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Theorem pj1rid 16846
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 465 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  T  e.  (SubGrp `  G )
)
3 subgrcl 16332 . . . . . 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 2457 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
76subgss 16328 . . . . . . 7  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
85, 7syl 16 . . . . . 6  |-  ( ph  ->  U  C_  ( Base `  G ) )
98sselda 3499 . . . . 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 16206 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  ( Base `  G ) )  -> 
(  .0.  .+  X
)  =  X )
134, 9, 12syl2anc 661 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (  .0.  .+  X )  =  X )
1413eqcomd 2465 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  =  (  .0.  .+  X
) )
15 pj1eu.s . . . 4  |-  .(+)  =  (
LSSum `  G )
16 pj1eu.z . . . 4  |-  Z  =  (Cntz `  G )
175adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  (SubGrp `  G )
)
18 pj1eu.4 . . . . 5  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
1918adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  ( T  i^i  U )  =  {  .0.  } )
20 pj1eu.5 . . . . 5  |-  ( ph  ->  T  C_  ( Z `  U ) )
2120adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  T  C_  ( Z `  U
) )
22 pj1f.p . . . 4  |-  P  =  ( proj1 `  G )
2315lsmub2 16803 . . . . . 6  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  U  C_  ( T  .(+)  U ) )
241, 5, 23syl2anc 661 . . . . 5  |-  ( ph  ->  U  C_  ( T  .(+) 
U ) )
2524sselda 3499 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  ( T  .(+)  U ) )
2611subg0cl 16335 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  .0.  e.  T )
272, 26syl 16 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  .0.  e.  T )
28 simpr 461 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
2910, 15, 11, 16, 2, 17, 19, 21, 22, 25, 27, 28pj1eq 16844 . . 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 459 1  |-  ( (
ph  /\  X  e.  U )  ->  (
( T P U ) `  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   0gc0g 14856   Grpcgrp 16179  SubGrpcsubg 16321  Cntzccntz 16479   LSSumclsm 16780   proj1cpj1 16781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-lsm 16782  df-pj1 16783
This theorem is referenced by:  dpjidcl  17233  dpjidclOLD  17240
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