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Theorem paddss12 30630
Description: Subset law for projective subspace sum. (unss12 3360 analog.) (Contributed by NM, 7-Mar-2012.)
Hypotheses
Ref Expression
padd0.a  |-  A  =  ( Atoms `  K )
padd0.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddss12  |-  ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  ->  (
( X  C_  Y  /\  Z  C_  W )  ->  ( X  .+  Z )  C_  ( Y  .+  W ) ) )

Proof of Theorem paddss12
StepHypRef Expression
1 simpl1 958 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  K  e.  B
)
2 simpl2 959 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Y  C_  A
)
3 sstr 3200 . . . . . . . 8  |-  ( ( Z  C_  W  /\  W  C_  A )  ->  Z  C_  A )
43ancoms 439 . . . . . . 7  |-  ( ( W  C_  A  /\  Z  C_  W )  ->  Z  C_  A )
54ad2ant2l 726 . . . . . 6  |-  ( ( ( Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W
) )  ->  Z  C_  A )
653adantl1 1111 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Z  C_  A
)
71, 2, 63jca 1132 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A
) )
8 simprl 732 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  X  C_  Y
)
9 padd0.a . . . . 5  |-  A  =  ( Atoms `  K )
10 padd0.p . . . . 5  |-  .+  =  ( + P `  K
)
119, 10paddss1 30628 . . . 4  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
127, 8, 11sylc 56 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) )
139, 10paddss2 30629 . . . . . 6  |-  ( ( K  e.  B  /\  W  C_  A  /\  Y  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
14133com23 1157 . . . . 5  |-  ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
1514imp 418 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  Z  C_  W )  -> 
( Y  .+  Z
)  C_  ( Y  .+  W ) )
1615adantrl 696 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) )
1712, 16sstrd 3202 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  W ) )
1817ex 423 1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  ->  (
( X  C_  Y  /\  Z  C_  W )  ->  ( X  .+  Z )  C_  ( 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   + Pcpadd 30606
This theorem is referenced by:  paddssw1  30654  paddunN  30738  pl42lem2N  30791
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-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-padd 30607
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