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Theorem hlatexch3N 35620
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 994 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  K  e.  HL )
2 simp21 1027 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  P  e.  A )
3 simp22 1028 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  Q  e.  A )
4 eqid 2454 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
5 hlatexch4.j . . . . . 6  |-  .\/  =  ( join `  K )
6 hlatexch4.a . . . . . 6  |-  A  =  ( Atoms `  K )
74, 5, 6hlatlej2 35516 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q ( le `  K ) ( P 
.\/  Q ) )
81, 2, 3, 7syl3anc 1226 . . . 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 1029 . . . . . 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 35516 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  R ( le `  K ) ( P 
.\/  R ) )
111, 2, 9, 10syl3anc 1226 . . . . 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 1023 . . . . 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 4465 . . . 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 35504 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
15143ad2ant1 1015 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  ( P 
.\/  Q )  =  ( P  .\/  R
) ) )  ->  K  e.  Lat )
16 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1716, 6atbase 35430 . . . . . 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 35430 . . . . . 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 35507 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
221, 2, 3, 21syl3anc 1226 . . . . 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 15894 . . . . 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 1228 . . . 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 919 . . 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 1022 . . . 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 35617 . . . 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 1244 . . 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 2462 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   Latclat 15877   Atomscatm 35404   HLchlt 35491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492
This theorem is referenced by: (None)
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