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Theorem paddasslem13 30643
Description: Lemma for paddass 30649. 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 1006 . . 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 1033 . . . 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 1034 . . . 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 30625 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
71, 2, 3, 6syl3anc 1182 . . 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 1035 . . 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 30626 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  Y ) 
C_  A  /\  Z  C_  A )  ->  ( X  .+  Y )  C_  ( ( X  .+  Y )  .+  Z
) )
101, 7, 8, 9syl3anc 1182 . 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 30175 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
121, 11syl 15 . . 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 738 . . 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 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 ) ) ) )  ->  y  e.  Y )
15 simpl3l 1010 . . 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 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
17 paddasslem.l . . . 4  |-  .<_  =  ( le `  K )
1816, 4atbase 30101 . . . . 5  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
1915, 18syl 15 . . . 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 3194 . . . . . 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 30101 . . . . . 6  |-  ( x  e.  A  ->  x  e.  ( Base `  K
) )
2220, 21syl 15 . . . . 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 1011 . . . . . 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 30101 . . . . . 6  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
2523, 24syl 15 . . . . 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 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  r  e.  ( Base `  K
) )  ->  (
x  .\/  r )  e.  ( Base `  K
) )
2812, 22, 25, 27syl3anc 1182 . . . 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 3194 . . . . . 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 30101 . . . . . 6  |-  ( y  e.  A  ->  y  e.  ( Base `  K
) )
3129, 30syl 15 . . . . 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 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  (
x  .\/  y )  e.  ( Base `  K
) )
3312, 22, 31, 32syl3anc 1182 . . . 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 741 . . . 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 14182 . . . . . 6  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  x  .<_  ( x  .\/  y
) )
3612, 22, 31, 35syl3anc 1182 . . . . 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 740 . . . . 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 14184 . . . . . 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 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 )  /\  r  .<_  ( x  .\/  y
) )  <->  ( x  .\/  r )  .<_  ( x 
.\/  y ) ) )
4036, 37, 39mpbi2and 887 . . . 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 14180 . . 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 30613 . . 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 1210 . 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 3194 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162   + Pcpadd 30606
This theorem is referenced by:  paddasslem14  30644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-join 14126  df-lat 14168  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-padd 30607
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