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Theorem pj1ghm 15337
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 2438 . 2  |-  ( Base `  ( Gs  ( T  .(+)  U ) ) )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) )
2 eqid 2438 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 ovex 6108 . . 3  |-  ( T 
.(+)  U )  e.  _V
4 eqid 2438 . . . 4  |-  ( Gs  ( T  .(+)  U )
)  =  ( Gs  ( T  .(+)  U )
)
5 pj1eu.a . . . 4  |-  .+  =  ( +g  `  G )
64, 5ressplusg 13573 . . 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 15290 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  e.  (SubGrp `  G ) )
148, 9, 10, 13syl3anc 1185 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  (SubGrp `  G
) )
154subggrp 14949 . . 3  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
1614, 15syl 16 . 2  |-  ( ph  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
17 subgrcl 14951 . . 3  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
188, 17syl 16 . 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 15331 . . . 4  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
232subgss 14947 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
248, 23syl 16 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
25 fss 5601 . . . 4  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  T  C_  ( Base `  G
) )  ->  ( T P U ) : ( T  .(+)  U ) --> ( Base `  G
) )
2622, 24, 25syl2anc 644 . . 3  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> ( Base `  G ) )
274subgbas 14950 . . . . 5  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2814, 27syl 16 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2928feq2d 5583 . . 3  |-  ( ph  ->  ( ( T P U ) : ( T  .(+)  U ) --> ( Base `  G )  <->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> ( Base `  G ) ) )
3026, 29mpbid 203 . 2  |-  ( ph  ->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> (
Base `  G )
)
3128eleq2d 2505 . . . . 5  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  <->  x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3228eleq2d 2505 . . . . 5  |-  ( ph  ->  ( y  e.  ( T  .(+)  U )  <->  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3331, 32anbi12d 693 . . . 4  |-  ( ph  ->  ( ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  <->  ( x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) )  /\  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) ) )
3433biimpar 473 . . 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 15333 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  x  =  ( ( ( T P U ) `  x )  .+  (
( U P T ) `  x ) ) )
3635adantrr 699 . . . . . . 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 15333 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( T  .(+)  U ) )  ->  y  =  ( ( ( T P U ) `  y )  .+  (
( U P T ) `  y ) ) )
3837adantrl 698 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  y  =  ( ( ( T P U ) `
 y )  .+  ( ( U P T ) `  y
) ) )
3936, 38oveq12d 6101 . . . . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  e.  (SubGrp `  G )
)
41 grpmnd 14819 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
4240, 17, 413syl 19 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  G  e.  Mnd )
4340, 23syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  C_  ( Base `  G
) )
44 simpl 445 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  x  e.  ( T  .(+)  U ) )
45 ffvelrn 5870 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  x
)  e.  T )
4622, 44, 45syl2an 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  T )
4743, 46sseldd 3351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  ( Base `  G
) )
48 simpr 449 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  y  e.  ( T  .(+)  U ) )
49 ffvelrn 5870 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  y
)  e.  T )
5022, 48, 49syl2an 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  T )
5143, 50sseldd 3351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Base `  G
) )
529adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  U  e.  (SubGrp `  G )
)
532subgss 14947 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
5452, 53syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  U  C_  ( Base `  G
) )
555, 11, 19, 12, 8, 9, 20, 10, 21pj2f 15332 . . . . . . . . 9  |-  ( ph  ->  ( U P T ) : ( T 
.(+)  U ) --> U )
56 ffvelrn 5870 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  x
)  e.  U )
5755, 44, 56syl2an 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  U )
5854, 57sseldd 3351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  ( Base `  G
) )
59 ffvelrn 5870 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  y
)  e.  U )
6055, 48, 59syl2an 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  U )
6154, 60sseldd 3351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  ( Base `  G
) )
6210adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  C_  ( Z `  U
) )
6362, 50sseldd 3351 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Z `  U ) )
645, 12cntzi 15130 . . . . . . . 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 644 . . . . . . 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 14703 . . . . . 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 2473 . . . . 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 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  ( T  i^i  U )  =  {  .0.  } )
695subgcl 14956 . . . . . . . 8  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  ->  ( x  .+  y )  e.  ( T  .(+)  U )
)
70693expb 1155 . . . . . . 7  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
7114, 70sylan 459 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
725subgcl 14956 . . . . . . 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 1185 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( T P U ) `  x
)  .+  ( ( T P U ) `  y ) )  e.  T )
745subgcl 14956 . . . . . . 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 1185 . . . . . 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 15334 . . . . 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 203 . . . 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 447 . . 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 458 . 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 15017 1  |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U ) )  GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322   {csn 3816   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471   ↾s cress 13472   +g cplusg 13531   0gc0g 13725   Mndcmnd 14686   Grpcgrp 14687  SubGrpcsubg 14940    GrpHom cghm 15005  Cntzccntz 15116   LSSumclsm 15270   proj
1cpj1 15271
This theorem is referenced by:  pj1ghm2  15338  dpjghm  15623  pj1lmhm  16174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-ghm 15006  df-cntz 15118  df-lsm 15272  df-pj1 15273
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