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Theorem padd4N 33792
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 988 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  K  e.  HL )
2 simp2r 1015 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Y  C_  A
)
3 simp3l 1016 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Z  C_  A
)
4 simp3r 1017 . . . 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 33791 . . . 4  |-  ( ( K  e.  HL  /\  ( Y  C_  A  /\  Z  C_  A  /\  W  C_  A ) )  -> 
( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
81, 2, 3, 4, 7syl13anc 1221 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z 
.+  ( Y  .+  W ) ) )
98oveq2d 6208 . 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 1014 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  X  C_  A
)
115, 6paddssat 33766 . . . 4  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  W  C_  A )  ->  ( Z  .+  W )  C_  A )
121, 3, 4, 11syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Z  .+  W )  C_  A
)
135, 6paddass 33790 . . 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 1221 . 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 33766 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  W  C_  A )  ->  ( Y  .+  W )  C_  A )
161, 2, 4, 15syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  W )  C_  A
)
175, 6paddass 33790 . . 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 1221 . 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 2502 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 965    = wceq 1370    e. wcel 1758    C_ wss 3428   ` cfv 5518  (class class class)co 6192   Atomscatm 33216   HLchlt 33303   +Pcpadd 33747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-padd 33748
This theorem is referenced by:  paddclN  33794
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