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Theorem pmodN 35314
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 3676 . 2  |-  ( X  i^i  ( ( X  i^i  Z )  .+  Y ) )  =  ( ( ( X  i^i  Z )  .+  Y )  i^i  X
)
2 hllat 34828 . . . . 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 1004 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A
)
5 inss2 3704 . . . . 5  |-  ( X  i^i  Z )  C_  Z
6 simpr3 1005 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A
)
75, 6syl5ss 3500 . . . 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 35277 . . . 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 1229 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Y  .+  ( X  i^i  Z ) )  =  ( ( X  i^i  Z ) 
.+  Y ) )
1211ineq2d 3685 . 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 3676 . . . 4  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
1413oveq2i 6292 . . 3  |-  ( ( X  i^i  Z ) 
.+  ( X  i^i  Y ) )  =  ( ( X  i^i  Z
)  .+  ( Y  i^i  X ) )
15 inss2 3704 . . . . 5  |-  ( X  i^i  Y )  C_  Y
1615, 4syl5ss 3500 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( X  i^i  Y )  C_  A )
178, 9paddcom 35277 . . . 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 1229 . . 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 1003 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  e.  S
)
207, 4, 193jca 1177 . . . 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 3703 . . . . 5  |-  ( X  i^i  Z )  C_  X
22 pmod.s . . . . . 6  |-  S  =  ( PSubSp `  K )
238, 22, 9pmod1i 35312 . . . . 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 2510 . 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 2510 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 974    = wceq 1383    e. wcel 1804    i^i cin 3460    C_ wss 3461   ` cfv 5578  (class class class)co 6281   Latclat 15549   Atomscatm 34728   HLchlt 34815   PSubSpcpsubsp 34960   +Pcpadd 35259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-psubsp 34967  df-padd 35260
This theorem is referenced by: (None)
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