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Theorem latj4 15254
Description: Rearrangement of lattice join of 4 classes. (chj4 24761 analog.) (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
latjass.b  |-  B  =  ( Base `  K
)
latjass.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latj4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Y )  .\/  ( Z 
.\/  W ) )  =  ( ( X 
.\/  Z )  .\/  ( Y  .\/  W ) ) )

Proof of Theorem latj4
StepHypRef Expression
1 simp1 981 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  K  e.  Lat )
2 simp2r 1008 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Y  e.  B )
3 simp3l 1009 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Z  e.  B )
4 simp3r 1010 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  W  e.  B )
5 latjass.b . . . . 5  |-  B  =  ( Base `  K
)
6 latjass.j . . . . 5  |-  .\/  =  ( join `  K )
75, 6latj12 15249 . . . 4  |-  ( ( K  e.  Lat  /\  ( Y  e.  B  /\  Z  e.  B  /\  W  e.  B
) )  ->  ( Y  .\/  ( Z  .\/  W ) )  =  ( Z  .\/  ( Y 
.\/  W ) ) )
81, 2, 3, 4, 7syl13anc 1213 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  ( Z  .\/  W ) )  =  ( Z  .\/  ( Y  .\/  W ) ) )
98oveq2d 6096 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .\/  ( Y  .\/  ( Z  .\/  W ) ) )  =  ( X  .\/  ( Z  .\/  ( Y  .\/  W ) ) ) )
10 simp2l 1007 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  X  e.  B )
115, 6latjcl 15204 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  W  e.  B )  ->  ( Z  .\/  W
)  e.  B )
121, 3, 4, 11syl3anc 1211 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Z  .\/  W
)  e.  B )
135, 6latjass 15248 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  ( Z  .\/  W
)  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  ( Z  .\/  W ) )  =  ( X 
.\/  ( Y  .\/  ( Z  .\/  W ) ) ) )
141, 10, 2, 12, 13syl13anc 1213 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Y )  .\/  ( Z 
.\/  W ) )  =  ( X  .\/  ( Y  .\/  ( Z 
.\/  W ) ) ) )
155, 6latjcl 15204 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  .\/  W
)  e.  B )
161, 2, 4, 15syl3anc 1211 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  .\/  W
)  e.  B )
175, 6latjass 15248 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .\/  W
)  e.  B ) )  ->  ( ( X  .\/  Z )  .\/  ( Y  .\/  W ) )  =  ( X 
.\/  ( Z  .\/  ( Y  .\/  W ) ) ) )
181, 10, 3, 16, 17syl13anc 1213 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Z )  .\/  ( Y 
.\/  W ) )  =  ( X  .\/  ( Z  .\/  ( Y 
.\/  W ) ) ) )
199, 14, 183eqtr4d 2475 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .\/  Y )  .\/  ( Z 
.\/  W ) )  =  ( ( X 
.\/  Z )  .\/  ( Y  .\/  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   joincjn 15097   Latclat 15198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-poset 15099  df-lub 15127  df-glb 15128  df-join 15129  df-meet 15130  df-lat 15199
This theorem is referenced by:  latj4rot  15255  latjjdi  15256  latjjdir  15257  hlatj4  32591  arglem1N  33407  cdleme11  33487  cdleme20l2  33538
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