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

Theorem resghm2b 15777
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 15761 . . 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 15761 . . 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 15715 . . . . . 6  |-  ( X  e.  (SubGrp `  T
)  ->  X  e.  (SubMnd `  T ) )
6 resghm2.u . . . . . . 7  |-  U  =  ( Ts  X )
76resmhm2b 15501 . . . . . 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 15698 . . . . . . 7  |-  ( X  e.  (SubGrp `  T
)  ->  T  e.  Grp )
1110adantr 465 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  ->  T  e.  Grp )
12 ghmmhmb 15770 . . . . . 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 2510 . . . 4  |-  ( ( S  e.  Grp  /\  ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X ) )  ->  ( F  e.  ( S  GrpHom  T )  <-> 
F  e.  ( S MndHom  T ) ) )
156subggrp 15696 . . . . . . 7  |-  ( X  e.  (SubGrp `  T
)  ->  U  e.  Grp )
1615adantr 465 . . . . . 6  |-  ( ( X  e.  (SubGrp `  T )  /\  ran  F 
C_  X )  ->  U  e.  Grp )
17 ghmmhmb 15770 . . . . . 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 2510 . . . 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 1369    e. wcel 1756    C_ wss 3340   ran crn 4853   ` cfv 5430  (class class class)co 6103   ↾s cress 14187   Grpcgrp 15422   MndHom cmhm 15474  SubMndcsubmnd 15475  SubGrpcsubg 15687    GrpHom cghm 15756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-0g 14392  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-subg 15690  df-ghm 15757
This theorem is referenced by:  cayley  15931  pj1ghm2  16213  dpjghm2  16575  reslmhm2b  17147
  Copyright terms: Public domain W3C validator