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Theorem pj1ghm 16325
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  =  ( proj1 `  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 2454 . 2  |-  ( Base `  ( Gs  ( T  .(+)  U ) ) )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) )
2 eqid 2454 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 ovex 6228 . . 3  |-  ( T 
.(+)  U )  e.  _V
4 eqid 2454 . . . 4  |-  ( Gs  ( T  .(+)  U )
)  =  ( Gs  ( T  .(+)  U )
)
5 pj1eu.a . . . 4  |-  .+  =  ( +g  `  G )
64, 5ressplusg 14403 . . 3  |-  ( ( T  .(+)  U )  e.  _V  ->  .+  =  ( +g  `  ( Gs  ( T  .(+)  U )
) ) )
73, 6ax-mp 5 . 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 16278 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  e.  (SubGrp `  G ) )
148, 9, 10, 13syl3anc 1219 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  (SubGrp `  G
) )
154subggrp 15807 . . 3  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
1614, 15syl 16 . 2  |-  ( ph  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
17 subgrcl 15809 . . 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  =  ( proj1 `  G )
225, 11, 19, 12, 8, 9, 20, 10, 21pj1f 16319 . . . 4  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
232subgss 15805 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
248, 23syl 16 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
25 fss 5678 . . . 4  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  T  C_  ( Base `  G
) )  ->  ( T P U ) : ( T  .(+)  U ) --> ( Base `  G
) )
2622, 24, 25syl2anc 661 . . 3  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> ( Base `  G ) )
274subgbas 15808 . . . . 5  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2814, 27syl 16 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2928feq2d 5658 . . 3  |-  ( ph  ->  ( ( T P U ) : ( T  .(+)  U ) --> ( Base `  G )  <->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> ( Base `  G ) ) )
3026, 29mpbid 210 . 2  |-  ( ph  ->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> (
Base `  G )
)
3128eleq2d 2524 . . . . 5  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  <->  x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3228eleq2d 2524 . . . . 5  |-  ( ph  ->  ( y  e.  ( T  .(+)  U )  <->  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3331, 32anbi12d 710 . . . 4  |-  ( ph  ->  ( ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  <->  ( x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) )  /\  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) ) )
3433biimpar 485 . . 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 16321 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  x  =  ( ( ( T P U ) `  x )  .+  (
( U P T ) `  x ) ) )
3635adantrr 716 . . . . . . 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 16321 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( T  .(+)  U ) )  ->  y  =  ( ( ( T P U ) `  y )  .+  (
( U P T ) `  y ) ) )
3837adantrl 715 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  y  =  ( ( ( T P U ) `
 y )  .+  ( ( U P T ) `  y
) ) )
3936, 38oveq12d 6221 . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  e.  (SubGrp `  G )
)
41 grpmnd 15673 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
4240, 17, 413syl 20 . . . . . . 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 457 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  x  e.  ( T  .(+)  U ) )
45 ffvelrn 5953 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  x
)  e.  T )
4622, 44, 45syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  T )
4743, 46sseldd 3468 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  ( Base `  G
) )
48 simpr 461 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  y  e.  ( T  .(+)  U ) )
49 ffvelrn 5953 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  y
)  e.  T )
5022, 48, 49syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  T )
5143, 50sseldd 3468 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Base `  G
) )
529adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  U  e.  (SubGrp `  G )
)
532subgss 15805 . . . . . . . . 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 16320 . . . . . . . . 9  |-  ( ph  ->  ( U P T ) : ( T 
.(+)  U ) --> U )
56 ffvelrn 5953 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  x
)  e.  U )
5755, 44, 56syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  U )
5854, 57sseldd 3468 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  ( Base `  G
) )
59 ffvelrn 5953 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  y
)  e.  U )
6055, 48, 59syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  U )
6154, 60sseldd 3468 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  ( Base `  G
) )
6210adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  C_  ( Z `  U
) )
6362, 50sseldd 3468 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Z `  U ) )
645, 12cntzi 15970 . . . . . . . 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 661 . . . . . . 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 15549 . . . . . 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 2498 . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  ( T  i^i  U )  =  {  .0.  } )
695subgcl 15814 . . . . . . . 8  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  ->  ( x  .+  y )  e.  ( T  .(+)  U )
)
70693expb 1189 . . . . . . 7  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
7114, 70sylan 471 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
725subgcl 15814 . . . . . . 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 1219 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( T P U ) `  x
)  .+  ( ( T P U ) `  y ) )  e.  T )
745subgcl 15814 . . . . . . 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 1219 . . . . . 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 16322 . . . . 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 210 . . . 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 459 . . 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 470 . 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 15879 1  |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U ) )  GrpHom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439   {csn 3988   -->wf 5525   ` cfv 5529  (class class class)co 6203   Basecbs 14296   ↾s cress 14297   +g cplusg 14361   0gc0g 14501   Mndcmnd 15532   Grpcgrp 15533  SubGrpcsubg 15798    GrpHom cghm 15867  Cntzccntz 15956   LSSumclsm 16258   proj1cpj1 16259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-0g 14503  df-mnd 15538  df-submnd 15588  df-grp 15668  df-minusg 15669  df-sbg 15670  df-subg 15801  df-ghm 15868  df-cntz 15958  df-lsm 16260  df-pj1 16261
This theorem is referenced by:  pj1ghm2  16326  dpjghm  16694  pj1lmhm  17314
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