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Theorem pj1id 15294
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  =  ( proj 1 `  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 14912 . . . . . . 7  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
4 eqid 2412 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
54subgss 14908 . . . . . . 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 14908 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
97, 8syl 16 . . . . . 6  |-  ( ph  ->  U  C_  ( Base `  G ) )
103, 6, 93jca 1134 . . . . 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  =  ( proj 1 `  G )
144, 11, 12, 13pj1val 15290 . . . . 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 458 . . . 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 15291 . . . . 5  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
21 riotacl2 6530 . . . . 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 2486 . . 3  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  e.  {
x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) } )
24 oveq1 6055 . . . . . . 7  |-  ( x  =  ( ( T P U ) `  X )  ->  (
x  .+  y )  =  ( ( ( T P U ) `
 X )  .+  y ) )
2524eqeq2d 2423 . . . . . 6  |-  ( x  =  ( ( T P U ) `  X )  ->  ( X  =  ( x  .+  y )  <->  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2625rexbidv 2695 . . . . 5  |-  ( x  =  ( ( T P U ) `  X )  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2726elrab 3060 . . . 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 451 . . 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 734 . . 3  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( ( ( T P U ) `  X
)  .+  y )
)
313ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  G  e.  Grp )
329ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  U  C_  ( Base `  G ) )
336ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  T  C_  ( Base `  G ) )
34 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  e.  ( T  .(+)  U )
)
3512, 17lsmcom2 15252 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
361, 7, 19, 35syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
3736ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
3834, 37eleqtrd 2488 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  e.  ( U  .(+)  T )
)
394, 11, 12, 13pj1val 15290 . . . . . 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 1187 . . . . 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 15292 . . . . . . . . 9  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
4241ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( T P U ) : ( T  .(+)  U ) --> T )
4342, 34ffvelrnd 5838 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( T P U ) `  X )  e.  T
)
4419ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  T  C_  ( Z `  U )
)
4544, 43sseldd 3317 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( T P U ) `  X )  e.  ( Z `  U ) )
46 simprl 733 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  y  e.  U
)
4711, 17cntzi 15091 . . . . . . . . 9  |-  ( ( ( ( T P U ) `  X
)  e.  ( Z `
 U )  /\  y  e.  U )  ->  ( ( ( T P U ) `  X )  .+  y
)  =  ( y 
.+  ( ( T P U ) `  X ) ) )
4845, 46, 47syl2anc 643 . . . . . . . 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 2444 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( y  .+  ( ( T P U ) `
 X ) ) )
50 oveq2 6056 . . . . . . . . 9  |-  ( v  =  ( ( T P U ) `  X )  ->  (
y  .+  v )  =  ( y  .+  ( ( T P U ) `  X
) ) )
5150eqeq2d 2423 . . . . . . . 8  |-  ( v  =  ( ( T P U ) `  X )  ->  ( X  =  ( y  .+  v )  <->  X  =  ( y  .+  (
( T P U ) `  X ) ) ) )
5251rspcev 3020 . . . . . . 7  |-  ( ( ( ( T P U ) `  X
)  e.  T  /\  X  =  ( y  .+  ( ( T P U ) `  X
) ) )  ->  E. v  e.  T  X  =  ( y  .+  v ) )
5343, 49, 52syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  E. v  e.  T  X  =  ( y  .+  v ) )
54 simpll 731 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ph )
55 incom 3501 . . . . . . . . . 10  |-  ( U  i^i  T )  =  ( T  i^i  U
)
5655, 18syl5eq 2456 . . . . . . . . 9  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
5717, 1, 7, 19cntzrecd 15273 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( Z `  T ) )
5811, 12, 16, 17, 7, 1, 56, 57pj1eu 15291 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( U  .(+)  T ) )  ->  E! u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )
5954, 38, 58syl2anc 643 . . . . . . 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 6055 . . . . . . . . . 10  |-  ( u  =  y  ->  (
u  .+  v )  =  ( y  .+  v ) )
6160eqeq2d 2423 . . . . . . . . 9  |-  ( u  =  y  ->  ( X  =  ( u  .+  v )  <->  X  =  ( y  .+  v
) ) )
6261rexbidv 2695 . . . . . . . 8  |-  ( u  =  y  ->  ( E. v  e.  T  X  =  ( u  .+  v )  <->  E. v  e.  T  X  =  ( y  .+  v
) ) )
6362riota2 6539 . . . . . . 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 643 . . . . . 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 202 . . . . 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 2444 . . . 4  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( U P T ) `  X )  =  y )
6766oveq2d 6064 . . 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 2447 . 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 2802 1  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  X  =  ( ( ( T P U ) `  X )  .+  (
( U P T ) `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   E!wreu 2676   {crab 2678    i^i cin 3287    C_ wss 3288   {csn 3782   -->wf 5417   ` cfv 5421  (class class class)co 6048   iota_crio 6509   Basecbs 13432   +g cplusg 13492   0gc0g 13686   Grpcgrp 14648  SubGrpcsubg 14901  Cntzccntz 15077   LSSumclsm 15231   proj
1cpj1 15232
This theorem is referenced by:  pj1eq  15295  pj1ghm  15298  pj1lmhm  16135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-mnd 14653  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-cntz 15079  df-lsm 15233  df-pj1 15234
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