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Theorem pj1ghm 16590
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 2441 . 2  |-  ( Base `  ( Gs  ( T  .(+)  U ) ) )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) )
2 eqid 2441 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 ovex 6305 . . 3  |-  ( T 
.(+)  U )  e.  _V
4 eqid 2441 . . . 4  |-  ( Gs  ( T  .(+)  U )
)  =  ( Gs  ( T  .(+)  U )
)
5 pj1eu.a . . . 4  |-  .+  =  ( +g  `  G )
64, 5ressplusg 14611 . . 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 16543 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  e.  (SubGrp `  G ) )
148, 9, 10, 13syl3anc 1227 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  (SubGrp `  G
) )
154subggrp 16073 . . 3  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
1614, 15syl 16 . 2  |-  ( ph  ->  ( Gs  ( T  .(+)  U ) )  e.  Grp )
17 subgrcl 16075 . . 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 16584 . . . 4  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> T )
232subgss 16071 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
248, 23syl 16 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
2522, 24fssd 5726 . . 3  |-  ( ph  ->  ( T P U ) : ( T 
.(+)  U ) --> ( Base `  G ) )
264subgbas 16074 . . . . 5  |-  ( ( T  .(+)  U )  e.  (SubGrp `  G )  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2714, 26syl 16 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Gs  ( T  .(+)  U ) ) ) )
2827feq2d 5704 . . 3  |-  ( ph  ->  ( ( T P U ) : ( T  .(+)  U ) --> ( Base `  G )  <->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> ( Base `  G ) ) )
2925, 28mpbid 210 . 2  |-  ( ph  ->  ( T P U ) : ( Base `  ( Gs  ( T  .(+)  U ) ) ) --> (
Base `  G )
)
3027eleq2d 2511 . . . . 5  |-  ( ph  ->  ( x  e.  ( T  .(+)  U )  <->  x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3127eleq2d 2511 . . . . 5  |-  ( ph  ->  ( y  e.  ( T  .(+)  U )  <->  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )
3230, 31anbi12d 710 . . . 4  |-  ( ph  ->  ( ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  <->  ( x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) )  /\  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) ) )
3332biimpar 485 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) )  /\  y  e.  ( Base `  ( Gs  ( T  .(+)  U ) ) ) ) )  ->  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )
345, 11, 19, 12, 8, 9, 20, 10, 21pj1id 16586 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( T  .(+)  U ) )  ->  x  =  ( ( ( T P U ) `  x )  .+  (
( U P T ) `  x ) ) )
3534adantrr 716 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  x  =  ( ( ( T P U ) `
 x )  .+  ( ( U P T ) `  x
) ) )
365, 11, 19, 12, 8, 9, 20, 10, 21pj1id 16586 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( T  .(+)  U ) )  ->  y  =  ( ( ( T P U ) `  y )  .+  (
( U P T ) `  y ) ) )
3736adantrl 715 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  y  =  ( ( ( T P U ) `
 y )  .+  ( ( U P T ) `  y
) ) )
3835, 37oveq12d 6295 . . . . . 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 ) ) ) )
398adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  e.  (SubGrp `  G )
)
40 grpmnd 15931 . . . . . . . 8  |-  ( G  e.  Grp  ->  G  e.  Mnd )
4139, 17, 403syl 20 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  G  e.  Mnd )
4239, 23syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  C_  ( Base `  G
) )
43 simpl 457 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  x  e.  ( T  .(+)  U ) )
44 ffvelrn 6010 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  x
)  e.  T )
4522, 43, 44syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  T )
4642, 45sseldd 3487 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  x )  e.  ( Base `  G
) )
47 simpr 461 . . . . . . . . 9  |-  ( ( x  e.  ( T 
.(+)  U )  /\  y  e.  ( T  .(+)  U ) )  ->  y  e.  ( T  .(+)  U ) )
48 ffvelrn 6010 . . . . . . . . 9  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  y
)  e.  T )
4922, 47, 48syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  T )
5042, 49sseldd 3487 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Base `  G
) )
519adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  U  e.  (SubGrp `  G )
)
522subgss 16071 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
5351, 52syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  U  C_  ( Base `  G
) )
545, 11, 19, 12, 8, 9, 20, 10, 21pj2f 16585 . . . . . . . . 9  |-  ( ph  ->  ( U P T ) : ( T 
.(+)  U ) --> U )
55 ffvelrn 6010 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  x  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  x
)  e.  U )
5654, 43, 55syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  U )
5753, 56sseldd 3487 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  x )  e.  ( Base `  G
) )
58 ffvelrn 6010 . . . . . . . . 9  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  y
)  e.  U )
5954, 47, 58syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  U )
6053, 59sseldd 3487 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( U P T ) `  y )  e.  ( Base `  G
) )
6110adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  T  C_  ( Z `  U
) )
6261, 49sseldd 3487 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  y )  e.  ( Z `  U ) )
635, 12cntzi 16236 . . . . . . . 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 ) ) )
6462, 56, 63syl2anc 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 ) ) )
652, 5, 41, 46, 50, 57, 60, 64mnd4g 15806 . . . . . 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 ) ) ) )
6638, 65eqtr4d 2485 . . . . 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 ) ) ) )
6720adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  ( T  i^i  U )  =  {  .0.  } )
685subgcl 16080 . . . . . . . 8  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
)  ->  ( x  .+  y )  e.  ( T  .(+)  U )
)
69683expb 1196 . . . . . . 7  |-  ( ( ( T  .(+)  U )  e.  (SubGrp `  G
)  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
7014, 69sylan 471 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
x  .+  y )  e.  ( T  .(+)  U ) )
715subgcl 16080 . . . . . . 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 )
7239, 45, 49, 71syl3anc 1227 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( T P U ) `  x
)  .+  ( ( T P U ) `  y ) )  e.  T )
735subgcl 16080 . . . . . . 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 )
7451, 56, 59, 73syl3anc 1227 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( ( U P T ) `  x
)  .+  ( ( U P T ) `  y ) )  e.  U )
755, 11, 19, 12, 39, 51, 67, 61, 21, 70, 72, 74pj1eq 16587 . . . . 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
) ) ) ) )
7666, 75mpbid 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 ) ) ) )
7776simpld 459 . . 3  |-  ( (
ph  /\  ( x  e.  ( T  .(+)  U )  /\  y  e.  ( T  .(+)  U )
) )  ->  (
( T P U ) `  ( x 
.+  y ) )  =  ( ( ( T P U ) `
 x )  .+  ( ( T P U ) `  y
) ) )
7833, 77syldan 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 ) ) )
791, 2, 7, 5, 16, 18, 29, 78isghmd 16145 1  |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U ) )  GrpHom  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093    i^i cin 3457    C_ wss 3458   {csn 4010   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   +g cplusg 14569   0gc0g 14709   Mndcmnd 15788   Grpcgrp 15922  SubGrpcsubg 16064    GrpHom cghm 16133  Cntzccntz 16222   LSSumclsm 16523   proj1cpj1 16524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-grp 15926  df-minusg 15927  df-sbg 15928  df-subg 16067  df-ghm 16134  df-cntz 16224  df-lsm 16525  df-pj1 16526
This theorem is referenced by:  pj1ghm2  16591  dpjghm  16980  pj1lmhm  17614
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