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Theorem paddss2 30304
Description: Subset law for projective subspace sum. (unss2 3482 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 3306 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim2d 814 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  Z  \/  p  e.  X
)  ->  ( p  e.  Z  \/  p  e.  Y ) ) )
3 ssrexv 3372 . . . . . . . 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 2781 . . . . . . 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 549 . . . . . 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 812 . . . . 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 453 . . . 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 960 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
9 simpl3 962 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
10 sstr 3320 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
11103ad2antr2 1123 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
1211ancoms 440 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
13 eqid 2408 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
14 eqid 2408 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
15 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
16 padd0.p . . . . . 6  |-  .+  =  ( + P `  K
)
1713, 14, 15, 16elpadd 30285 . . . . 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 1184 . . . 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 961 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Y  C_  A )
2013, 14, 15, 16elpadd 30285 . . . . 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 1184 . . . 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 260 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Z  .+  X )  ->  p  e.  ( Z  .+  Y ) ) )
2322ssrdv 3318 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( Z  .+  X
)  C_  ( Z  .+  Y ) )
2423ex 424 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2671    C_ wss 3284   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   Atomscatm 29750   + Pcpadd 30281
This theorem is referenced by:  paddss12  30305  pmod1i  30334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-padd 30282
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