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Theorem cvrval4N 29603
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 29460 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
5 eqid 2283 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
6 cvrval4.j . . . . . . 7  |-  .\/  =  ( join `  K )
7 cvrval4.a . . . . . . 7  |-  A  =  ( Atoms `  K )
81, 5, 6, 3, 7cvrval3 29602 . . . . . 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 447 . . . . . . 7  |-  ( ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
109reximi 2650 . . . . . 6  |-  ( E. p  e.  A  ( -.  p ( le
`  K ) X  /\  ( X  .\/  p )  =  Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
118, 10syl6bi 219 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1211imp 418 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  E. p  e.  A  ( X  .\/  p )  =  Y )
134, 12jca 518 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) )
1413ex 423 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) ) )
15 simp1r 980 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  Y )
16 simp3 957 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  ( X  .\/  p )  =  Y )
1715, 16breqtrrd 4049 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  .<  ( X  .\/  p
) )
18 simp1l1 1048 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  K  e.  HL )
19 simp1l2 1049 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  X  e.  B )
20 simp2 956 . . . . . . . . . 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 29599 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( -.  p ( le `  K ) X  <->  X C ( X 
.\/  p ) ) )
2218, 19, 20, 21syl3anc 1182 . . . . . . . . 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 29600 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  X C
( X  .\/  p
) ) )
2418, 19, 20, 23syl3anc 1182 . . . . . . . . 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 247 . . . . . . . 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 223 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A  /\  ( X  .\/  p )  =  Y )  ->  -.  p ( le `  K ) X )
2726, 16jca 518 . . . . . 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 1150 . . . . 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 2653 . . . 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 586 . . 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 225 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y )  ->  X C Y ) )
3214, 31impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   ltcplt 14075   joincjn 14078    <o ccvr 29452   Atomscatm 29453   HLchlt 29540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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