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Theorem pmod2iN 33493
Description: Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmod.a  |-  A  =  ( Atoms `  K )
pmod.s  |-  S  =  ( PSubSp `  K )
pmod.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
pmod2iN  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Z  C_  X  ->  ( ( X  i^i  Y )  .+  Z )  =  ( X  i^i  ( Y 
.+  Z ) ) ) )

Proof of Theorem pmod2iN
StepHypRef Expression
1 incom 3543 . . . . . 6  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
21oveq1i 6101 . . . . 5  |-  ( ( X  i^i  Y ) 
.+  Z )  =  ( ( Y  i^i  X )  .+  Z )
3 hllat 33008 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1009 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  K  e.  Lat )
5 simp22 1022 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  Y  C_  A )
6 ssinss1 3578 . . . . . . 7  |-  ( Y 
C_  A  ->  ( Y  i^i  X )  C_  A )
75, 6syl 16 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Y  i^i  X
)  C_  A )
8 simp23 1023 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  Z  C_  A )
9 pmod.a . . . . . . 7  |-  A  =  ( Atoms `  K )
10 pmod.p . . . . . . 7  |-  .+  =  ( +P `  K
)
119, 10paddcom 33457 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Y  i^i  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  i^i  X
)  .+  Z )  =  ( Z  .+  ( Y  i^i  X ) ) )
124, 7, 8, 11syl3anc 1218 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( Y  i^i  X )  .+  Z )  =  ( Z  .+  ( Y  i^i  X ) ) )
132, 12syl5eq 2487 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( X  i^i  Y )  .+  Z )  =  ( Z  .+  ( Y  i^i  X ) ) )
14 simp21 1021 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  X  e.  S )
158, 5, 143jca 1168 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Z  C_  A  /\  Y  C_  A  /\  X  e.  S )
)
16 pmod.s . . . . . . 7  |-  S  =  ( PSubSp `  K )
179, 16, 10pmod1i 33492 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Z  C_  A  /\  Y  C_  A  /\  X  e.  S ) )  -> 
( Z  C_  X  ->  ( ( Z  .+  Y )  i^i  X
)  =  ( Z 
.+  ( Y  i^i  X ) ) ) )
18173impia 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( Z  C_  A  /\  Y  C_  A  /\  X  e.  S )  /\  Z  C_  X )  ->  (
( Z  .+  Y
)  i^i  X )  =  ( Z  .+  ( Y  i^i  X ) ) )
1915, 18syld3an2 1265 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( Z  .+  Y )  i^i  X
)  =  ( Z 
.+  ( Y  i^i  X ) ) )
209, 10paddcom 33457 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  =  ( Y  .+  Z
) )
214, 8, 5, 20syl3anc 1218 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Z  .+  Y
)  =  ( Y 
.+  Z ) )
2221ineq1d 3551 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( Z  .+  Y )  i^i  X
)  =  ( ( Y  .+  Z )  i^i  X ) )
2313, 19, 223eqtr2d 2481 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( X  i^i  Y )  .+  Z )  =  ( ( Y 
.+  Z )  i^i 
X ) )
24 incom 3543 . . 3  |-  ( ( Y  .+  Z )  i^i  X )  =  ( X  i^i  ( Y  .+  Z ) )
2523, 24syl6eq 2491 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( X  i^i  Y )  .+  Z )  =  ( X  i^i  ( Y  .+  Z ) ) )
26253expia 1189 1  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Z  C_  X  ->  ( ( X  i^i  Y )  .+  Z )  =  ( X  i^i  ( Y 
.+  Z ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3327    C_ wss 3328   ` cfv 5418  (class class class)co 6091   Latclat 15215   Atomscatm 32908   HLchlt 32995   PSubSpcpsubsp 33140   +Pcpadd 33439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-psubsp 33147  df-padd 33440
This theorem is referenced by: (None)
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