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Theorem paddass 29294
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 445 . . . 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 29293 . . . 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 28820 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
105, 6paddcom 29269 . . . . . . 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 5834 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( ( Y  .+  X ) 
.+  Z ) )
145, 6paddssat 29270 . . . . . 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 29269 . . . . . 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 2316 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( Z 
.+  ( Y  .+  X ) ) )
205, 6paddcom 29269 . . . . . . 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 5835 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( X  .+  ( Z  .+  Y ) ) )
245, 6paddssat 29270 . . . . . 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 29269 . . . . . 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 2316 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( ( Z 
.+  Y )  .+  X ) )
308, 19, 293sstr4d 3222 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  C_  ( X  .+  ( Y  .+  Z
) ) )
315, 6paddasslem18 29293 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) ) 
C_  ( ( X 
.+  Y )  .+  Z ) )
3230, 31eqssd 3197 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 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    C_ wss 3153   ` cfv 5221  (class class class)co 5819   Latclat 14145   Atomscatm 28720   HLchlt 28807   + Pcpadd 29251
This theorem is referenced by:  padd12N  29295  padd4N  29296  pmodl42N  29307  pmapjlln1  29311
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-padd 29252
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