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Theorem paddasslem13 33150
Description: Lemma for paddass 33156. 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 1056 . . 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 1083 . . . 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 1084 . . . 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 33132 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  C_  A )
71, 2, 3, 6syl3anc 1264 . . 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 1085 . . 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 33133 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  Y ) 
C_  A  /\  Z  C_  A )  ->  ( X  .+  Y )  C_  ( ( X  .+  Y )  .+  Z
) )
101, 7, 8, 9syl3anc 1264 . 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 32682 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
121, 11syl 17 . . 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 770 . . 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 771 . . 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 1060 . . 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 2420 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
17 paddasslem.l . . . 4  |-  .<_  =  ( le `  K )
1816, 4atbase 32608 . . . . 5  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
1915, 18syl 17 . . . 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 3462 . . . . . 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 32608 . . . . . 6  |-  ( x  e.  A  ->  x  e.  ( Base `  K
) )
2220, 21syl 17 . . . . 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 1061 . . . . . 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 32608 . . . . . 6  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
2523, 24syl 17 . . . . 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 16249 . . . . 5  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  r  e.  ( Base `  K
) )  ->  (
x  .\/  r )  e.  ( Base `  K
) )
2812, 22, 25, 27syl3anc 1264 . . . 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 3462 . . . . . 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 32608 . . . . . 6  |-  ( y  e.  A  ->  y  e.  ( Base `  K
) )
3129, 30syl 17 . . . . 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 16249 . . . . 5  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  (
x  .\/  y )  e.  ( Base `  K
) )
3312, 22, 31, 32syl3anc 1264 . . . 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 773 . . . 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 16258 . . . . . 6  |-  ( ( K  e.  Lat  /\  x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  x  .<_  ( x  .\/  y
) )
3612, 22, 31, 35syl3anc 1264 . . . . 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 772 . . . . 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 16260 . . . . . 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 1266 . . . . 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 929 . . . 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 16256 . . 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 33120 . . 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 1292 . 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 3462 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616    C_ wss 3433   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15081   lecple 15157   joincjn 16141   Latclat 16243   Atomscatm 32582   HLchlt 32669   +Pcpadd 33113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-poset 16143  df-lub 16172  df-glb 16173  df-join 16174  df-meet 16175  df-lat 16244  df-ats 32586  df-atl 32617  df-cvlat 32641  df-hlat 32670  df-padd 33114
This theorem is referenced by:  paddasslem14  33151
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