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Theorem paddss2 30078
Description: Subset law for projective subspace sum. (unss2 3434 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 3260 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim2d 813 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  Z  \/  p  e.  X
)  ->  ( p  e.  Z  \/  p  e.  Y ) ) )
3 ssrexv 3324 . . . . . . . 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 2739 . . . . . . 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 548 . . . . . 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 811 . . . . 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 452 . . . 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 959 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
9 simpl3 961 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
10 sstr 3273 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
11103ad2antr2 1122 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
1211ancoms 439 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
13 eqid 2366 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
14 eqid 2366 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
15 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
16 padd0.p . . . . . 6  |-  .+  =  ( + P `  K
)
1713, 14, 15, 16elpadd 30059 . . . . 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 1183 . . . 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 960 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Y  C_  A )
2013, 14, 15, 16elpadd 30059 . . . . 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 1183 . . . 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 259 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Z  .+  X )  ->  p  e.  ( Z  .+  Y ) ) )
2322ssrdv 3271 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( Z  .+  X
)  C_  ( Z  .+  Y ) )
2423ex 423 1  |-  ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  ->  ( X  C_  Y  ->  ( Z  .+  X )  C_  ( Z  .+  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   E.wrex 2629    C_ wss 3238   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   Atomscatm 29524   + Pcpadd 30055
This theorem is referenced by:  paddss12  30079  pmod1i  30108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-padd 30056
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