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Theorem paddss2 33428
Description: Subset law for projective subspace sum. (unss2 3617 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 3438 . . . . . . 7  |-  ( X 
C_  Y  ->  (
p  e.  X  ->  p  e.  Y )
)
21orim2d 856 . . . . . 6  |-  ( X 
C_  Y  ->  (
( p  e.  Z  \/  p  e.  X
)  ->  ( p  e.  Z  \/  p  e.  Y ) ) )
3 ssrexv 3506 . . . . . . . 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 2873 . . . . . . 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 573 . . . . . 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 854 . . . . 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 472 . . . 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 1017 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  K  e.  B )
9 simpl3 1019 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Z  C_  A )
10 sstr 3452 . . . . . . 7  |-  ( ( X  C_  Y  /\  Y  C_  A )  ->  X  C_  A )
11103ad2antr2 1180 . . . . . 6  |-  ( ( X  C_  Y  /\  ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A ) )  ->  X  C_  A
)
1211ancoms 459 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  X  C_  A )
13 eqid 2462 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
14 eqid 2462 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
15 padd0.a . . . . . 6  |-  A  =  ( Atoms `  K )
16 padd0.p . . . . . 6  |-  .+  =  ( +P `  K
)
1713, 14, 15, 16elpadd 33409 . . . . 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 1276 . . . 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 1018 . . . . 5  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  ->  Y  C_  A )
2013, 14, 15, 16elpadd 33409 . . . . 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 1276 . . . 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 276 . . 3  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( p  e.  ( Z  .+  X )  ->  p  e.  ( Z  .+  Y ) ) )
2322ssrdv 3450 . 2  |-  ( ( ( K  e.  B  /\  Y  C_  A  /\  Z  C_  A )  /\  X  C_  Y )  -> 
( Z  .+  X
)  C_  ( Z  .+  Y ) )
2423ex 440 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 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   E.wrex 2750    C_ wss 3416   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   lecple 15246   joincjn 16238   Atomscatm 32874   +Pcpadd 33405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-padd 33406
This theorem is referenced by:  paddss12  33429  pmod1i  33458
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