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Theorem pmod2iN 34938
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 3696 . . . . . 6  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
21oveq1i 6304 . . . . 5  |-  ( ( X  i^i  Y ) 
.+  Z )  =  ( ( Y  i^i  X )  .+  Z )
3 hllat 34453 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1017 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  K  e.  Lat )
5 simp22 1030 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  Y  C_  A )
6 ssinss1 3731 . . . . . . 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 1031 . . . . . 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 34902 . . . . . 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 1228 . . . . 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 2520 . . . 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 1029 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  X  e.  S )
158, 5, 143jca 1176 . . . . 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 34937 . . . . . 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 1193 . . . . 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 1275 . . . 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 34902 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  =  ( Y  .+  Z
) )
214, 8, 5, 20syl3anc 1228 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Z  .+  Y
)  =  ( Y 
.+  Z ) )
2221ineq1d 3704 . . . 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 2514 . . 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 3696 . . 3  |-  ( ( Y  .+  Z )  i^i  X )  =  ( X  i^i  ( Y  .+  Z ) )
2523, 24syl6eq 2524 . 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 1198 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 973    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   ` cfv 5593  (class class class)co 6294   Latclat 15544   Atomscatm 34353   HLchlt 34440   PSubSpcpsubsp 34585   +Pcpadd 34884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795  df-poset 15445  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-covers 34356  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441  df-psubsp 34592  df-padd 34885
This theorem is referenced by: (None)
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