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Theorem padd4N 30639
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 958 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  K  e.  HL )
2 simp2r 985 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Y  C_  A
)
3 simp3l 986 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Z  C_  A
)
4 simp3r 987 . . . 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 30638 . . . 4  |-  ( ( K  e.  HL  /\  ( Y  C_  A  /\  Z  C_  A  /\  W  C_  A ) )  -> 
( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
81, 2, 3, 4, 7syl13anc 1187 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z 
.+  ( Y  .+  W ) ) )
98oveq2d 6099 . 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 984 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  X  C_  A
)
115, 6paddssat 30613 . . . 4  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  W  C_  A )  ->  ( Z  .+  W )  C_  A )
121, 3, 4, 11syl3anc 1185 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Z  .+  W )  C_  A
)
135, 6paddass 30637 . . 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 1187 . 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 30613 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  W  C_  A )  ->  ( Y  .+  W )  C_  A )
161, 2, 4, 15syl3anc 1185 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  W )  C_  A
)
175, 6paddass 30637 . . 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 1187 . 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 2480 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Atomscatm 30063   HLchlt 30150   + Pcpadd 30594
This theorem is referenced by:  paddclN  30641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-padd 30595
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