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Theorem pmodN 33415
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 3625 . 2  |-  ( X  i^i  ( ( X  i^i  Z )  .+  Y ) )  =  ( ( ( X  i^i  Z )  .+  Y )  i^i  X
)
2 hllat 32929 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
32adantr 467 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  Lat )
4 simpr2 1015 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A
)
5 inss2 3653 . . . . 5  |-  ( X  i^i  Z )  C_  Z
6 simpr3 1016 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A
)
75, 6syl5ss 3443 . . . 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 33378 . . . 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 1268 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Y  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z ) 
.+  Y ) )
1211ineq2d 3634 . 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 3625 . . . 4  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
1413oveq2i 6301 . . 3  |-  ( ( X  i^i  Z ) 
.+  ( X  i^i  Y ) )  =  ( ( X  i^i  Z
)  .+  ( Y  i^i  X ) )
15 inss2 3653 . . . . 5  |-  ( X  i^i  Y )  C_  Y
1615, 4syl5ss 3443 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  Y )  C_  A )
178, 9paddcom 33378 . . . 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 1268 . . 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 1014 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  e.  S
)
207, 4, 193jca 1188 . . . 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 3652 . . . . 5  |-  ( X  i^i  Z )  C_  X
22 pmod.s . . . . . 6  |-  S  =  ( PSubSp `  K )
238, 22, 9pmod1i 33413 . . . . 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 20 . . . 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 473 . . 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 2511 . 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 2511 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 371    /\ w3a 985    = wceq 1444    e. wcel 1887    i^i cin 3403    C_ wss 3404   ` cfv 5582  (class class class)co 6290   Latclat 16291   Atomscatm 32829   HLchlt 32916   PSubSpcpsubsp 33061   +Pcpadd 33360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-lat 16292  df-covers 32832  df-ats 32833  df-atl 32864  df-cvlat 32888  df-hlat 32917  df-psubsp 33068  df-padd 33361
This theorem is referenced by: (None)
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