Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddss2 Structured version   Unicode version

Theorem paddss2 32816
Description: Subset law for projective subspace sum. (unss2 3613 analog.) (Contributed by NM, 7-Mar-2012.)
Hypotheses
Ref Expression
padd0.a  |-  A  =  ( Atoms `  K )
padd0.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
paddss2  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( Z  .+  X )  C_  ( Z  .+  Y ) ) )

Proof of Theorem paddss2
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3435 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim2d 841 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  Z  \/  p  e.  X
)  ->  ( p  e.  Z  \/  p  e.  Y ) ) )
3 ssrexv 3503 . . . . . . . 8  |-  ( X 
C_  Y  ->  ( E. r  e.  X  p ( le `  K ) ( q ( join `  K
) r )  ->  E. r  e.  Y  p ( le `  K ) ( q ( join `  K
) r ) ) )
43reximdv 2877 . . . . . . 7  |-  ( X 
C_  Y  ->  ( E. q  e.  Z  E. r  e.  X  p ( le `  K ) ( q ( join `  K
) r )  ->  E. q  e.  Z  E. r  e.  Y  p ( le `  K ) ( q ( join `  K
) r ) ) )
54anim2d 563 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  A  /\  E. q  e.  Z  E. r  e.  X  p ( le `  K ) ( q ( join `  K
) r ) )  ->  ( p  e.  A  /\  E. q  e.  Z  E. r  e.  Y  p ( le `  K ) ( q ( join `  K
) r ) ) ) )
62, 5orim12d 839 . . . . 5  |-  ( X 
C_  Y  ->  (
( ( p  e.  Z  \/  p  e.  X )  \/  (
p  e.  A  /\  E. q  e.  Z  E. r  e.  X  p
( le `  K
) ( q (
join `  K )
r ) ) )  ->  ( ( p  e.  Z  \/  p  e.  Y )  \/  (
p  e.  A  /\  E. q  e.  Z  E. r  e.  Y  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
76adantl 464 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( ( ( p  e.  Z  \/  p  e.  X )  \/  (
p  e.  A  /\  E. q  e.  Z  E. r  e.  X  p
( le `  K
) ( q (
join `  K )
r ) ) )  ->  ( ( p  e.  Z  \/  p  e.  Y )  \/  (
p  e.  A  /\  E. q  e.  Z  E. r  e.  Y  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
8 simpl1 1000 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
9 simpl3 1002 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
10 sstr 3449 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
11103ad2antr2 1163 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
1211ancoms 451 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
13 eqid 2402 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
14 eqid 2402 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
15 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
16 padd0.p . . . . . 6  |-  .+  =  ( +P `  K
)
1713, 14, 15, 16elpadd 32797 . . . . 5  |-  ( ( K  e.  B  /\  Z  C_  A  /\  X  C_  A )  ->  (
p  e.  ( Z 
.+  X )  <->  ( (
p  e.  Z  \/  p  e.  X )  \/  ( p  e.  A  /\  E. q  e.  Z  E. r  e.  X  p ( le `  K ) ( q ( join `  K
) r ) ) ) ) )
188, 9, 12, 17syl3anc 1230 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Z  .+  X )  <-> 
( ( p  e.  Z  \/  p  e.  X )  \/  (
p  e.  A  /\  E. q  e.  Z  E. r  e.  X  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
19 simpl2 1001 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Y  C_  A )
2013, 14, 15, 16elpadd 32797 . . . . 5  |-  ( ( K  e.  B  /\  Z  C_  A  /\  Y  C_  A )  ->  (
p  e.  ( Z 
.+  Y )  <->  ( (
p  e.  Z  \/  p  e.  Y )  \/  ( p  e.  A  /\  E. q  e.  Z  E. r  e.  Y  p ( le `  K ) ( q ( join `  K
) r ) ) ) ) )
218, 9, 19, 20syl3anc 1230 . . . 4  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Z  .+  Y )  <-> 
( ( p  e.  Z  \/  p  e.  Y )  \/  (
p  e.  A  /\  E. q  e.  Z  E. r  e.  Y  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
227, 18, 213imtr4d 268 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Z  .+  X )  ->  p  e.  ( Z  .+  Y ) ) )
2322ssrdv 3447 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( Z  .+  X
)  C_  ( Z  .+  Y ) )
2423ex 432 1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( Z  .+  X )  C_  ( Z  .+  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2754    C_ wss 3413   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   lecple 14808   joincjn 15789   Atomscatm 32262   +Pcpadd 32793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-padd 32794
This theorem is referenced by:  paddss12  32817  pmod1i  32846
  Copyright terms: Public domain W3C validator