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Theorem pj1eq 16696
Description: Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.)
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 )
pj1eq.5  |-  ( ph  ->  X  e.  ( T 
.(+)  U ) )
pj1eq.6  |-  ( ph  ->  B  e.  T )
pj1eq.7  |-  ( ph  ->  C  e.  U )
Assertion
Ref Expression
pj1eq  |-  ( ph  ->  ( X  =  ( B  .+  C )  <-> 
( ( ( T P U ) `  X )  =  B  /\  ( ( U P T ) `  X )  =  C ) ) )

Proof of Theorem pj1eq
StepHypRef Expression
1 pj1eq.5 . . . 4  |-  ( ph  ->  X  e.  ( T 
.(+)  U ) )
2 pj1eu.a . . . . 5  |-  .+  =  ( +g  `  G )
3 pj1eu.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
4 pj1eu.o . . . . 5  |-  .0.  =  ( 0g `  G )
5 pj1eu.z . . . . 5  |-  Z  =  (Cntz `  G )
6 pj1eu.2 . . . . 5  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
7 pj1eu.3 . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
8 pj1eu.4 . . . . 5  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
9 pj1eu.5 . . . . 5  |-  ( ph  ->  T  C_  ( Z `  U ) )
10 pj1f.p . . . . 5  |-  P  =  ( proj1 `  G )
112, 3, 4, 5, 6, 7, 8, 9, 10pj1id 16695 . . . 4  |-  ( (
ph  /\  X  e.  ( T  .(+)  U ) )  ->  X  =  ( ( ( T P U ) `  X )  .+  (
( U P T ) `  X ) ) )
121, 11mpdan 668 . . 3  |-  ( ph  ->  X  =  ( ( ( T P U ) `  X ) 
.+  ( ( U P T ) `  X ) ) )
1312eqeq1d 2445 . 2  |-  ( ph  ->  ( X  =  ( B  .+  C )  <-> 
( ( ( T P U ) `  X )  .+  (
( U P T ) `  X ) )  =  ( B 
.+  C ) ) )
142, 3, 4, 5, 6, 7, 8, 9, 10pj1f 16693 . . . 4  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
1514, 1ffvelrnd 6017 . . 3  |-  ( ph  ->  ( ( T P U ) `  X
)  e.  T )
16 pj1eq.6 . . 3  |-  ( ph  ->  B  e.  T )
172, 3, 4, 5, 6, 7, 8, 9, 10pj2f 16694 . . . 4  |-  ( ph  ->  ( U P T ) : ( T 
.(+)  U ) --> U )
1817, 1ffvelrnd 6017 . . 3  |-  ( ph  ->  ( ( U P T ) `  X
)  e.  U )
19 pj1eq.7 . . 3  |-  ( ph  ->  C  e.  U )
202, 4, 5, 6, 7, 8, 9, 15, 16, 18, 19subgdisjb 16689 . 2  |-  ( ph  ->  ( ( ( ( T P U ) `
 X )  .+  ( ( U P T ) `  X
) )  =  ( B  .+  C )  <-> 
( ( ( T P U ) `  X )  =  B  /\  ( ( U P T ) `  X )  =  C ) ) )
2113, 20bitrd 253 1  |-  ( ph  ->  ( X  =  ( B  .+  C )  <-> 
( ( ( T P U ) `  X )  =  B  /\  ( ( U P T ) `  X )  =  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    i^i cin 3460    C_ wss 3461   {csn 4014   ` cfv 5578  (class class class)co 6281   +g cplusg 14678   0gc0g 14818  SubGrpcsubg 16173  Cntzccntz 16331   LSSumclsm 16632   proj1cpj1 16633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-0g 14820  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-grp 16035  df-minusg 16036  df-sbg 16037  df-subg 16176  df-cntz 16333  df-lsm 16634  df-pj1 16635
This theorem is referenced by:  pj1lid  16697  pj1rid  16698  pj1ghm  16699  lsmhash  16701  dpjidcl  17085  dpjidclOLD  17092  pj1lmhm  17724
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