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Theorem paddss12 30305
Description: Subset law for projective subspace sum. (unss12 3483 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 960 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  K  e.  B
)
2 simpl2 961 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Y  C_  A
)
3 sstr 3320 . . . . . . . 8  |-  ( ( Z  C_  W  /\  W  C_  A )  ->  Z  C_  A )
43ancoms 440 . . . . . . 7  |-  ( ( W  C_  A  /\  Z  C_  W )  ->  Z  C_  A )
54ad2ant2l 727 . . . . . 6  |-  ( ( ( Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W
) )  ->  Z  C_  A )
653adantl1 1113 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Z  C_  A
)
71, 2, 63jca 1134 . . . 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 733 . . . 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 30303 . . . 4  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
127, 8, 11sylc 58 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) )
139, 10paddss2 30304 . . . . . 6  |-  ( ( K  e.  B  /\  W  C_  A  /\  Y  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
14133com23 1159 . . . . 5  |-  ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
1514imp 419 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  Z  C_  W )  -> 
( Y  .+  Z
)  C_  ( Y  .+  W ) )
1615adantrl 697 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) )
1712, 16sstrd 3322 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  W ) )
1817ex 424 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3284   ` cfv 5417  (class class class)co 6044   Atomscatm 29750   + Pcpadd 30281
This theorem is referenced by:  paddssw1  30329  paddunN  30413  pl42lem2N  30466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-padd 30282
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