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Theorem paddcom 33776
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 3603 . . . 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 991 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  K  e.  Lat )
4 simpl2 992 . . . . . . . . . 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 3460 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  A )
7 eqid 2452 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
8 padd0.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
97, 8atbase 33253 . . . . . . . . 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 993 . . . . . . . . . 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 3460 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  A )
147, 8atbase 33253 . . . . . . . . 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 2452 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
177, 16latjcom 15343 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  q  e.  ( Base `  K )  /\  r  e.  ( Base `  K
) )  ->  (
q ( join `  K
) r )  =  ( r ( join `  K ) q ) )
183, 10, 15, 17syl3anc 1219 . . . . . . 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 4407 . . . . . 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 2868 . . . . 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 2982 . . . . 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 3064 . . 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 3614 . 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 2452 . . 3  |-  ( le
`  K )  =  ( le `  K
)
26 padd0.p . . 3  |-  .+  =  ( +P `  K
)
2725, 16, 8, 26paddval 33761 . 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 33761 . . 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 1194 . 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 2503 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 965    = wceq 1370    e. wcel 1758   E.wrex 2797   {crab 2800    u. cun 3429    C_ wss 3431   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   Latclat 15329   Atomscatm 33227   +Pcpadd 33758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-lub 15258  df-join 15260  df-lat 15330  df-ats 33231  df-padd 33759
This theorem is referenced by:  paddass  33801  padd12N  33802  pmod2iN  33812  pmodN  33813  pmapjat2  33817
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