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Theorem paddass 35975
Description: Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
paddass.a  |-  A  =  ( Atoms `  K )
paddass.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
paddass  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )

Proof of Theorem paddass
StepHypRef Expression
1 simpl 455 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  HL )
2 simpr3 1002 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A )
3 simpr2 1001 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A )
4 simpr1 1000 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A )
5 paddass.a . . . . 5  |-  A  =  ( Atoms `  K )
6 paddass.p . . . . 5  |-  .+  =  ( +P `  K
)
75, 6paddasslem18 35974 . . . 4  |-  ( ( K  e.  HL  /\  ( Z  C_  A  /\  Y  C_  A  /\  X  C_  A ) )  -> 
( Z  .+  ( Y  .+  X ) ) 
C_  ( ( Z 
.+  Y )  .+  X ) )
81, 2, 3, 4, 7syl13anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Z  .+  ( Y  .+  X ) ) 
C_  ( ( Z 
.+  Y )  .+  X ) )
9 hllat 35501 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
105, 6paddcom 35950 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
119, 10syl3an1 1259 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
12113adant3r3 1205 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  Y
)  =  ( Y 
.+  X ) )
1312oveq1d 6211 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( ( Y  .+  X ) 
.+  Z ) )
145, 6paddssat 35951 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  A )  ->  ( Y  .+  X )  C_  A )
151, 3, 4, 14syl3anc 1226 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Y  .+  X
)  C_  A )
165, 6paddcom 35950 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Y  .+  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Z  .+  ( Y  .+  X ) ) )
179, 16syl3an1 1259 . . . . 5  |-  ( ( K  e.  HL  /\  ( Y  .+  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Z  .+  ( Y  .+  X ) ) )
181, 15, 2, 17syl3anc 1226 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( Y  .+  X )  .+  Z
)  =  ( Z 
.+  ( Y  .+  X ) ) )
1913, 18eqtrd 2423 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( Z 
.+  ( Y  .+  X ) ) )
205, 6paddcom 35950 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
219, 20syl3an1 1259 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
22213adant3r1 1203 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Y  .+  Z
)  =  ( Z 
.+  Y ) )
2322oveq2d 6212 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( X  .+  ( Z  .+  Y ) ) )
245, 6paddssat 35951 . . . . . 6  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  C_  A )
251, 2, 3, 24syl3anc 1226 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Z  .+  Y
)  C_  A )
265, 6paddcom 35950 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  ( Z  .+  Y )  C_  A )  ->  ( X  .+  ( Z  .+  Y ) )  =  ( ( Z  .+  Y )  .+  X
) )
279, 26syl3an1 1259 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Z  .+  Y )  C_  A )  ->  ( X  .+  ( Z  .+  Y ) )  =  ( ( Z  .+  Y )  .+  X
) )
281, 4, 25, 27syl3anc 1226 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Z  .+  Y ) )  =  ( ( Z 
.+  Y )  .+  X ) )
2923, 28eqtrd 2423 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( ( Z 
.+  Y )  .+  X ) )
308, 19, 293sstr4d 3460 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  C_  ( X  .+  ( Y  .+  Z
) ) )
315, 6paddasslem18 35974 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) ) 
C_  ( ( X 
.+  Y )  .+  Z ) )
3230, 31eqssd 3434 1  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    C_ wss 3389   ` cfv 5496  (class class class)co 6196   Latclat 15792   Atomscatm 35401   HLchlt 35488   +Pcpadd 35932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-lat 15793  df-clat 15855  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489  df-padd 35933
This theorem is referenced by:  padd12N  35976  padd4N  35977  pmodl42N  35988  pmapjlln1  35992
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