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Theorem pj1ghm 15061
Description: The left projection function is a group homomorphism. (Contributed 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  =  ( proj 1 `  G )
Assertion
Ref Expression
pj1ghm  |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U ) )  GrpHom  G ) )

Proof of Theorem pj1ghm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2316 . 2  |-  ( Base `  ( Gs  ( T  .(+)  U ) ) )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) )
2 eqid 2316 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 ovex 5925 . . 3  |-  ( T 
.(+)  U )  e.  _V
4 eqid 2316 . . . 4  |-  ( Gs  ( T  .(+)  U )
)  =  ( Gs  ( T  .(+)  U )
)
5 pj1eu.a . . . 4  |-  .+  =  ( +g  `  G )
64, 5ressplusg 13297 . . 3  |-  ( ( T  .(+)  U )  e.  _V  ->  .+  =  ( +g  `  ( Gs  ( T  .(+)  U )
) ) )
73, 6ax-mp 8 . 2  |-  .+  =  ( +g  `  ( Gs  ( T  .(+)  U )
) )
8 pj1eu.2 . . . 4  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
9 pj1eu.3 . . . 4  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
10 pj1eu.5 . . . 4  |-  ( ph  ->  T  C_  ( Z `  U ) )
11 pj1eu.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
12 pj1eu.z . . . . 5  |-  Z  =  (Cntz `  G )
1311, 12lsmsubg 15014 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  e.  (SubGrp `  G ) )
148, 9, 10, 13syl3anc 1182 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  (SubGrp `  G
) )
154subggrp 14673 . . 3  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
1614, 15syl 15 . 2  |-  ( ph  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
17 subgrcl 14675 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
188, 17syl 15 . 2  |-  ( ph  ->  G  e.  Grp )
19 pj1eu.o . . . . 5  |-  .0.  =  ( 0g `  G )
20 pj1eu.4 . . . . 5  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
21 pj1f.p . . . . 5  |-  P  =  ( proj 1 `  G )
225, 11, 19, 12, 8, 9, 20, 10, 21pj1f 15055 . . . 4  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
232subgss 14671 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
248, 23syl 15 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
25 fss 5435 . . . 4  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  T  C_  ( Base `  G
) )  ->  ( T P U ) : ( T  .(+)  U ) --> ( Base `  G
) )
2622, 24, 25syl2anc 642 . . 3  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> ( Base `  G ) )
274subgbas 14674 . . . . 5  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2814, 27syl 15 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2928feq2d 5417 . . 3  |-  ( ph  ->  ( ( T P U ) : ( T  .(+)  U ) --> ( Base `  G )  <->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> ( Base `  G ) ) )
3026, 29mpbid 201 . 2  |-  ( ph  ->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> (
Base `  G )
)
3128eleq2d 2383 . . . . 5  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  <->  x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3228eleq2d 2383 . . . . 5  |-  ( ph  ->  ( y  e.  ( T  .(+)  U )  <->  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3331, 32anbi12d 691 . . . 4  |-  ( ph  ->  ( ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  <->  ( x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) )  /\  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) ) )
3433biimpar 471 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) )  /\  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )  ->  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )
355, 11, 19, 12, 8, 9, 20, 10, 21pj1id 15057 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  x  =  ( ( ( T P U ) `  x )  .+  (
( U P T ) `  x ) ) )
3635adantrr 697 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  x  =  ( ( ( T P U ) `
 x )  .+  ( ( U P T ) `  x
) ) )
375, 11, 19, 12, 8, 9, 20, 10, 21pj1id 15057 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( T  .(+)  U ) )  ->  y  =  ( ( ( T P U ) `  y )  .+  (
( U P T ) `  y ) ) )
3837adantrl 696 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  y  =  ( ( ( T P U ) `
 y )  .+  ( ( U P T ) `  y
) ) )
3936, 38oveq12d 5918 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  =  ( ( ( ( T P U ) `  x ) 
.+  ( ( U P T ) `  x ) )  .+  ( ( ( T P U ) `  y )  .+  (
( U P T ) `  y ) ) ) )
408adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  e.  (SubGrp `  G )
)
41 grpmnd 14543 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
4240, 17, 413syl 18 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  G  e.  Mnd )
4340, 23syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  C_  ( Base `  G
) )
44 simpl 443 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  x  e.  ( T  .(+)  U ) )
45 ffvelrn 5701 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  x
)  e.  T )
4622, 44, 45syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  T )
4743, 46sseldd 3215 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  ( Base `  G
) )
48 simpr 447 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  y  e.  ( T  .(+)  U ) )
