Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmod2iN Unicode version

Theorem pmod2iN 30343
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 3501 . . . . . 6  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
21oveq1i 6058 . . . . 5  |-  ( ( X  i^i  Y ) 
.+  Z )  =  ( ( Y  i^i  X )  .+  Z )
3 hllat 29858 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 978 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  K  e.  Lat )
5 simp22 991 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  Y  C_  A )
6 ssinss1 3537 . . . . . . 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 992 . . . . . 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 30307 . . . . . 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 1184 . . . . 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 2456 . . . 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 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  X  e.  S )
158, 5, 143jca 1134 . . . . 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 30342 . . . . . 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 1150 . . . . 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 1231 . . . 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 30307 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  =  ( Y  .+  Z
) )
214, 8, 5, 20syl3anc 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Z  .+  Y
)  =  ( Y 
.+  Z ) )
2221ineq1d 3509 . . . 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 2450 . . 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 3501 . . 3  |-  ( ( Y  .+  Z )  i^i  X )  =  ( X  i^i  ( Y  .+  Z ) )
2523, 24syl6eq 2460 . 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 1155 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3287    C_ wss 3288   ` cfv 5421  (class class class)co 6048   Latclat 14437   Atomscatm 29758   HLchlt 29845   PSubSpcpsubsp 29990   + Pcpadd 30289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-join 14396  df-lat 14438  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-psubsp 29997  df-padd 30290
  Copyright terms: Public domain W3C validator