MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resghm2b Structured version   Unicode version

Theorem resghm2b 16157
Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resghm2.u  |-  U  =  ( Ts  X )
Assertion
Ref Expression
resghm2b  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )

Proof of Theorem resghm2b
StepHypRef Expression
1 ghmgrp1 16141 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
21a1i 11 . 2  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
)
3 ghmgrp1 16141 . . 3  |-  ( F  e.  ( S  GrpHom  U )  ->  S  e.  Grp )
43a1i 11 . 2  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  U )  ->  S  e.  Grp )
)
5 subgsubm 16095 . . . . . 6  |-  ( X  e.  (SubGrp `  T
)  ->  X  e.  (SubMnd `  T ) )
6 resghm2.u . . . . . . 7  |-  U  =  ( Ts  X )
76resmhm2b 15864 . . . . . 6  |-  ( ( X  e.  (SubMnd `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )
85, 7sylan 471 . . . . 5  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )
98adantl 466 . . . 4  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S MndHom  T )  <-> 
F  e.  ( S MndHom  U ) ) )
10 subgrcl 16078 . . . . . . 7  |-  ( X  e.  (SubGrp `  T
)  ->  T  e.  Grp )
1110adantr 465 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  ->  T  e.  Grp )
12 ghmmhmb 16150 . . . . . 6  |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T
) )
1311, 12sylan2 474 . . . . 5  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( S  GrpHom  T )  =  ( S MndHom  T ) )
1413eleq2d 2537 . . . 4  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S  GrpHom  T )  <-> 
F  e.  ( S MndHom  T ) ) )
156subggrp 16076 . . . . . . 7  |-  ( X  e.  (SubGrp `  T
)  ->  U  e.  Grp )
1615adantr 465 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  ->  U  e.  Grp )
17 ghmmhmb 16150 . . . . . 6  |-  ( ( S  e.  Grp  /\  U  e.  Grp )  ->  ( S  GrpHom  U )  =  ( S MndHom  U
) )
1816, 17sylan2 474 . . . . 5  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( S  GrpHom  U )  =  ( S MndHom  U ) )
1918eleq2d 2537 . . . 4  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S  GrpHom  U )  <-> 
F  e.  ( S MndHom  U ) ) )
209, 14, 193bitr4d 285 . . 3  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S  GrpHom  T )  <-> 
F  e.  ( S 
GrpHom  U ) ) )
2120expcom 435 . 2  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( S  e.  Grp  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) ) )
222, 4, 21pm5.21ndd 354 1  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  -> 
( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   ran crn 5006   ` cfv 5594  (class class class)co 6295   ↾s cress 14508   MndHom cmhm 15837  SubMndcsubmnd 15838   Grpcgrp 15925  SubGrpcsubg 16067    GrpHom cghm 16136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-subg 16070  df-ghm 16137
This theorem is referenced by:  cayley  16311  pj1ghm2  16595  dpjghm2  16985  reslmhm2b  17571  m2cpmghm  19114
  Copyright terms: Public domain W3C validator