49 ffvelrn 5701 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  y
)  e.  T )
5022, 48, 49syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  T )
5143, 50sseldd 3215 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Base `  G
) )
529adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  U  e.  (SubGrp `  G )
)
532subgss 14671 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
5452, 53syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  U  C_  ( Base `  G
) )
555, 11, 19, 12, 8, 9, 20, 10, 21pj2f 15056 . . . . . . . . 9  |-  ( ph  ->  ( U P T ) : ( T 
.(+)  U ) --> U )
56 ffvelrn 5701 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  x
)  e.  U )
5755, 44, 56syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  U )
5854, 57sseldd 3215 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  ( Base `  G
) )
59 ffvelrn 5701 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  y
)  e.  U )
6055, 48, 59syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  U )
6154, 60sseldd 3215 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  ( Base `  G
) )
6210adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  C_  ( Z `  U
) )
6362, 50sseldd 3215 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Z `  U ) )
645, 12cntzi 14854 . . . . . . . 8  |-  ( ( ( ( T P U ) `  y
)  e.  ( Z `
 U )  /\  ( ( U P T ) `  x
)  e.  U )  ->  ( ( ( T P U ) `
 y )  .+  ( ( U P T ) `  x
) )  =  ( ( ( U P T ) `  x
)  .+  ( ( T P U ) `  y ) ) )
6563, 57, 64syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( T P U ) `  y
)  .+  ( ( U P T ) `  x ) )  =  ( ( ( U P T ) `  x )  .+  (
( T P U ) `  y ) ) )
662, 5, 42, 47, 51, 58, 61, 65mnd4g 14427 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( ( T P U ) `  x )  .+  (
( T P U ) `  y ) )  .+  ( ( ( U P T ) `  x ) 
.+  ( ( U P T ) `  y ) ) )  =  ( ( ( ( T P U ) `  x ) 
.+  ( ( U P T ) `  x ) )  .+  ( ( ( T P U ) `  y )  .+  (
( U P T ) `  y ) ) ) )
6739, 66eqtr4d 2351 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  =  ( ( ( ( T P U ) `  x ) 
.+  ( ( T P U ) `  y ) )  .+  ( ( ( U P T ) `  x )  .+  (
( U P T ) `  y ) ) ) )
6820adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  ( T  i^i  U )  =  {  .0.  } )
695subgcl 14680 . . . . . . . 8  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  ->  ( x  .+  y )  e.  ( T  .(+)  U )
)
70693expb 1152 . . . . . . 7  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
7114, 70sylan 457 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
725subgcl 14680 . . . . . . 7  |-  ( ( T  e.  (SubGrp `  G )  /\  (
( T P U ) `  x )  e.  T  /\  (
( T P U ) `  y )  e.  T )  -> 
( ( ( T P U ) `  x )  .+  (
( T P U ) `  y ) )  e.  T )
7340, 46, 50, 72syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( T P U ) `  x
)  .+  ( ( T P U ) `  y ) )  e.  T )
745subgcl 14680 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  G )  /\  (
( U P T ) `  x )  e.  U  /\  (
( U P T ) `  y )  e.  U )  -> 
( ( ( U P T ) `  x )  .+  (
( U P T ) `  y ) )  e.  U )
7552, 57, 60, 74syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( U P T ) `  x
)  .+  ( ( U P T ) `  y ) )  e.  U )
765, 11, 19, 12, 40, 52, 68, 62, 21, 71, 73, 75pj1eq 15058 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( x  .+  y
)  =  ( ( ( ( T P U ) `  x
)  .+  ( ( T P U ) `  y ) )  .+  ( ( ( U P T ) `  x )  .+  (
( U P T ) `  y ) ) )  <->  ( (
( T P U ) `  ( x 
.+  y ) )  =  ( ( ( T P U ) `
 x )  .+  ( ( T P U ) `  y
) )  /\  (
( U P T ) `  ( x 
.+  y ) )  =  ( ( ( U P T ) `
 x )  .+  ( ( U P T ) `  y
) ) ) ) )
7767, 76mpbid 201 . . . 4  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( T P U ) `  (
x  .+  y )
)  =  ( ( ( T P U ) `  x ) 
.+  ( ( T P U ) `  y ) )  /\  ( ( U P T ) `  (
x  .+  y )
)  =  ( ( ( U P T ) `  x ) 
.+  ( ( U P T ) `  y ) ) ) )
7877simpld 445 . . 3  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  ( x 
.+  y ) )  =  ( ( ( T P U ) `
 x )  .+  ( ( T P U ) `  y
) ) )
7934, 78syldan 456 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) )  /\  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )  ->  ( ( T P U ) `  ( x  .+  y ) )  =  ( ( ( T P U ) `  x ) 
.+  ( ( T P U ) `  y ) ) )
801, 2, 7, 5, 16, 18, 30, 79isghmd 14741 1  |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U ) )  GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822    i^i cin 3185    C_ wss 3186   {csn 3674   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195   ↾s cress 13196   +g cplusg 13255   0gc0g 13449   Mndcmnd 14410   Grpcgrp 14411  SubGrpcsubg 14664    GrpHom cghm 14729  Cntzccntz 14840   LSSumclsm 14994   proj
1cpj1 14995
This theorem is referenced by:  pj1ghm2  15062  dpjghm  15347  pj1lmhm  15902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-0g 13453  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-ghm 14730  df-cntz 14842  df-lsm 14996  df-pj1 14997
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