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Theorem paddss12 30008
Description: Subset law for projective subspace sum. (unss12 3347 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 3187 . . . . . . . 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 30006 . . . 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 30007 . . . . . 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 3189 . 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 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Atomscatm 29453   + Pcpadd 29984
This theorem is referenced by:  paddssw1  30032  paddunN  30116  pl42lem2N  30169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-padd 29985
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