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Theorem padd4N 30651
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 955 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  K  e.  HL )
2 simp2r 982 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Y  C_  A
)
3 simp3l 983 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  Z  C_  A
)
4 simp3r 984 . . . 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 30650 . . . 4  |-  ( ( K  e.  HL  /\  ( Y  C_  A  /\  Z  C_  A  /\  W  C_  A ) )  -> 
( Y  .+  ( Z  .+  W ) )  =  ( Z  .+  ( Y  .+  W ) ) )
81, 2, 3, 4, 7syl13anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  ( Z  .+  W ) )  =  ( Z 
.+  ( Y  .+  W ) ) )
98oveq2d 5890 . 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 981 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  X  C_  A
)
115, 6paddssat 30625 . . . 4  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  W  C_  A )  ->  ( Z  .+  W )  C_  A )
121, 3, 4, 11syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Z  .+  W )  C_  A
)
135, 6paddass 30649 . . 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 1184 . 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 30625 . . . 4  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  W  C_  A )  ->  ( Y  .+  W )  C_  A )
161, 2, 4, 15syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A )  /\  ( Z  C_  A  /\  W  C_  A ) )  ->  ( Y  .+  W )  C_  A
)
175, 6paddass 30649 . . 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 1184 . 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 2338 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Atomscatm 30075   HLchlt 30162   + Pcpadd 30606
This theorem is referenced by:  paddclN  30653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-padd 30607
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