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Theorem paddssw2 30739
Description: Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
Hypotheses
Ref Expression
paddssw.a  |-  A  =  ( Atoms `  K )
paddssw.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddssw2  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  C_  Z  ->  ( X  C_  Z  /\  Y  C_  Z ) ) )

Proof of Theorem paddssw2
StepHypRef Expression
1 paddssw.a . . . . . 6  |-  A  =  ( Atoms `  K )
2 paddssw.p . . . . . 6  |-  .+  =  ( + P `  K
)
31, 2sspadd1 30710 . . . . 5  |-  ( ( K  e.  B  /\  X  C_  A  /\  Y  C_  A )  ->  X  C_  ( X  .+  Y
) )
433adant3r3 1165 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  ( X  .+  Y ) )
5 sstr 3342 . . . 4  |-  ( ( X  C_  ( X  .+  Y )  /\  ( X  .+  Y )  C_  Z )  ->  X  C_  Z )
64, 5sylan 459 . . 3  |-  ( ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  ( X  .+  Y )  C_  Z
)  ->  X  C_  Z
)
76ex 425 . 2  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  C_  Z  ->  X  C_  Z )
)
8 simpl 445 . . . . 5  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  K  e.  B )
9 simpr2 965 . . . . 5  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  A )
10 simpr1 964 . . . . 5  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A )
111, 2sspadd2 30711 . . . . 5  |-  ( ( K  e.  B  /\  Y  C_  A  /\  X  C_  A )  ->  Y  C_  ( X  .+  Y
) )
128, 9, 10, 11syl3anc 1185 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  ->  Y  C_  ( X  .+  Y ) )
13 sstr 3342 . . . 4  |-  ( ( Y  C_  ( X  .+  Y )  /\  ( X  .+  Y )  C_  Z )  ->  Y  C_  Z )
1412, 13sylan 459 . . 3  |-  ( ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  /\  ( X  .+  Y )  C_  Z
)  ->  Y  C_  Z
)
1514ex 425 . 2  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  C_  Z  ->  Y  C_  Z )
)
167, 15jcad 521 1  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  C_  Z  ->  ( X  C_  Z  /\  Y  C_  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    C_ wss 3306   ` cfv 5483  (class class class)co 6110   Atomscatm 30159   + Pcpadd 30690
This theorem is referenced by:  paddss  30740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-padd 30691
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