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Theorem paddss1 33769
Description: Subset law for projective subspace sum. (unss1 3625 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 3450 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim1d 835 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  X  \/  p  e.  Z
)  ->  ( p  e.  Y  \/  p  e.  Z ) ) )
3 ssrexv 3517 . . . . . . 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 565 . . . . . 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 834 . . . . 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 466 . . . 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 991 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
8 sstr 3464 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983ad2antr2 1154 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
109ancoms 453 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
11 simpl3 993 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
12 eqid 2451 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2451 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
14 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
15 padd0.p . . . . . 6  |-  .+  =  ( +P `  K
)
1612, 13, 14, 15elpadd 33751 . . . . 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 1219 . . . 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 33751 . . . . 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 465 . . . 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 268 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( X  .+  Z )  ->  p  e.  ( Y  .+  Z ) ) )
2120ssrdv 3462 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( X  .+  Z
)  C_  ( Y  .+  Z ) )
2221ex 434 1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( X  .+  Z )  C_  ( Y  .+  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2796    C_ wss 3428   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   lecple 14349   joincjn 15218   Atomscatm 33216   +Pcpadd 33747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-padd 33748
This theorem is referenced by:  paddss12  33771  paddasslem12  33783  pmod1i  33800  pl42lem3N  33933
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