  A  ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
118, 10syl6bi 220 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1211imp 419 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
134, 12jca 519 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1413ex 424 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) ) )
15 simp1r 982 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  Y )
16 simp3 959 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
1715, 16breqtrrd 4238 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  ( X  .\/  p
) )
18 simp1l1 1050 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  K  e.  HL )
19 simp1l2 1051 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  e.  B )
20 simp2 958 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  p  e.  A )
211, 5, 6, 3, 7cvr1 30207 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( -.  p ( le `  K ) X  <->  X C ( X 
.\/  p ) ) )
2218, 19, 20, 21syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( -.  p ( le `  K ) X  <->  X C
( X  .\/  p
) ) )
231, 2, 6, 3, 7cvr2N 30208 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  X C
( X  .\/  p
) ) )
2418, 19, 20, 23syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( X  .<  ( X  .\/  p )  <->  X C
( X  .\/  p
) ) )
2522, 24bitr4d 248 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( -.  p ( le `  K ) X  <->  X  .<  ( X  .\/  p ) ) )
2617, 25mpbird 224 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  -.  p ( le `  K ) X )
2726, 16jca 519 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( -.  p ( le `  K ) X  /\  ( X  .\/  p )  =  Y ) )
28273exp 1152 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  ( p  e.  A  ->  ( ( X  .\/  p )  =  Y  ->  ( -.  p ( le `  K ) X  /\  ( X  .\/  p )  =  Y ) ) ) )
2928reximdvai 2816 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  ( E. p  e.  A  ( X  .\/  p )  =  Y  ->  E. p  e.  A  ( -.  p ( le `  K ) X  /\  ( X  .\/  p )  =  Y ) ) )
3029expimpd 587 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y )  ->  E. p  e.  A  ( -.  p ( le `  K ) X  /\  ( X  .\/  p )  =  Y ) ) )
3130, 8sylibrd 226 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y )  ->  X C Y ) )
3214, 31impbid 184 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   ltcplt 14398   joincjn 14401    <o ccvr 30060   Atomscatm 30061   HLchlt 30148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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