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Theorem paddasslem13 30556
Description: Lemma for paddass 30562. The case when  r 
.<_  ( x  .\/  y
). (Unlike the proof in Maeda and Maeda, we don't need  x  =/=  y.) (Contributed by NM, 11-Jan-2012.)
Hypotheses
Ref Expression
paddasslem.l  |-  .<_  =  ( le `  K )
paddasslem.j  |-  .\/  =  ( join `  K )
paddasslem.a  |-  A  =  ( Atoms `  K )
paddasslem.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddasslem13  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z
) )

Proof of Theorem paddasslem13
StepHypRef Expression
1 simpl1l 1008 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  K  e.  HL )
2 simpl21 1035 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  X  C_  A
)
3 simpl22 1036 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  Y  C_  A
)
4 paddasslem.a . . . . 5  |-  A  =  ( Atoms `  K )
5 paddasslem.p . . . . 5  |-  .+  =  ( + P `  K
)
64, 5paddssat 30538 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
71, 2, 3, 6syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  ( X  .+  Y )  C_  A
)
8 simpl23 1037 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  Z  C_  A
)
94, 5sspadd1 30539 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  Y ) 
C_  A  /\  Z  C_  A )  ->  ( X  .+  Y )  C_  ( ( X  .+  Y )  .+  Z
) )
101, 7, 8, 9syl3anc 1184 . 2  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  ( X  .+  Y )  C_  (
( X  .+  Y
)  .+  Z )
)
11 hllat 30088 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
121, 11syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  K  e.  Lat )
13 simprll 739 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  x  e.  X )
14 simprlr 740 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  y  e.  Y )
15 simpl3l 1012 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  e.  A )
16 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
17 paddasslem.l . . . 4  |-  .<_  =  ( le `  K )
1816, 4atbase 30014 . . . . 5  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
1915, 18syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  e.  ( Base `  K )
)
202, 13sseldd 3341 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  x  e.  A )
2116, 4atbase 30014 . . . . . 6  |-  ( x  e.  A  ->  x  e.  ( Base `  K
) )
2220, 21syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  x  e.  ( Base `  K )
)
23 simpl3r 1013 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  r  e.  A )
2416, 4atbase 30014 . . . . . 6  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
2523, 24syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  r  e.  ( Base `  K )
)
26 paddasslem.j . . . . . 6  |-  .\/  =  ( join `  K )
2716, 26latjcl 14471 . . . . 5  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  r  e.  ( Base `  K
) )  ->  (
x  .\/  r )  e.  ( Base `  K
) )
2812, 22, 25, 27syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  ( x  .\/  r )  e.  (
Base `  K )
)
293, 14sseldd 3341 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  y  e.  A )
3016, 4atbase 30014 . . . . . 6  |-  ( y  e.  A  ->  y  e.  ( Base `  K
) )
3129, 30syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  y  e.  ( Base `  K )
)
3216, 26latjcl 14471 . . . . 5  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  (
x  .\/  y )  e.  ( Base `  K
) )
3312, 22, 31, 32syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  ( x  .\/  y )  e.  (
Base `  K )
)
34 simprrr 742 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  .<_  ( x  .\/  r ) )
3516, 17, 26latlej1 14481 . . . . . 6  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  x  .<_  ( x  .\/  y
) )
3612, 22, 31, 35syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  x  .<_  ( x  .\/  y ) )
37 simprrl 741 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  r  .<_  ( x  .\/  y ) )
3816, 17, 26latjle12 14483 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( x  e.  ( Base `  K )  /\  r  e.  ( Base `  K )  /\  (
x  .\/  y )  e.  ( Base `  K
) ) )  -> 
( ( x  .<_  ( x  .\/  y )  /\  r  .<_  ( x 
.\/  y ) )  <-> 
( x  .\/  r
)  .<_  ( x  .\/  y ) ) )
3912, 22, 25, 33, 38syl13anc 1186 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  ( (
x  .<_  ( x  .\/  y )  /\  r  .<_  ( x  .\/  y
) )  <->  ( x  .\/  r )  .<_  ( x 
.\/  y ) ) )
4036, 37, 39mpbi2and 888 . . . 4  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  ( x  .\/  r )  .<_  ( x 
.\/  y ) )
4116, 17, 12, 19, 28, 33, 34, 40lattrd 14479 . . 3  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  .<_  ( x  .\/  y ) )
4217, 26, 4, 5elpaddri 30526 . . 3  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( x  e.  X  /\  y  e.  Y
)  /\  ( p  e.  A  /\  p  .<_  ( x  .\/  y
) ) )  ->  p  e.  ( X  .+  Y ) )
4312, 2, 3, 13, 14, 15, 41, 42syl322anc 1212 . 2  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  e.  ( X  .+  Y ) )
4410, 43sseldd 3341 1  |-  ( ( ( ( K  e.  HL  /\  p  =/=  z )  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A )  /\  (
p  e.  A  /\  r  e.  A )
)  /\  ( (
x  e.  X  /\  y  e.  Y )  /\  ( r  .<_  ( x 
.\/  y )  /\  p  .<_  ( x  .\/  r ) ) ) )  ->  p  e.  ( ( X  .+  Y )  .+  Z
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29988   HLchlt 30075   + Pcpadd 30519
This theorem is referenced by:  paddasslem14  30557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-lub 14423  df-join 14425  df-lat 14467  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-padd 30520
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