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Theorem padd4N 35266
Description: Rearrangement of 4 terms in a 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
padd4N  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( ( X 
.+  Y )  .+  ( Z  .+  W ) )  =  ( ( X  .+  Z ) 
.+  ( Y  .+  W ) ) )

Proof of Theorem padd4N
StepHypRef Expression
1 simp1 995 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  K  e.  HL )
2 simp2r 1022 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Y  C_  A
)
3 simp3l 1023 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Z  C_  A
)
4 simp3r 1024 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  W  C_  A
)
5 paddass.a . . . . 5  |-  A  =  ( Atoms `  K )
6 paddass.p . . . . 5  |-  .+  =  ( +P `  K
)
75, 6padd12N 35265 . . . 4  |-  ( ( K  e.  HL  /\  ( Y  C_  A  /\  Z  C_  A  /\  W  C_  A ) )  -> 
( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
81, 2, 3, 4, 7syl13anc 1229 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z 
.+  ( Y  .+  W ) ) )
98oveq2d 6293 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( X  .+  ( Y  .+  ( Z 
.+  W ) ) )  =  ( X 
.+  ( Z  .+  ( Y  .+  W ) ) ) )
10 simp2l 1021 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  X  C_  A
)
115, 6paddssat 35240 . . . 4  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  W  C_  A )  ->  ( Z  .+  W )  C_  A )
121, 3, 4, 11syl3anc 1227 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Z  .+  W )  C_  A
)
135, 6paddass 35264 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  ( Z  .+  W )  C_  A ) )  -> 
( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( X  .+  ( Y  .+  ( Z 
.+  W ) ) ) )
141, 10, 2, 12, 13syl13anc 1229 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( ( X 
.+  Y )  .+  ( Z  .+  W ) )  =  ( X 
.+  ( Y  .+  ( Z  .+  W ) ) ) )
155, 6paddssat 35240 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  W  C_  A )  ->  ( Y  .+  W )  C_  A )
161, 2, 4, 15syl3anc 1227 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  W )  C_  A
)
175, 6paddass 35264 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Z  C_  A  /\  ( Y  .+  W )  C_  A ) )  -> 
( ( X  .+  Z )  .+  ( Y  .+  W ) )  =  ( X  .+  ( Z  .+  ( Y 
.+  W ) ) ) )
181, 10, 3, 16, 17syl13anc 1229 . 2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( ( X 
.+  Z )  .+  ( Y  .+  W ) )  =  ( X 
.+  ( Z  .+  ( Y  .+  W ) ) ) )
199, 14, 183eqtr4d 2492 1  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( ( X 
.+  Y )  .+  ( Z  .+  W ) )  =  ( ( X  .+  Z ) 
.+  ( Y  .+  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    C_ wss 3458   ` cfv 5574  (class class class)co 6277   Atomscatm 34690   HLchlt 34777   +Pcpadd 35221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-clat 15607  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-padd 35222
This theorem is referenced by:  paddclN  35268
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