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Theorem cvrval4N 29421
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 29278 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
5 eqid 2316 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
6 cvrval4.j . . . . . . 7  |-  .\/  =  ( join `  K )
7 cvrval4.a . . . . . . 7  |-  A  =  ( Atoms `  K )
81, 5, 6, 3, 7cvrval3 29420 . . . . . 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 2684 . . . . . 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 4086 . . . . . . . 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 29417 . . . . . . . . . 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 29418 . . . . . . . . . 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 2687 . . . 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 1633    e. wcel 1701   E.wrex 2578   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   ltcplt 14124   joincjn 14127    <o ccvr 29270   Atomscatm 29271   HLchlt 29358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359
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