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Theorem paddss 30644
Description: Subset law for projective subspace sum. (unss 3523 analog.) (Contributed by NM, 7-Mar-2012.)
Hypotheses
Ref Expression
paddss.a  |-  A  =  ( Atoms `  K )
paddss.s  |-  S  =  ( PSubSp `  K )
paddss.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddss  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( ( X  C_  Z  /\  Y  C_  Z
)  <->  ( X  .+  Y )  C_  Z
) )

Proof of Theorem paddss
StepHypRef Expression
1 simpl 445 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  K  e.  B )
2 simpr1 964 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  X  C_  A )
3 simpr2 965 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Y  C_  A )
4 paddss.a . . . . . 6  |-  A  =  ( Atoms `  K )
5 paddss.s . . . . . 6  |-  S  =  ( PSubSp `  K )
64, 5psubssat 30553 . . . . 5  |-  ( ( K  e.  B  /\  Z  e.  S )  ->  Z  C_  A )
763ad2antr3 1125 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Z  C_  A )
8 paddss.p . . . . 5  |-  .+  =  ( + P `  K
)
94, 8paddssw1 30642 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  C_  Z  /\  Y  C_  Z
)  ->  ( X  .+  Y )  C_  ( Z  .+  Z ) ) )
101, 2, 3, 7, 9syl13anc 1187 . . 3  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( ( X  C_  Z  /\  Y  C_  Z
)  ->  ( X  .+  Y )  C_  ( Z  .+  Z ) ) )
115, 8paddidm 30640 . . . . 5  |-  ( ( K  e.  B  /\  Z  e.  S )  ->  ( Z  .+  Z
)  =  Z )
12113ad2antr3 1125 . . . 4  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( Z  .+  Z
)  =  Z )
1312sseq2d 3378 . . 3  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( ( X  .+  Y )  C_  ( Z  .+  Z )  <->  ( X  .+  Y )  C_  Z
) )
1410, 13sylibd 207 . 2  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( ( X  C_  Z  /\  Y  C_  Z
)  ->  ( X  .+  Y )  C_  Z
) )
154, 8paddssw2 30643 . . 3  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  C_  A ) )  -> 
( ( X  .+  Y )  C_  Z  ->  ( X  C_  Z  /\  Y  C_  Z ) ) )
161, 2, 3, 7, 15syl13anc 1187 . 2  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( ( X  .+  Y )  C_  Z  ->  ( X  C_  Z  /\  Y  C_  Z ) ) )
1714, 16impbid 185 1  |-  ( ( K  e.  B  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( ( X  C_  Z  /\  Y  C_  Z
)  <->  ( X  .+  Y )  C_  Z
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Atomscatm 30063   PSubSpcpsubsp 30295   + Pcpadd 30594
This theorem is referenced by:  pmodlem1  30645  pclunN  30697  osumcllem1N  30755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-psubsp 30302  df-padd 30595
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