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Theorem pj1id 16196
Description: Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
pj1eu.a  |-  .+  =  ( +g  `  G )
pj1eu.s  |-  .(+)  =  (
LSSum `  G )
pj1eu.o  |-  .0.  =  ( 0g `  G )
pj1eu.z  |-  Z  =  (Cntz `  G )
pj1eu.2  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
pj1eu.3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pj1eu.4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
pj1eu.5  |-  ( ph  ->  T  C_  ( Z `  U ) )
pj1f.p  |-  P  =  ( proj1 `  G )
Assertion
Ref Expression
pj1id  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  X  =  ( ( ( T P U ) `  X )  .+  (
( U P T ) `  X ) ) )

Proof of Theorem pj1id
Dummy variables  v  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pj1eu.2 . . . . . . 7  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
2 subgrcl 15686 . . . . . . 7  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
4 eqid 2443 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
54subgss 15682 . . . . . . 7  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
61, 5syl 16 . . . . . 6  |-  ( ph  ->  T  C_  ( Base `  G ) )
7 pj1eu.3 . . . . . . 7  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
84subgss 15682 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
97, 8syl 16 . . . . . 6  |-  ( ph  ->  U  C_  ( Base `  G ) )
103, 6, 93jca 1168 . . . . 5  |-  ( ph  ->  ( G  e.  Grp  /\  T  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) ) )
11 pj1eu.a . . . . . 6  |-  .+  =  ( +g  `  G )
12 pj1eu.s . . . . . 6  |-  .(+)  =  (
LSSum `  G )
13 pj1f.p . . . . . 6  |-  P  =  ( proj1 `  G )
144, 11, 12, 13pj1val 16192 . . . . 5  |-  ( ( ( G  e.  Grp  /\  T  C_  ( Base `  G )  /\  U  C_  ( Base `  G
) )  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  =  (
iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
1510, 14sylan 471 . . . 4  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  =  (
iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
16 pj1eu.o . . . . . 6  |-  .0.  =  ( 0g `  G )
17 pj1eu.z . . . . . 6  |-  Z  =  (Cntz `  G )
18 pj1eu.4 . . . . . 6  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
19 pj1eu.5 . . . . . 6  |-  ( ph  ->  T  C_  ( Z `  U ) )
2011, 12, 16, 17, 1, 7, 18, 19pj1eu 16193 . . . . 5  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
21 riotacl2 6066 . . . . 5  |-  ( E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y )  ->  ( iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )  e. 
{ x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) } )
2220, 21syl 16 . . . 4  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  ( iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )  e.  { x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) } )
2315, 22eqeltrd 2517 . . 3  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  e.  {
x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) } )
24 oveq1 6098 . . . . . . 7  |-  ( x  =  ( ( T P U ) `  X )  ->  (
x  .+  y )  =  ( ( ( T P U ) `
 X )  .+  y ) )
2524eqeq2d 2454 . . . . . 6  |-  ( x  =  ( ( T P U ) `  X )  ->  ( X  =  ( x  .+  y )  <->  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2625rexbidv 2736 . . . . 5  |-  ( x  =  ( ( T P U ) `  X )  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2726elrab 3117 . . . 4  |-  ( ( ( T P U ) `  X )  e.  { x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) }  <->  ( ( ( T P U ) `
 X )  e.  T  /\  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2827simprbi 464 . . 3  |-  ( ( ( T P U ) `  X )  e.  { x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) }  ->  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) )
2923, 28syl 16 . 2  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) )
30 simprr 756 . . 3  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( ( ( T P U ) `  X
)  .+  y )
)
313ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  G  e.  Grp )
329ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  U  C_  ( Base `  G ) )
336ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  T  C_  ( Base `  G ) )
34 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  e.  ( T  .(+)  U )
)
3512, 17lsmcom2 16154 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
361, 7, 19, 35syl3anc 1218 . . . . . . . 8  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
3736ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
3834, 37eleqtrd 2519 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  e.  ( U  .(+)  T )
