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Theorem pj1ghm 15012
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 2283 . 2  |-  ( Base `  ( Gs  ( T  .(+)  U ) ) )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) )
2 eqid 2283 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 ovex 5883 . . 3  |-  ( T 
.(+)  U )  e.  _V
4 eqid 2283 . . . 4  |-  ( Gs  ( T  .(+)  U )
)  =  ( Gs  ( T  .(+)  U )
)
5 pj1eu.a . . . 4  |-  .+  =  ( +g  `  G )
64, 5ressplusg 13250 . . 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 14965 . . . 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 14624 . . 3  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
1614, 15syl 15 . 2  |-  ( ph  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
17 subgrcl 14626 . . 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 15006 . . . 4  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
232subgss 14622 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
248, 23syl 15 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
25 fss 5397 . . . 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 14625 . . . . 5  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2814, 27syl 15 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2928feq2d 5380 . . 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 2350 . . . . 5  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  <->  x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3228eleq2d 2350 . . . . 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 15008 . . . . . . . 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 15008 . . . . . . . 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 5876 . . . . . 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 14494 . . . . . . . 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 5663 . . . . . . . . 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 3181 . . . . . . 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 5663 . . . . . . . . 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 3181 . . . . . . 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 14622 . . . . . . . . 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 15007 . . . . . . . . 9  |-  ( ph  ->  ( U P T ) : ( T 
.(+)  U ) --> U )
56 ffvelrn 5663 . . . . . . . . 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 3181 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  ( Base `  G
) )
59 ffvelrn 5663 . . . . . . . . 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 3181 . . . . . . 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 3181 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Z `  U ) )
645, 12cntzi 14805 . . . . . . . 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 14378 . . . . . 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 2318 . . . . 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 14631 . . . . . . . 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 14631 . . . . . . 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 14631 . . . . . . 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 15009 . . . . 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 14692 1  |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U ) )  GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362  SubGrpcsubg 14615    GrpHom cghm 14680  Cntzccntz 14791   LSSumclsm 14945   proj
1cpj1 14946
This theorem is referenced by:  pj1ghm2  15013  dpjghm  15298  pj1lmhm  15853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-cntz 14793  df-lsm 14947  df-pj1 14948
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