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Theorem paddidm 29280
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
StepHypRef Expression
1 simpl 445 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  K  e.  B )
2 eqid 2258 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 paddidm.s . . . . . 6  |-  S  =  ( PSubSp `  K )
42, 3psubssat 29193 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( Atoms `  K ) )
5 eqid 2258 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
6 eqid 2258 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
7 paddidm.p . . . . . 6  |-  .+  =  ( + P `  K
)
85, 6, 2, 7elpadd 29238 . . . . 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 1187 . . . 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 12 . . . . 5  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( ( p  e.  X  \/  p  e.  X )  ->  p  e.  X ) )
125, 6, 2, 3psubspi 29186 . . . . . . 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 1172 . . . . . 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 576 . . . . 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 3160 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  C_  X )
182, 7sspadd1 29254 . . 3  |-  ( ( K  e.  B  /\  X  C_  ( Atoms `  K
)  /\  X  C_  ( Atoms `  K ) )  ->  X  C_  ( X  .+  X ) )
191, 4, 4, 18syl3anc 1187 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  ( X  .+  X ) )
2017, 19eqssd 3171 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( X  .+  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2519    C_ wss 3127   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   lecple 13178   joincjn 14041   Atomscatm 28703   PSubSpcpsubsp 28935   + Pcpadd 29234
This theorem is referenced by:  paddclN  29281  paddss  29284  pmod1i  29287
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-psubsp 28942  df-padd 29235
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