)
394, 11, 12, 13pj1val 16192 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  U  C_  ( Base `  G )  /\  T  C_  ( Base `  G
) )  /\  X  e.  ( U  .(+)  T ) )  ->  ( ( U P T ) `  X )  =  (
iota_ u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) ) )
4031, 32, 33, 38, 39syl31anc 1221 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( U P T ) `  X )  =  (
iota_ u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) ) )
4111, 12, 16, 17, 1, 7, 18, 19, 13pj1f 16194 . . . . . . . . 9  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
4241ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( T P U ) : ( T  .(+)  U ) --> T )
4342, 34ffvelrnd 5844 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( T P U ) `  X )  e.  T
)
4419ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  T  C_  ( Z `  U )
)
4544, 43sseldd 3357 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( T P U ) `  X )  e.  ( Z `  U ) )
46 simprl 755 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  y  e.  U
)
4711, 17cntzi 15847 . . . . . . . . 9  |-  ( ( ( ( T P U ) `  X
)  e.  ( Z `
 U )  /\  y  e.  U )  ->  ( ( ( T P U ) `  X )  .+  y
)  =  ( y 
.+  ( ( T P U ) `  X ) ) )
4845, 46, 47syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( ( T P U ) `
 X )  .+  y )  =  ( y  .+  ( ( T P U ) `
 X ) ) )
4930, 48eqtrd 2475 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( y  .+  ( ( T P U ) `
 X ) ) )
50 oveq2 6099 . . . . . . . . 9  |-  ( v  =  ( ( T P U ) `  X )  ->  (
y  .+  v )  =  ( y  .+  ( ( T P U ) `  X
) ) )
5150eqeq2d 2454 . . . . . . . 8  |-  ( v  =  ( ( T P U ) `  X )  ->  ( X  =  ( y  .+  v )  <->  X  =  ( y  .+  (
( T P U ) `  X ) ) ) )
5251rspcev 3073 . . . . . . 7  |-  ( ( ( ( T P U ) `  X
)  e.  T  /\  X  =  ( y  .+  ( ( T P U ) `  X
) ) )  ->  E. v  e.  T  X  =  ( y  .+  v ) )
5343, 49, 52syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  E. v  e.  T  X  =  ( y  .+  v ) )
54 simpll 753 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ph )
55 incom 3543 . . . . . . . . . 10  |-  ( U  i^i  T )  =  ( T  i^i  U
)
5655, 18syl5eq 2487 . . . . . . . . 9  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
5717, 1, 7, 19cntzrecd 16175 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( Z `  T ) )
5811, 12, 16, 17, 7, 1, 56, 57pj1eu 16193 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( U  .(+)  T ) )  ->  E! u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )
5954, 38, 58syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  E! u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )
60 oveq1 6098 . . . . . . . . . 10  |-  ( u  =  y  ->  (
u  .+  v )  =  ( y  .+  v ) )
6160eqeq2d 2454 . . . . . . . . 9  |-  ( u  =  y  ->  ( X  =  ( u  .+  v )  <->  X  =  ( y  .+  v
) ) )
6261rexbidv 2736 . . . . . . . 8  |-  ( u  =  y  ->  ( E. v  e.  T  X  =  ( u  .+  v )  <->  E. v  e.  T  X  =  ( y  .+  v
) ) )
6362riota2 6075 . . . . . . 7  |-  ( ( y  e.  U  /\  E! u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )  -> 
( E. v  e.  T  X  =  ( y  .+  v )  <-> 
( iota_ u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )  =  y ) )
6446, 59, 63syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( E. v  e.  T  X  =  ( y  .+  v
)  <->  ( iota_ u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )  =  y ) )
6553, 64mpbid 210 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( iota_ u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )  =  y )
6640, 65eqtrd 2475 . . . 4  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( U P T ) `  X )  =  y )
6766oveq2d 6107 . . 3  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( ( T P U ) `
 X )  .+  ( ( U P T ) `  X
) )  =  ( ( ( T P U ) `  X
)  .+  y )
)
6830, 67eqtr4d 2478 . 2  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( ( ( T P U ) `  X
)  .+  ( ( U P T ) `  X ) ) )
6929, 68rexlimddv 2845 1  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  X  =  ( ( ( T P U ) `  X )  .+  (
( U P T ) `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716   E!wreu 2717   {crab 2719    i^i cin 3327    C_ wss 3328   {csn 3877   -->wf 5414   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   0gc0g 14378   Grpcgrp 15410  SubGrpcsubg 15675  Cntzccntz 15833   LSSumclsm 16133   proj1cpj1 16134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-lsm 16135  df-pj1 16136
This theorem is referenced by:  pj1eq  16197  pj1ghm  16200  pj1lmhm  17181
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