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

Theorem cvrval4N 33366
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 33223 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
5 eqid 2451 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
6 cvrval4.j . . . . . . 7  |-  .\/  =  ( join `  K )
7 cvrval4.a . . . . . . 7  |-  A  =  ( Atoms `  K )
81, 5, 6, 3, 7cvrval3 33365 . . . . . 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 461 . . . . . . 7  |-  ( ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
109reximi 2921 . . . . . 6  |-  ( E. p  e.  A  ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
118, 10syl6bi 228 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1211imp 429 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
134, 12jca 532 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1413ex 434 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) ) )
15 simp1r 1013 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  Y )
16 simp3 990 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
1715, 16breqtrrd 4418 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  ( X  .\/  p
) )
18 simp1l1 1081 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  K  e.  HL )
19 simp1l2 1082 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  e.  B )
20 simp2 989 . . . . . . . . . 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 33362 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( -.  p ( le `  K ) X  <->  X C ( X 
.\/  p ) ) )
2218, 19, 20, 21syl3anc 1219 . . . . . . . . 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 33363 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  X C
( X  .\/  p
) ) )
2418, 19, 20, 23syl3anc 1219 . . . . . . . . 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 256 . . . . . . . 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 232 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  -.  p ( le `  K ) X )
2726, 16jca 532 . . . . . 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 1187 . . . . 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 2924 . . . 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 603 . . 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 234 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y )  ->  X C Y ) )
3214, 31impbid 191 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2796   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   ltcplt 15215   joincjn 15218    <o ccvr 33215   Atomscatm 33216   HLchlt 33303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator