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Theorem paddss12 33455
Description: Subset law for projective subspace sum. (unss12 3597 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 1033 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  K  e.  B
)
2 simpl2 1034 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Y  C_  A
)
3 sstr 3426 . . . . . . . 8  |-  ( ( Z  C_  W  /\  W  C_  A )  ->  Z  C_  A )
43ancoms 460 . . . . . . 7  |-  ( ( W  C_  A  /\  Z  C_  W )  ->  Z  C_  A )
54ad2ant2l 760 . . . . . 6  |-  ( ( ( Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W
) )  ->  Z  C_  A )
653adantl1 1186 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  Z  C_  A
)
71, 2, 63jca 1210 . . . 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 772 . . . 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 33453 . . . 4  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
127, 8, 11sylc 61 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) )
139, 10paddss2 33454 . . . . . 6  |-  ( ( K  e.  B  /\  W  C_  A  /\  Y  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
14133com23 1237 . . . . 5  |-  ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  ->  ( Z  C_  W  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) ) )
1514imp 436 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  Z  C_  W )  -> 
( Y  .+  Z
)  C_  ( Y  .+  W ) )
1615adantrl 730 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( Y  .+  Z )  C_  ( Y  .+  W ) )
1712, 16sstrd 3428 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  W  C_  A )  /\  ( X  C_  Y  /\  Z  C_  W ) )  ->  ( X  .+  Z )  C_  ( Y  .+  W ) )
1817ex 441 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 setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    C_ wss 3390   ` cfv 5589  (class class class)co 6308   Atomscatm 32900   +Pcpadd 33431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-padd 33432
This theorem is referenced by:  paddssw1  33479  paddunN  33563  pl42lem2N  33616
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