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Theorem paddcom 30002
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 3319 . . . 4  |-  ( X  u.  Y )  =  ( Y  u.  X
)
21a1i 10 . . 3  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  u.  Y )  =  ( Y  u.  X ) )
3 simpl1 958 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  K  e.  Lat )
4 simpl2 959 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  X  C_  A )
5 simprl 732 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  X )
64, 5sseldd 3181 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  A )
7 eqid 2283 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
8 padd0.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
97, 8atbase 29479 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
106, 9syl 15 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  q  e.  ( Base `  K
) )
11 simpl3 960 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  Y  C_  A )
12 simprr 733 . . . . . . . . . 10  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  Y )
1311, 12sseldd 3181 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  A )
147, 8atbase 29479 . . . . . . . . 9  |-  ( r  e.  A  ->  r  e.  ( Base `  K
) )
1513, 14syl 15 . . . . . . . 8  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( q  e.  X  /\  r  e.  Y
) )  ->  r  e.  ( Base `  K
) )
16 eqid 2283 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
177, 16latjcom 14165 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  q  e.  ( Base `  K )  /\  r  e.  ( Base `  K
) )  ->  (
q ( join `  K
) r )  =  ( r ( join `  K ) q ) )
183, 10, 15, 17syl3anc 1182 . . . . . . 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 4035 . . . . . 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 2584 . . . . 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 2701 . . . . 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 252 . . . 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 2780 . . 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 3330 . 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 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
26 padd0.p . . 3  |-  .+  =  ( + P `  K
)
2725, 16, 8, 26paddval 29987 . 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 29987 . . 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 1157 . 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 2325 1  |-  ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    u. cun 3150    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   + Pcpadd 29984
This theorem is referenced by:  paddass  30027  padd12N  30028  pmod2iN  30038  pmodN  30039  pmapjat2  30043
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-join 14110  df-lat 14152  df-ats 29457  df-padd 29985
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