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Theorem pj1id 17342
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 16815 . . . . . . 7  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
31, 2syl 17 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
4 eqid 2450 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
54subgss 16811 . . . . . . 7  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
61, 5syl 17 . . . . . 6  |-  ( ph  ->  T  C_  ( Base `  G ) )
7 pj1eu.3 . . . . . . 7  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
84subgss 16811 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
97, 8syl 17 . . . . . 6  |-  ( ph  ->  U  C_  ( Base `  G ) )
103, 6, 93jca 1187 . . . . 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 17338 . . . . 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 474 . . . 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 17339 . . . . 5  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
21 riotacl2 6263 . . . . 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 17 . . . 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 2528 . . 3  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  e.  {
x  e.  T  |  E. y  e.  U  X  =  ( x  .+  y ) } )
24 oveq1 6295 . . . . . . 7  |-  ( x  =  ( ( T P U ) `  X )  ->  (
x  .+  y )  =  ( ( ( T P U ) `
 X )  .+  y ) )
2524eqeq2d 2460 . . . . . 6  |-  ( x  =  ( ( T P U ) `  X )  ->  ( X  =  ( x  .+  y )  <->  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2625rexbidv 2900 . . . . 5  |-  ( x  =  ( ( T P U ) `  X )  ->  ( E. y  e.  U  X  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) ) )
2726elrab 3195 . . . 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 466 . . 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 17 . 2  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  E. y  e.  U  X  =  ( ( ( T P U ) `  X )  .+  y
) )
30 simprr 765 . . 3  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( ( ( T P U ) `  X
)  .+  y )
)
313ad2antrr 731 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  G  e.  Grp )
329ad2antrr 731 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  U  C_  ( Base `  G ) )
336ad2antrr 731 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  T  C_  ( Base `  G ) )
34 simplr 761 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  e.  ( T  .(+)  U )
)
3512, 17lsmcom2 17300 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
361, 7, 19, 35syl3anc 1267 . . . . . . . 8  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
3736ad2antrr 731 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
3834, 37eleqtrd 2530 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  e.  ( U  .(+)  T )
)
394, 11, 12, 13pj1val 17338 . . . . . 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 1270 . . . . 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 17340 . . . . . . . . 9  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
4241ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( T P U ) : ( T  .(+)  U ) --> T )
4342, 34ffvelrnd 6021 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( T P U ) `  X )  e.  T
)
4419ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  T  C_  ( Z `  U )
)
4544, 43sseldd 3432 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( T P U ) `  X )  e.  ( Z `  U ) )
46 simprl 763 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  y  e.  U
)
4711, 17cntzi 16976 . . . . . . . . 9  |-  ( ( ( ( T P U ) `  X
)  e.  ( Z `
 U )  /\  y  e.  U )  ->  ( ( ( T P U ) `  X )  .+  y
)  =  ( y 
.+  ( ( T P U ) `  X ) ) )
4845, 46, 47syl2anc 666 . . . . . . . 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 2484 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  X  =  ( y  .+  ( ( T P U ) `
 X ) ) )
50 oveq2 6296 . . . . . . . . 9  |-  ( v  =  ( ( T P U ) `  X )  ->  (
y  .+  v )  =  ( y  .+  ( ( T P U ) `  X
) ) )
5150eqeq2d 2460 . . . . . . . 8  |-  ( v  =  ( ( T P U ) `  X )  ->  ( X  =  ( y  .+  v )  <->  X  =  ( y  .+  (
( T P U ) `  X ) ) ) )
5251rspcev 3149 . . . . . . 7  |-  ( ( ( ( T P U ) `  X
)  e.  T  /\  X  =  ( y  .+  ( ( T P U ) `  X
) ) )  ->  E. v  e.  T  X  =  ( y  .+  v ) )
5343, 49, 52syl2anc 666 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  E. v  e.  T  X  =  ( y  .+  v ) )
54 simpll 759 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ph )
55 incom 3624 . . . . . . . . . 10  |-  ( U  i^i  T )  =  ( T  i^i  U
)
5655, 18syl5eq 2496 . . . . . . . . 9  |-  ( ph  ->  ( U  i^i  T
)  =  {  .0.  } )
5717, 1, 7, 19cntzrecd 17321 . . . . . . . . 9  |-  ( ph  ->  U  C_  ( Z `  T ) )
5811, 12, 16, 17, 7, 1, 56, 57pj1eu 17339 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( U  .(+)  T ) )  ->  E! u  e.  U  E. v  e.  T  X  =  ( u  .+  v ) )
5954, 38, 58syl2anc 666 . . . . . . 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 6295 . . . . . . . . . 10  |-  ( u  =  y  ->  (
u  .+  v )  =  ( y  .+  v ) )
6160eqeq2d 2460 . . . . . . . . 9  |-  ( u  =  y  ->  ( X  =  ( u  .+  v )  <->  X  =  ( y  .+  v
) ) )
6261rexbidv 2900 . . . . . . . 8  |-  ( u  =  y  ->  ( E. v  e.  T  X  =  ( u  .+  v )  <->  E. v  e.  T  X  =  ( y  .+  v
) ) )
6362riota2 6272 . . . . . . 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 666 . . . . . 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 214 . . . . 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 2484 . . . 4  |-  ( ( ( ph  /\  X  e.  ( T  .(+)  U ) )  /\  ( y  e.  U  /\  X  =  ( ( ( T P U ) `
 X )  .+  y ) ) )  ->  ( ( U P T ) `  X )  =  y )
6766oveq2d 6304 . . 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 2487 . 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 2882 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   E.wrex 2737   E!wreu 2738   {crab 2740    i^i cin 3402    C_ wss 3403   {csn 3967   -->wf 5577   ` cfv 5581   iota_crio 6249  (class class class)co 6288   Basecbs 15114   +g cplusg 15183   0gc0g 15331   Grpcgrp 16662  SubGrpcsubg 16804  Cntzccntz 16962   LSSumclsm 17279   proj1cpj1 17280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-grp 16666  df-minusg 16667  df-sbg 16668  df-subg 16807  df-cntz 16964  df-lsm 17281  df-pj1 17282
This theorem is referenced by:  pj1eq  17343  pj1ghm  17346  pj1lmhm  18316
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