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Theorem paddss1 35938
Description: Subset law for projective subspace sum. (unss1 3659 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 3483 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim1d 837 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  X  \/  p  e.  Z
)  ->  ( p  e.  Y  \/  p  e.  Z ) ) )
3 ssrexv 3551 . . . . . . 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 563 . . . . . 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 836 . . . . 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 464 . . . 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 997 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
8 sstr 3497 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
983ad2antr2 1160 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
109ancoms 451 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
11 simpl3 999 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
12 eqid 2454 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2454 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
14 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
15 padd0.p . . . . . 6  |-  .+  =  ( +P `  K
)
1612, 13, 14, 15elpadd 35920 . . . . 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 1226 . . . 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 35920 . . . . 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 463 . . . 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 3495 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( X  .+  Z
)  C_  ( Y  .+  Z ) )
2221ex 432 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   Atomscatm 35385   +Pcpadd 35916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-padd 35917
This theorem is referenced by:  paddss12  35940  paddasslem12  35952  pmod1i  35969  pl42lem3N  36102
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