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Theorem paddss12 30690
Description: Subset law for projective subspace sum. (unss12 3521 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 961 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  K  e.  B
)
2 simpl2 962 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Y  C_  A
)
3 sstr 3358 . . . . . . . 8  |-  ( ( Z  C_  W  /\  W  C_  A )  ->  Z  C_  A )
43ancoms 441 . . . . . . 7  |-  ( ( W  C_  A  /\  Z  C_  W )  ->  Z  C_  A )
54ad2ant2l 728 . . . . . 6  |-  ( ( ( Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W
) )  ->  Z  C_  A )
653adantl1 1114 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Z  C_  A
)
71, 2, 63jca 1135 . . . 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 734 . . . 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 30688 . . . 4  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
127, 8, 11sylc 59 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) )
139, 10paddss2 30689 . . . . . 6  |-  ( ( K  e.  B  /\  W  C_  A  /\  Y  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
14133com23 1160 . . . . 5  |-  ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
1514imp 420 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  Z  C_  W )  -> 
( Y  .+  Z
)  C_  ( Y  .+  W ) )
1615adantrl 698 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) )
1712, 16sstrd 3360 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  W ) )
1817ex 425 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5457  (class class class)co 6084   Atomscatm 30135   + Pcpadd 30666
This theorem is referenced by:  paddssw1  30714  paddunN  30798  pl42lem2N  30851
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 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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-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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-padd 30667
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