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Theorem pmodN 29966
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 3478 . 2  |-  ( X  i^i  ( ( X  i^i  Z )  .+  Y ) )  =  ( ( ( X  i^i  Z )  .+  Y )  i^i  X
)
2 hllat 29480 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
32adantr 452 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  Lat )
4 simpr2 964 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A
)
5 inss2 3507 . . . . 5  |-  ( X  i^i  Z )  C_  Z
6 simpr3 965 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A
)
75, 6syl5ss 3304 . . . 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 29929 . . . 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 1184 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Y  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z ) 
.+  Y ) )
1211ineq2d 3487 . 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 3478 . . . 4  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
1413oveq2i 6033 . . 3  |-  ( ( X  i^i  Z ) 
.+  ( X  i^i  Y ) )  =  ( ( X  i^i  Z
)  .+  ( Y  i^i  X ) )
15 inss2 3507 . . . . 5  |-  ( X  i^i  Y )  C_  Y
1615, 4syl5ss 3304 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  Y )  C_  A )
178, 9paddcom 29929 . . . 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 1184 . . 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 963 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  e.  S
)
207, 4, 193jca 1134 . . . 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 3506 . . . . 5  |-  ( X  i^i  Z )  C_  X
22 pmod.s . . . . . 6  |-  S  =  ( PSubSp `  K )
238, 22, 9pmod1i 29964 . . . . 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 457 . . 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 2447 . 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 2447 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    i^i cin 3264    C_ wss 3265   ` cfv 5396  (class class class)co 6022   Latclat 14403   Atomscatm 29380   HLchlt 29467   PSubSpcpsubsp 29612   + Pcpadd 29911
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-join 14362  df-lat 14404  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-psubsp 29619  df-padd 29912
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