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Theorem paddass 29954
Description: Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be non-empty. (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 444 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  HL )
2 simpr3 965 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A )
3 simpr2 964 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A )
4 simpr1 963 . . . 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 29953 . . . 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 1186 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Z  .+  ( Y  .+  X ) ) 
C_  ( ( Z 
.+  Y )  .+  X ) )
9 hllat 29480 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
105, 6paddcom 29929 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
119, 10syl3an1 1217 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
12113adant3r3 1164 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  Y
)  =  ( Y 
.+  X ) )
1312oveq1d 6037 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( ( Y  .+  X ) 
.+  Z ) )
145, 6paddssat 29930 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  A )  ->  ( Y  .+  X )  C_  A )
151, 3, 4, 14syl3anc 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Y  .+  X
)  C_  A )
165, 6paddcom 29929 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Y  .+  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Z  .+  ( Y  .+  X ) ) )
179, 16syl3an1 1217 . . . . 5  |-  ( ( K  e.  HL  /\  ( Y  .+  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Z  .+  ( Y  .+  X ) ) )
181, 15, 2, 17syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( Y  .+  X )  .+  Z
)  =  ( Z 
.+  ( Y  .+  X ) ) )
1913, 18eqtrd 2421 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( Z 
.+  ( Y  .+  X ) ) )
205, 6paddcom 29929 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
219, 20syl3an1 1217 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
22213adant3r1 1162 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Y  .+  Z
)  =  ( Z 
.+  Y ) )
2322oveq2d 6038 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( X  .+  ( Z  .+  Y ) ) )
245, 6paddssat 29930 . . . . . 6  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  C_  A )
251, 2, 3, 24syl3anc 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Z  .+  Y
)  C_  A )
265, 6paddcom 29929 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  ( Z  .+  Y )  C_  A )  ->  ( X  .+  ( Z  .+  Y ) )  =  ( ( Z  .+  Y )  .+  X
) )
279, 26syl3an1 1217 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Z  .+  Y )  C_  A )  ->  ( X  .+  ( Z  .+  Y ) )  =  ( ( Z  .+  Y )  .+  X
) )
281, 4, 25, 27syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Z  .+  Y ) )  =  ( ( Z 
.+  Y )  .+  X ) )
2923, 28eqtrd 2421 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( ( Z 
.+  Y )  .+  X ) )
308, 19, 293sstr4d 3336 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  C_  ( X  .+  ( Y  .+  Z
) ) )
315, 6paddasslem18 29953 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) ) 
C_  ( ( X 
.+  Y )  .+  Z ) )
3230, 31eqssd 3310 1  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3265   ` cfv 5396  (class class class)co 6022   Latclat 14403   Atomscatm 29380   HLchlt 29467   + Pcpadd 29911
This theorem is referenced by:  padd12N  29955  padd4N  29956  pmodl42N  29967  pmapjlln1  29971
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-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-padd 29912
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