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Theorem pmodN 30661
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 3374 . 2  |-  ( X  i^i  ( ( X  i^i  Z )  .+  Y ) )  =  ( ( ( X  i^i  Z )  .+  Y )  i^i  X
)
2 hllat 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
32adantr 451 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  Lat )
4 simpr2 962 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A
)
5 inss2 3403 . . . . 5  |-  ( X  i^i  Z )  C_  Z
6 simpr3 963 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A
)
75, 6syl5ss 3203 . . . 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 30624 . . . 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 1182 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Y  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z ) 
.+  Y ) )
1211ineq2d 3383 . 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 3374 . . . 4  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
1413oveq2i 5885 . . 3  |-  ( ( X  i^i  Z ) 
.+  ( X  i^i  Y ) )  =  ( ( X  i^i  Z
)  .+  ( Y  i^i  X ) )
15 inss2 3403 . . . . 5  |-  ( X  i^i  Y )  C_  Y
1615, 4syl5ss 3203 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  Y )  C_  A )
178, 9paddcom 30624 . . . 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 1182 . . 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 961 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  e.  S
)
207, 4, 193jca 1132 . . . 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 3402 . . . . 5  |-  ( X  i^i  Z )  C_  X
22 pmod.s . . . . . 6  |-  S  =  ( PSubSp `  K )
238, 22, 9pmod1i 30659 . . . . 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 16 . . . 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 456 . . 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 2354 . 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 2354 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Latclat 14167   Atomscatm 30075   HLchlt 30162   PSubSpcpsubsp 30307   + Pcpadd 30606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-join 14126  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-padd 30607
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