Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddasslem2 Structured version   Unicode version

Theorem paddasslem2 33823
Description: Lemma for paddass 33840. (Contributed by NM, 8-Jan-2012.)
Hypotheses
Ref Expression
paddasslem.l  |-  .<_  =  ( le `  K )
paddasslem.j  |-  .\/  =  ( join `  K )
paddasslem.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
paddasslem2  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  z  .<_  ( r 
.\/  y ) )

Proof of Theorem paddasslem2
StepHypRef Expression
1 simp1l 1012 . . . 4  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  K  e.  HL )
2 simp1r 1013 . . . . 5  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  r  e.  A
)
3 simp23 1023 . . . . 5  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  z  e.  A
)
4 simp22 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  y  e.  A
)
52, 3, 43jca 1168 . . . 4  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  ( r  e.  A  /\  z  e.  A  /\  y  e.  A ) )
6 simp21 1021 . . . . 5  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  x  e.  A
)
7 simp3l 1016 . . . . 5  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  -.  r  .<_  ( x  .\/  y ) )
8 paddasslem.l . . . . . 6  |-  .<_  =  ( le `  K )
9 paddasslem.j . . . . . 6  |-  .\/  =  ( join `  K )
10 paddasslem.a . . . . . 6  |-  A  =  ( Atoms `  K )
118, 9, 10atnlej2 33382 . . . . 5  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  x  e.  A  /\  y  e.  A
)  /\  -.  r  .<_  ( x  .\/  y
) )  ->  r  =/=  y )
121, 2, 6, 4, 7, 11syl131anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  r  =/=  y
)
131, 5, 123jca 1168 . . 3  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  ( K  e.  HL  /\  ( r  e.  A  /\  z  e.  A  /\  y  e.  A )  /\  r  =/=  y ) )
14 simp3r 1017 . . 3  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  r  .<_  ( y 
.\/  z ) )
158, 9, 10hlatexch1 33397 . . 3  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  z  e.  A  /\  y  e.  A
)  /\  r  =/=  y )  ->  (
r  .<_  ( y  .\/  z )  ->  z  .<_  ( y  .\/  r
) ) )
1613, 14, 15sylc 60 . 2  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  z  .<_  ( y 
.\/  r ) )
17 hllat 33366 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
181, 17syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  K  e.  Lat )
19 eqid 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2019, 10atbase 33292 . . . 4  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
212, 20syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  r  e.  (
Base `  K )
)
2219, 10atbase 33292 . . . 4  |-  ( y  e.  A  ->  y  e.  ( Base `  K
) )
234, 22syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  y  e.  (
Base `  K )
)
2419, 9latjcom 15351 . . 3  |-  ( ( K  e.  Lat  /\  r  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  (
r  .\/  y )  =  ( y  .\/  r ) )
2518, 21, 23, 24syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  ( r  .\/  y )  =  ( y  .\/  r ) )
2616, 25breqtrrd 4429 1  |-  ( ( ( K  e.  HL  /\  r  e.  A )  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  /\  ( -.  r  .<_  ( x 
.\/  y )  /\  r  .<_  ( y  .\/  z ) ) )  ->  z  .<_  ( r 
.\/  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Latclat 15337   Atomscatm 33266   HLchlt 33353
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354
This theorem is referenced by:  paddasslem4  33825
  Copyright terms: Public domain W3C validator