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Theorem paddass 29157
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 968 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A )
3 simpr2 967 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A )
4 simpr1 966 . . . 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 29156 . . . 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 1189 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Z  .+  ( Y  .+  X ) ) 
C_  ( ( Z 
.+  Y )  .+  X ) )
9 hllat 28683 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
105, 6paddcom 29132 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
119, 10syl3an1 1220 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
12113adant3r3 1167 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  Y
)  =  ( Y 
.+  X ) )
1312oveq1d 5772 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( ( Y  .+  X ) 
.+  Z ) )
145, 6paddssat 29133 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  A )  ->  ( Y  .+  X )  C_  A )
151, 3, 4, 14syl3anc 1187 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Y  .+  X
)  C_  A )
165, 6paddcom 29132 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Y  .+  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Z  .+  ( Y  .+  X ) ) )
179, 16syl3an1 1220 . . . . 5  |-  ( ( K  e.  HL  /\  ( Y  .+  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  .+  X
)  .+  Z )  =  ( Z  .+  ( Y  .+  X ) ) )
181, 15, 2, 17syl3anc 1187 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( Y  .+  X )  .+  Z
)  =  ( Z 
.+  ( Y  .+  X ) ) )
1913, 18eqtrd 2288 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( Z 
.+  ( Y  .+  X ) ) )
205, 6paddcom 29132 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
219, 20syl3an1 1220 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  Z  C_  A )  ->  ( Y  .+  Z )  =  ( Z  .+  Y
) )
22213adant3r1 1165 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Y  .+  Z
)  =  ( Z 
.+  Y ) )
2322oveq2d 5773 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( X  .+  ( Z  .+  Y ) ) )
245, 6paddssat 29133 . . . . . 6  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  C_  A )
251, 2, 3, 24syl3anc 1187 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( Z  .+  Y
)  C_  A )
265, 6paddcom 29132 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  ( Z  .+  Y )  C_  A )  ->  ( X  .+  ( Z  .+  Y ) )  =  ( ( Z  .+  Y )  .+  X
) )
279, 26syl3an1 1220 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Z  .+  Y )  C_  A )  ->  ( X  .+  ( Z  .+  Y ) )  =  ( ( Z  .+  Y )  .+  X
) )
281, 4, 25, 27syl3anc 1187 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Z  .+  Y ) )  =  ( ( Z 
.+  Y )  .+  X ) )
2923, 28eqtrd 2288 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( ( Z 
.+  Y )  .+  X ) )
308, 19, 293sstr4d 3163 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  C_  ( X  .+  ( Y  .+  Z
) ) )
315, 6paddasslem18 29156 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) ) 
C_  ( ( X 
.+  Y )  .+  Z ) )
3230, 31eqssd 3138 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 939    = wceq 1619    e. wcel 1621    C_ wss 3094   ` cfv 4638  (class class class)co 5757   Latclat 14078   Atomscatm 28583   HLchlt 28670   + Pcpadd 29114
This theorem is referenced by:  padd12N  29158  padd4N  29159  pmodl42N  29170  pmapjlln1  29174
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-padd 29115
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