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Theorem paddidm 29955
Description: Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
Hypotheses
Ref Expression
paddidm.s  |-  S  =  ( PSubSp `  K )
paddidm.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
paddidm  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  =  X )

Proof of Theorem paddidm
Dummy variables  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  K  e.  B )
2 eqid 2387 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 paddidm.s . . . . . 6  |-  S  =  ( PSubSp `  K )
42, 3psubssat 29868 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( Atoms `  K ) )
5 eqid 2387 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
6 eqid 2387 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
7 paddidm.p . . . . . 6  |-  .+  =  ( + P `  K
)
85, 6, 2, 7elpadd 29913 . . . . 5  |-  ( ( K  e.  B  /\  X  C_  ( Atoms `  K
)  /\  X  C_  ( Atoms `  K ) )  ->  ( p  e.  ( X  .+  X
)  <->  ( ( p  e.  X  \/  p  e.  X )  \/  (
p  e.  ( Atoms `  K )  /\  E. q  e.  X  E. r  e.  X  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
91, 4, 4, 8syl3anc 1184 . . . 4  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( p  e.  ( X  .+  X )  <-> 
( ( p  e.  X  \/  p  e.  X )  \/  (
p  e.  ( Atoms `  K )  /\  E. q  e.  X  E. r  e.  X  p
( le `  K
) ( q (
join `  K )
r ) ) ) ) )
10 pm1.2 500 . . . . . 6  |-  ( ( p  e.  X  \/  p  e.  X )  ->  p  e.  X )
1110a1i 11 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( ( p  e.  X  \/  p  e.  X )  ->  p  e.  X ) )
125, 6, 2, 3psubspi 29861 . . . . . . 7  |-  ( ( ( K  e.  B  /\  X  e.  S  /\  p  e.  ( Atoms `  K ) )  /\  E. q  e.  X  E. r  e.  X  p ( le
`  K ) ( q ( join `  K
) r ) )  ->  p  e.  X
)
13123exp1 1169 . . . . . 6  |-  ( K  e.  B  ->  ( X  e.  S  ->  ( p  e.  ( Atoms `  K )  ->  ( E. q  e.  X  E. r  e.  X  p ( le `  K ) ( q ( join `  K
) r )  ->  p  e.  X )
) ) )
1413imp4b 574 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( ( p  e.  ( Atoms `  K )  /\  E. q  e.  X  E. r  e.  X  p ( le `  K ) ( q ( join `  K
) r ) )  ->  p  e.  X
) )
1511, 14jaod 370 . . . 4  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( ( ( p  e.  X  \/  p  e.  X )  \/  (
p  e.  ( Atoms `  K )  /\  E. q  e.  X  E. r  e.  X  p
( le `  K
) ( q (
join `  K )
r ) ) )  ->  p  e.  X
) )
169, 15sylbid 207 . . 3  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( p  e.  ( X  .+  X )  ->  p  e.  X
) )
1716ssrdv 3297 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  C_  X )
182, 7sspadd1 29929 . . 3  |-  ( ( K  e.  B  /\  X  C_  ( Atoms `  K
)  /\  X  C_  ( Atoms `  K ) )  ->  X  C_  ( X  .+  X ) )
191, 4, 4, 18syl3anc 1184 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( X  .+  X ) )
2017, 19eqssd 3308 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2650    C_ wss 3263   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   lecple 13463   joincjn 14328   Atomscatm 29378   PSubSpcpsubsp 29610   + Pcpadd 29909
This theorem is referenced by:  paddclN  29956  paddss  29959  pmod1i  29962
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-psubsp 29617  df-padd 29910
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