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Theorem padd12N 33502
Description: Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddass.a  |-  A  =  ( Atoms `  K )
paddass.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
padd12N  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )

Proof of Theorem padd12N
StepHypRef Expression
1 hllat 33027 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
21adantr 465 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  Lat )
3 simpr1 994 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A )
4 simpr2 995 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A )
5 paddass.a . . . . 5  |-  A  =  ( Atoms `  K )
6 paddass.p . . . . 5  |-  .+  =  ( +P `  K
)
75, 6paddcom 33476 . . . 4  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
82, 3, 4, 7syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  Y
)  =  ( Y 
.+  X ) )
98oveq1d 6121 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( ( Y  .+  X ) 
.+  Z ) )
105, 6paddass 33501 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  .+  Z
)  =  ( X 
.+  ( Y  .+  Z ) ) )
11 simpl 457 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  HL )
12 simpr3 996 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Z  C_  A )
135, 6paddass 33501 . . 3  |-  ( ( K  e.  HL  /\  ( Y  C_  A  /\  X  C_  A  /\  Z  C_  A ) )  -> 
( ( Y  .+  X )  .+  Z
)  =  ( Y 
.+  ( X  .+  Z ) ) )
1411, 4, 3, 12, 13syl13anc 1220 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( Y  .+  X )  .+  Z
)  =  ( Y 
.+  ( X  .+  Z ) ) )
159, 10, 143eqtr3d 2483 1  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3343   ` cfv 5433  (class class class)co 6106   Latclat 15230   Atomscatm 32927   HLchlt 33014   +Pcpadd 33458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-poset 15131  df-plt 15143  df-lub 15159  df-glb 15160  df-join 15161  df-meet 15162  df-p0 15224  df-lat 15231  df-clat 15293  df-oposet 32840  df-ol 32842  df-oml 32843  df-covers 32930  df-ats 32931  df-atl 32962  df-cvlat 32986  df-hlat 33015  df-padd 33459
This theorem is referenced by:  padd4N  33503  pmodl42N  33514
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