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Theorem paddcom 35239
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 3630 . . . 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 998 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  K  e.  Lat )
4 simpl2 999 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  X  C_  A )
5 simprl 755 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  X )
64, 5sseldd 3487 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  A )
7 eqid 2441 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
8 padd0.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
97, 8atbase 34716 . . . . . . . . 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 1000 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  Y  C_  A )
12 simprr 756 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  Y )
1311, 12sseldd 3487 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  A )
147, 8atbase 34716 . . . . . . . . 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 2441 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
177, 16latjcom 15558 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  q  e.  ( Base `  K )  /\  r  e.  ( Base `  K
) )  ->  (
q ( join `  K
) r )  =  ( r ( join `  K ) q ) )
183, 10, 15, 17syl3anc 1227 . . . . . . 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 4445 . . . . . 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 2958 . . . . 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 3003 . . . . 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 3085 . . 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 3641 . 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 2441 . . 3  |-  ( le
`  K )  =  ( le `  K
)
26 padd0.p . . 3  |-  .+  =  ( +P `  K
)
2725, 16, 8, 26paddval 35224 . 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 35224 . . 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 1201 . 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 2492 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 369    /\ w3a 972    = wceq 1381    e. wcel 1802   E.wrex 2792   {crab 2795    u. cun 3456    C_ wss 3458   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   joincjn 15442   Latclat 15544   Atomscatm 34690   +Pcpadd 35221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-lub 15473  df-join 15475  df-lat 15545  df-ats 34694  df-padd 35222
This theorem is referenced by:  paddass  35264  padd12N  35265  pmod2iN  35275  pmodN  35276  pmapjat2  35280
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