Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlatexch3N Structured version   Unicode version

Theorem hlatexch3N 33430
Description: Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlatexch4.j  |-  .\/  =  ( join `  K )
hlatexch4.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlatexch3N  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( P  .\/  Q
)  =  ( Q 
.\/  R ) )

Proof of Theorem hlatexch3N
StepHypRef Expression
1 simp1 988 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  K  e.  HL )
2 simp21 1021 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  P  e.  A )
3 simp22 1022 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  Q  e.  A )
4 eqid 2451 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
5 hlatexch4.j . . . . . 6  |-  .\/  =  ( join `  K )
6 hlatexch4.a . . . . . 6  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 33326 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q ( le `  K ) ( P 
.\/  Q ) )
81, 2, 3, 7syl3anc 1219 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  Q ( le `  K ) ( P 
.\/  Q ) )
9 simp23 1023 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  R  e.  A )
104, 5, 6hlatlej2 33326 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  R ( le `  K ) ( P 
.\/  R ) )
111, 2, 9, 10syl3anc 1219 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  R ( le `  K ) ( P 
.\/  R ) )
12 simp3r 1017 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( P  .\/  Q
)  =  ( P 
.\/  R ) )
1311, 12breqtrrd 4416 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  R ( le `  K ) ( P 
.\/  Q ) )
14 hllat 33314 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
15143ad2ant1 1009 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  K  e.  Lat )
16 eqid 2451 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1716, 6atbase 33240 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
183, 17syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  Q  e.  ( Base `  K ) )
1916, 6atbase 33240 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
209, 19syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  R  e.  ( Base `  K ) )
2116, 5, 6hlatjcl 33317 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
221, 2, 3, 21syl3anc 1219 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2316, 4, 5latjle12 15334 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( Q ( le `  K ) ( P  .\/  Q
)  /\  R ( le `  K ) ( P  .\/  Q ) )  <->  ( Q  .\/  R ) ( le `  K ) ( P 
.\/  Q ) ) )
2415, 18, 20, 22, 23syl13anc 1221 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( ( Q ( le `  K ) ( P  .\/  Q
)  /\  R ( le `  K ) ( P  .\/  Q ) )  <->  ( Q  .\/  R ) ( le `  K ) ( P 
.\/  Q ) ) )
258, 13, 24mpbi2and 912 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( Q  .\/  R
) ( le `  K ) ( P 
.\/  Q ) )
26 simp3l 1016 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  Q  =/=  R )
274, 5, 6ps-1 33427 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R
)  /\  ( P  e.  A  /\  Q  e.  A ) )  -> 
( ( Q  .\/  R ) ( le `  K ) ( P 
.\/  Q )  <->  ( Q  .\/  R )  =  ( P  .\/  Q ) ) )
281, 3, 9, 26, 2, 3, 27syl132anc 1237 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( ( Q  .\/  R ) ( le `  K ) ( P 
.\/  Q )  <->  ( Q  .\/  R )  =  ( P  .\/  Q ) ) )
2925, 28mpbid 210 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( Q  .\/  R
)  =  ( P 
.\/  Q ) )
3029eqcomd 2459 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  -> 
( P  .\/  Q
)  =  ( Q 
.\/  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   Latclat 15317   Atomscatm 33214   HLchlt 33301
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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator