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Theorem pj1id 16506
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 15994 . . . . . . 7  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
4 eqid 2460 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
54subgss 15990 . . . . . . 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 15990 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
97, 8syl 16 . . . . . 6  |-  ( ph  ->  U  C_  ( Base `  G ) )
103, 6, 93jca 1171 . . . . 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 16502 . . . . 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 16503 . . . . 5  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
21 riotacl2 6250 . . . . 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 2548 . . 3  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  e.  {
x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) } )
24 oveq1 6282 . . . . . . 7  |-  ( x  =  ( ( T P U ) `  X )  ->  (
x  .+  y )  =  ( ( ( T P U ) `
 X )  .+  y ) )
2524eqeq2d 2474 . . . . . 6  |-  ( x  =  ( ( T P U ) `  X )  ->  ( X  =  ( x  .+  y )  <->  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2625rexbidv 2966 . . . . 5  |-  ( x  =  ( ( T P U ) `  X )  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2726elrab 3254 . . . 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 16464 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
361, 7, 19, 35syl3anc 1223 . . . . . . . 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 2550 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  e.  ( U  .(+)  T )
)
394, 11, 12, 13pj1val 16502 . . . . . 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 1226 . . . . 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 16504 . . . . . . . . 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 6013 . . . . . . 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 3498 . . . . . . . . 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 16155 . . . . . . . . 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 2501 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( y  .+  ( ( T P U ) `
 X ) ) )
50 oveq2 6283 . . . . . . . . 9  |-  ( v  =  ( ( T P U ) `  X )  ->  (
y  .+  v )  =  ( y  .+  ( ( T P U ) `  X
) ) )
5150eqeq2d 2474 . . . . . . . 8  |-  ( v  =  ( ( T P U ) `  X )  ->  ( X  =  ( y  .+  v )  <->  X  =  ( y  .+  (
( T P U ) `  X ) ) ) )
5251rspcev 3207 . . . . . . 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 3684 . . . . . . . . . 10  |-  ( U  i^i  T )  =  ( T  i^i  U
)
5655, 18syl5eq 2513 . . . . . . . . 9  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
5717, 1, 7, 19cntzrecd 16485 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( Z `  T ) )
5811, 12, 16, 17, 7, 1, 56, 57pj1eu 16503 . . . . . . . 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 6282 . . . . . . . . . 10  |-  ( u  =  y  ->  (
u  .+  v )  =  ( y  .+  v ) )
6160eqeq2d 2474 . . . . . . . . 9  |-  ( u  =  y  ->  ( X  =  ( u  .+  v )  <->  X  =  ( y  .+  v
) ) )
6261rexbidv 2966 . . . . . . . 8  |-  ( u  =  y  ->  ( E. v  e.  T  X  =  ( u  .+  v )  <->  E. v  e.  T  X  =  ( y  .+  v
) ) )
6362riota2 6259 . . . . . . 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 2501 . . . 4  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( U P T ) `  X )  =  y )
6766oveq2d 6291 . . 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 2504 . 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 2952 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 968    = wceq 1374    e. wcel 1762   E.wrex 2808   E!wreu 2809   {crab 2811    i^i cin 3468    C_ wss 3469   {csn 4020   -->wf 5575   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   0gc0g 14684   Grpcgrp 15716  SubGrpcsubg 15983  Cntzccntz 16141   LSSumclsm 16443   proj1cpj1 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-lsm 16445  df-pj1 16446
This theorem is referenced by:  pj1eq  16507  pj1ghm  16510  pj1lmhm  17522
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