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Theorem paddss1 30676
Description: Subset law for projective subspace sum. (unss1 3518 analog.) (Contributed by NM, 7-Mar-2012.)
Hypotheses
Ref Expression
padd0.a  |-  A  =  ( Atoms `  K )
padd0.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddss1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )

Proof of Theorem paddss1
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3344 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim1d 814 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  X  \/  p  e.  Z
)  ->  ( p  e.  Y  \/  p  e.  Z ) ) )
3 ssrexv 3410 . . . . . . 7  |-  ( X 
C_  Y  ->  ( E. q  e.  X  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r )  ->  E. q  e.  Y  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) )
43anim2d 550 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) )  ->  ( p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) ) )
52, 4orim12d 813 . . . . 5  |-  ( X 
C_  Y  ->  (
( ( p  e.  X  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) )  ->  ( ( p  e.  Y  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
65adantl 454 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( ( ( p  e.  X  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) )  ->  ( ( p  e.  Y  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
7 simpl1 961 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
8 sstr 3358 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983ad2antr2 1124 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
109ancoms 441 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
11 simpl3 963 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
12 eqid 2438 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2438 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
14 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
15 padd0.p . . . . . 6  |-  .+  =  ( + P `  K
)
1612, 13, 14, 15elpadd 30658 . . . . 5  |-  ( ( K  e.  B  /\  X  C_  A  /\  Z  C_  A )  ->  (
p  e.  ( X 
.+  Z )  <->  ( (
p  e.  X  \/  p  e.  Z )  \/  ( p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) ) ) )
177, 10, 11, 16syl3anc 1185 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( X  .+  Z )  <-> 
( ( p  e.  X  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  X  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
1812, 13, 14, 15elpadd 30658 . . . . 5  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  (
p  e.  ( Y 
.+  Z )  <->  ( (
p  e.  Y  \/  p  e.  Z )  \/  ( p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p ( le `  K ) ( q ( join `  K
) r ) ) ) ) )
1918adantr 453 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Y  .+  Z )  <-> 
( ( p  e.  Y  \/  p  e.  Z )  \/  (
p  e.  A  /\  E. q  e.  Y  E. r  e.  Z  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
206, 17, 193imtr4d 261 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( X  .+  Z )  ->  p  e.  ( Y  .+  Z ) ) )
2120ssrdv 3356 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( X  .+  Z
)  C_  ( Y  .+  Z ) )
2221ex 425 1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30123   + Pcpadd 30654
This theorem is referenced by:  paddss12  30678  paddasslem12  30690  pmod1i  30707  pl42lem3N  30840
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-padd 30655
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