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Theorem padd4N 34511
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 991 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  K  e.  HL )
2 simp2r 1018 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Y  C_  A
)
3 simp3l 1019 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Z  C_  A
)
4 simp3r 1020 . . . 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 34510 . . . 4  |-  ( ( K  e.  HL  /\  ( Y  C_  A  /\  Z  C_  A  /\  W  C_  A ) )  -> 
( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
81, 2, 3, 4, 7syl13anc 1225 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z 
.+  ( Y  .+  W ) ) )
98oveq2d 6291 . 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 1017 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  X  C_  A
)
115, 6paddssat 34485 . . . 4  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  W  C_  A )  ->  ( Z  .+  W )  C_  A )
121, 3, 4, 11syl3anc 1223 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Z  .+  W )  C_  A
)
135, 6paddass 34509 . . 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 1225 . 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 34485 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  W  C_  A )  ->  ( Y  .+  W )  C_  A )
161, 2, 4, 15syl3anc 1223 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  W )  C_  A
)
175, 6paddass 34509 . . 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 1225 . 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 2511 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 968    = wceq 1374    e. wcel 1762    C_ wss 3469   ` cfv 5579  (class class class)co 6275   Atomscatm 33935   HLchlt 34022   +Pcpadd 34466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-padd 34467
This theorem is referenced by:  paddclN  34513
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