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Theorem paddcom 35950
Description: Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
Hypotheses
Ref Expression
padd0.a  |-  A  =  ( Atoms `  K )
padd0.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
paddcom  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )

Proof of Theorem paddcom
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uncom 3562 . . . 4  |-  ( X  u.  Y )  =  ( Y  u.  X
)
21a1i 11 . . 3  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  u.  Y )  =  ( Y  u.  X ) )
3 simpl1 997 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  K  e.  Lat )
4 simpl2 998 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  X  C_  A )
5 simprl 754 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  X )
64, 5sseldd 3418 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  A )
7 eqid 2382 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
8 padd0.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
97, 8atbase 35427 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
106, 9syl 16 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  ( Base `  K
) )
11 simpl3 999 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  Y  C_  A )
12 simprr 755 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  Y )
1311, 12sseldd 3418 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  A )
147, 8atbase 35427 . . . . . . . . 9  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
1513, 14syl 16 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  ( Base `  K
) )
16 eqid 2382 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
177, 16latjcom 15806 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  q  e.  ( Base `  K )  /\  r  e.  ( Base `  K
) )  ->  (
q ( join `  K
) r )  =  ( r ( join `  K ) q ) )
183, 10, 15, 17syl3anc 1226 . . . . . . 7  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  (
q ( join `  K
) r )  =  ( r ( join `  K ) q ) )
1918breq2d 4379 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  (
p ( le `  K ) ( q ( join `  K
) r )  <->  p ( le `  K ) ( r ( join `  K
) q ) ) )
20192rexbidva 2899 . . . . 5  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( E. q  e.  X  E. r  e.  Y  p ( le `  K ) ( q ( join `  K
) r )  <->  E. q  e.  X  E. r  e.  Y  p ( le `  K ) ( r ( join `  K
) q ) ) )
21 rexcom 2944 . . . . 5  |-  ( E. q  e.  X  E. r  e.  Y  p
( le `  K
) ( r (
join `  K )
q )  <->  E. r  e.  Y  E. q  e.  X  p ( le `  K ) ( r ( join `  K
) q ) )
2220, 21syl6bb 261 . . . 4  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( E. q  e.  X  E. r  e.  Y  p ( le `  K ) ( q ( join `  K
) r )  <->  E. r  e.  Y  E. q  e.  X  p ( le `  K ) ( r ( join `  K
) q ) ) )
2322rabbidv 3026 . . 3  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p
( le `  K
) ( q (
join `  K )
r ) }  =  { p  e.  A  |  E. r  e.  Y  E. q  e.  X  p ( le `  K ) ( r ( join `  K
) q ) } )
242, 23uneq12d 3573 . 2  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  (
( X  u.  Y
)  u.  { p  e.  A  |  E. q  e.  X  E. r  e.  Y  p
( le `  K
) ( q (
join `  K )
r ) } )  =  ( ( Y  u.  X )  u. 
{ p  e.  A  |  E. r  e.  Y  E. q  e.  X  p ( le `  K ) ( r ( join `  K
) q ) } ) )
25 eqid 2382 . . 3  |-  ( le
`  K )  =  ( le `  K
)
26 padd0.p . . 3  |-  .+  =  ( +P `  K
)
2725, 16, 8, 26paddval 35935 . 2  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( ( X  u.  Y )  u.  {
p  e.  A  |  E. q  e.  X  E. r  e.  Y  p ( le `  K ) ( q ( join `  K
) r ) } ) )
2825, 16, 8, 26paddval 35935 . . 3  |-  ( ( K  e.  Lat  /\  Y  C_  A  /\  X  C_  A )  ->  ( Y  .+  X )  =  ( ( Y  u.  X )  u.  {
p  e.  A  |  E. r  e.  Y  E. q  e.  X  p ( le `  K ) ( r ( join `  K
) q ) } ) )
29283com23 1200 . 2  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( Y  .+  X )  =  ( ( Y  u.  X )  u.  {
p  e.  A  |  E. r  e.  Y  E. q  e.  X  p ( le `  K ) ( r ( join `  K
) q ) } ) )
3024, 27, 293eqtr4d 2433 1  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   E.wrex 2733   {crab 2736    u. cun 3387    C_ wss 3389   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   Latclat 15792   Atomscatm 35401   +Pcpadd 35932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-lub 15721  df-join 15723  df-lat 15793  df-ats 35405  df-padd 35933
This theorem is referenced by:  paddass  35975  padd12N  35976  pmod2iN  35986  pmodN  35987  pmapjat2  35991
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