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Theorem pj1lid 15333
Description: The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-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  =  ( proj 1 `  G )
Assertion
Ref Expression
pj1lid  |-  ( (
ph  /\  X  e.  T )  ->  (
( T P U ) `  X )  =  X )

Proof of Theorem pj1lid
StepHypRef Expression
1 pj1eu.2 . . . . . . 7  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
21adantr 452 . . . . . 6  |-  ( (
ph  /\  X  e.  T )  ->  T  e.  (SubGrp `  G )
)
3 subgrcl 14949 . . . . . 6  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
42, 3syl 16 . . . . 5  |-  ( (
ph  /\  X  e.  T )  ->  G  e.  Grp )
5 eqid 2436 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
65subgss 14945 . . . . . . 7  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
71, 6syl 16 . . . . . 6  |-  ( ph  ->  T  C_  ( Base `  G ) )
87sselda 3348 . . . . 5  |-  ( (
ph  /\  X  e.  T )  ->  X  e.  ( Base `  G
) )
9 pj1eu.a . . . . . 6  |-  .+  =  ( +g  `  G )
10 pj1eu.o . . . . . 6  |-  .0.  =  ( 0g `  G )
115, 9, 10grprid 14836 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  ( Base `  G ) )  -> 
( X  .+  .0.  )  =  X )
124, 8, 11syl2anc 643 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  ( X  .+  .0.  )  =  X )
1312eqcomd 2441 . . 3  |-  ( (
ph  /\  X  e.  T )  ->  X  =  ( X  .+  .0.  ) )
14 pj1eu.s . . . 4  |-  .(+)  =  (
LSSum `  G )
15 pj1eu.z . . . 4  |-  Z  =  (Cntz `  G )
16 pj1eu.3 . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
1716adantr 452 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  U  e.  (SubGrp `  G )
)
18 pj1eu.4 . . . . 5  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
1918adantr 452 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  ( T  i^i  U )  =  {  .0.  } )
20 pj1eu.5 . . . . 5  |-  ( ph  ->  T  C_  ( Z `  U ) )
2120adantr 452 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  T  C_  ( Z `  U
) )
22 pj1f.p . . . 4  |-  P  =  ( proj 1 `  G )
2314lsmub1 15290 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  T  C_  ( T  .(+)  U ) )
241, 16, 23syl2anc 643 . . . . 5  |-  ( ph  ->  T  C_  ( T  .(+) 
U ) )
2524sselda 3348 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  X  e.  ( T  .(+)  U ) )
26 simpr 448 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  X  e.  T )
2710subg0cl 14952 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  .0.  e.  U )
2817, 27syl 16 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  .0.  e.  U )
299, 14, 10, 15, 2, 17, 19, 21, 22, 25, 26, 28pj1eq 15332 . . 3  |-  ( (
ph  /\  X  e.  T )  ->  ( X  =  ( X  .+  .0.  )  <->  ( (
( T P U ) `  X )  =  X  /\  (
( U P T ) `  X )  =  .0.  ) ) )
3013, 29mpbid 202 . 2  |-  ( (
ph  /\  X  e.  T )  ->  (
( ( T P U ) `  X
)  =  X  /\  ( ( U P T ) `  X
)  =  .0.  )
)
3130simpld 446 1  |-  ( (
ph  /\  X  e.  T )  ->  (
( T P U ) `  X )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   {csn 3814   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685  SubGrpcsubg 14938  Cntzccntz 15114   LSSumclsm 15268   proj
1cpj1 15269
This theorem is referenced by:  dpjlid  15619  pjfo  16942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-pj1 15271
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