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Theorem paddidm 30640
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 445 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  K  e.  B )
2 eqid 2438 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 paddidm.s . . . . . 6  |-  S  =  ( PSubSp `  K )
42, 3psubssat 30553 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( Atoms `  K ) )
5 eqid 2438 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
6 eqid 2438 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
7 paddidm.p . . . . . 6  |-  .+  =  ( + P `  K
)
85, 6, 2, 7elpadd 30598 . . . . 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 1185 . . . 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 pm1.2 501 . . . . . 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 30546 . . . . . . 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 1170 . . . . . 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 575 . . . . 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 371 . . . 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 208 . . 3  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( p  e.  ( X  .+  X )  ->  p  e.  X
) )
1716ssrdv 3356 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  C_  X )
182, 7sspadd1 30614 . . 3  |-  ( ( K  e.  B  /\  X  C_  ( Atoms `  K
)  /\  X  C_  ( Atoms `  K ) )  ->  X  C_  ( X  .+  X ) )
191, 4, 4, 18syl3anc 1185 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( X  .+  X ) )
2017, 19eqssd 3367 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30063   PSubSpcpsubsp 30295   + Pcpadd 30594
This theorem is referenced by:  paddclN  30641  paddss  30644  pmod1i  30647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-psubsp 30302  df-padd 30595
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