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Theorem paddidm 33126
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 458 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  K  e.  B )
2 eqid 2429 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 paddidm.s . . . . . 6  |-  S  =  ( PSubSp `  K )
42, 3psubssat 33039 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( Atoms `  K ) )
5 eqid 2429 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
6 eqid 2429 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
7 paddidm.p . . . . . 6  |-  .+  =  ( +P `  K
)
85, 6, 2, 7elpadd 33084 . . . . 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 1264 . . . 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 pm1.2 515 . . . . . 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 33032 . . . . . . 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 1221 . . . . . 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 593 . . . . 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 381 . . . 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 218 . . 3  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( p  e.  ( X  .+  X )  ->  p  e.  X
) )
1716ssrdv 3476 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  C_  X )
182, 7sspadd1 33100 . . 3  |-  ( ( K  e.  B  /\  X  C_  ( Atoms `  K
)  /\  X  C_  ( Atoms `  K ) )  ->  X  C_  ( X  .+  X ) )
191, 4, 4, 18syl3anc 1264 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( X  .+  X ) )
2017, 19eqssd 3487 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783    C_ wss 3442   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15160   joincjn 16144   Atomscatm 32549   PSubSpcpsubsp 32781   +Pcpadd 33080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-psubsp 32788  df-padd 33081
This theorem is referenced by:  paddclN  33127  paddss  33130  pmod1i  33133
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