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Theorem pj1rid 16192
Description: The left projection function is the zero operator on the right 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  =  ( proj1 `  G )
Assertion
Ref Expression
pj1rid  |-  ( (
ph  /\  X  e.  U )  ->  (
( T P U ) `  X )  =  .0.  )

Proof of Theorem pj1rid
StepHypRef Expression
1 pj1eu.2 . . . . . . 7  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
21adantr 462 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  T  e.  (SubGrp `  G )
)
3 subgrcl 15679 . . . . . 6  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
42, 3syl 16 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  G  e.  Grp )
5 pj1eu.3 . . . . . . 7  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
6 eqid 2441 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
76subgss 15675 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
85, 7syl 16 . . . . . 6  |-  ( ph  ->  U  C_  ( Base `  G ) )
98sselda 3353 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  ( Base `  G
) )
10 pj1eu.a . . . . . 6  |-  .+  =  ( +g  `  G )
11 pj1eu.o . . . . . 6  |-  .0.  =  ( 0g `  G )
126, 10, 11grplid 15561 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  ( Base `  G ) )  -> 
(  .0.  .+  X
)  =  X )
134, 9, 12syl2anc 656 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (  .0.  .+  X )  =  X )
1413eqcomd 2446 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  =  (  .0.  .+  X
) )
15 pj1eu.s . . . 4  |-  .(+)  =  (
LSSum `  G )
16 pj1eu.z . . . 4  |-  Z  =  (Cntz `  G )
175adantr 462 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  (SubGrp `  G )
)
18 pj1eu.4 . . . . 5  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
1918adantr 462 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  ( T  i^i  U )  =  {  .0.  } )
20 pj1eu.5 . . . . 5  |-  ( ph  ->  T  C_  ( Z `  U ) )
2120adantr 462 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  T  C_  ( Z `  U
) )
22 pj1f.p . . . 4  |-  P  =  ( proj1 `  G )
2315lsmub2 16149 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  U  C_  ( T  .(+)  U ) )
241, 5, 23syl2anc 656 . . . . 5  |-  ( ph  ->  U  C_  ( T  .(+) 
U ) )
2524sselda 3353 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  ( T  .(+)  U ) )
2611subg0cl 15682 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  .0.  e.  T )
272, 26syl 16 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  .0.  e.  T )
28 simpr 458 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
2910, 15, 11, 16, 2, 17, 19, 21, 22, 25, 27, 28pj1eq 16190 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( X  =  (  .0.  .+  X )  <->  ( (
( T P U ) `  X )  =  .0.  /\  (
( U P T ) `  X )  =  X ) ) )
3014, 29mpbid 210 . 2  |-  ( (
ph  /\  X  e.  U )  ->  (
( ( T P U ) `  X
)  =  .0.  /\  ( ( U P T ) `  X
)  =  X ) )
3130simpld 456 1  |-  ( (
ph  /\  X  e.  U )  ->  (
( T P U ) `  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    i^i cin 3324    C_ wss 3325   {csn 3874   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406  SubGrpcsubg 15668  Cntzccntz 15826   LSSumclsm 16126   proj1cpj1 16127
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-cntz 15828  df-lsm 16128  df-pj1 16129
This theorem is referenced by:  dpjidcl  16547  dpjidclOLD  16554
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