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Theorem cvrval4N 32946
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 32803 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
5 eqid 2423 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
6 cvrval4.j . . . . . . 7  |-  .\/  =  ( join `  K )
7 cvrval4.a . . . . . . 7  |-  A  =  ( Atoms `  K )
81, 5, 6, 3, 7cvrval3 32945 . . . . . 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 463 . . . . . . 7  |-  ( ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
109reximi 2894 . . . . . 6  |-  ( E. p  e.  A  ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
118, 10syl6bi 232 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1211imp 431 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
134, 12jca 535 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1413ex 436 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) ) )
15 simp1r 1031 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  Y )
16 simp3 1008 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
1715, 16breqtrrd 4449 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  ( X  .\/  p
) )
18 simp1l1 1099 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  K  e.  HL )
19 simp1l2 1100 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  e.  B )
20 simp2 1007 . . . . . . . . . 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 32942 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( -.  p ( le `  K ) X  <->  X C ( X 
.\/  p ) ) )
2218, 19, 20, 21syl3anc 1265 . . . . . . . . 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 32943 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  X C
( X  .\/  p
) ) )
2418, 19, 20, 23syl3anc 1265 . . . . . . . . 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 260 . . . . . . . 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 236 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  -.  p ( le `  K ) X )
2726, 16jca 535 . . . . . 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 1205 . . . . 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 2898 . . . 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 607 . . 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 238 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y )  ->  X C Y ) )
3214, 31impbid 194 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   E.wrex 2777   class class class wbr 4422   ` cfv 5600  (class class class)co 6304   Basecbs 15118   lecple 15194   ltcplt 16183   joincjn 16186    <o ccvr 32795   Atomscatm 32796   HLchlt 32883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4219  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6266  df-ov 6307  df-oprab 6308  df-preset 16170  df-poset 16188  df-plt 16201  df-lub 16217  df-glb 16218  df-join 16219  df-meet 16220  df-p0 16282  df-lat 16289  df-clat 16351  df-oposet 32709  df-ol 32711  df-oml 32712  df-covers 32799  df-ats 32800  df-atl 32831  df-cvlat 32855  df-hlat 32884
This theorem is referenced by: (None)
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