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Theorem pmodN 34664
Description: The modular law for projective subspaces. (Contributed by NM, 26-Mar-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
pmodN  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  ( Y  .+  ( X  i^i  Z ) ) )  =  ( ( X  i^i  Y ) 
.+  ( X  i^i  Z ) ) )

Proof of Theorem pmodN
StepHypRef Expression
1 incom 3691 . 2  |-  ( X  i^i  ( ( X  i^i  Z )  .+  Y ) )  =  ( ( ( X  i^i  Z )  .+  Y )  i^i  X
)
2 hllat 34178 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
32adantr 465 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  Lat )
4 simpr2 1003 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A
)
5 inss2 3719 . . . . 5  |-  ( X  i^i  Z )  C_  Z
6 simpr3 1004 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A
)
75, 6syl5ss 3515 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  Z )  C_  A )
8 pmod.a . . . . 5  |-  A  =  ( Atoms `  K )
9 pmod.p . . . . 5  |-  .+  =  ( +P `  K
)
108, 9paddcom 34627 . . . 4  |-  ( ( K  e.  Lat  /\  Y  C_  A  /\  ( X  i^i  Z )  C_  A )  ->  ( Y  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z
)  .+  Y )
)
113, 4, 7, 10syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Y  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z ) 
.+  Y ) )
1211ineq2d 3700 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  ( Y  .+  ( X  i^i  Z ) ) )  =  ( X  i^i  ( ( X  i^i  Z )  .+  Y ) ) )
13 incom 3691 . . . 4  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
1413oveq2i 6295 . . 3  |-  ( ( X  i^i  Z ) 
.+  ( X  i^i  Y ) )  =  ( ( X  i^i  Z
)  .+  ( Y  i^i  X ) )
15 inss2 3719 . . . . 5  |-  ( X  i^i  Y )  C_  Y
1615, 4syl5ss 3515 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  Y )  C_  A )
178, 9paddcom 34627 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  i^i  Y ) 
C_  A  /\  ( X  i^i  Z )  C_  A )  ->  (
( X  i^i  Y
)  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z )  .+  ( X  i^i  Y ) ) )
183, 16, 7, 17syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( ( X  i^i  Y )  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z ) 
.+  ( X  i^i  Y ) ) )
19 simpr1 1002 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  e.  S
)
207, 4, 193jca 1176 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( ( X  i^i  Z )  C_  A  /\  Y  C_  A  /\  X  e.  S
) )
21 inss1 3718 . . . . 5  |-  ( X  i^i  Z )  C_  X
22 pmod.s . . . . . 6  |-  S  =  ( PSubSp `  K )
238, 22, 9pmod1i 34662 . . . . 5  |-  ( ( K  e.  HL  /\  ( ( X  i^i  Z )  C_  A  /\  Y  C_  A  /\  X  e.  S ) )  -> 
( ( X  i^i  Z )  C_  X  ->  ( ( ( X  i^i  Z )  .+  Y )  i^i  X )  =  ( ( X  i^i  Z )  .+  ( Y  i^i  X ) ) ) )
2421, 23mpi 17 . . . 4  |-  ( ( K  e.  HL  /\  ( ( X  i^i  Z )  C_  A  /\  Y  C_  A  /\  X  e.  S ) )  -> 
( ( ( X  i^i  Z )  .+  Y )  i^i  X
)  =  ( ( X  i^i  Z ) 
.+  ( Y  i^i  X ) ) )
2520, 24syldan 470 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( ( ( X  i^i  Z ) 
.+  Y )  i^i 
X )  =  ( ( X  i^i  Z
)  .+  ( Y  i^i  X ) ) )
2614, 18, 253eqtr4a 2534 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( ( X  i^i  Y )  .+  ( X  i^i  Z ) )  =  ( ( ( X  i^i  Z
)  .+  Y )  i^i  X ) )
271, 12, 263eqtr4a 2534 1  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  ( Y  .+  ( X  i^i  Z ) ) )  =  ( ( X  i^i  Y ) 
.+  ( X  i^i  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   ` cfv 5588  (class class class)co 6284   Latclat 15532   Atomscatm 34078   HLchlt 34165   PSubSpcpsubsp 34310   +Pcpadd 34609
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-lat 15533  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-psubsp 34317  df-padd 34610
This theorem is referenced by: (None)
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