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

Theorem rhmima 16895
Description: The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
rhmima  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubRing `  N ) )

Proof of Theorem rhmima
StepHypRef Expression
1 rhmghm 16814 . . 3  |-  ( F  e.  ( M RingHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
2 subrgsubg 16870 . . 3  |-  ( X  e.  (SubRing `  M
)  ->  X  e.  (SubGrp `  M ) )
3 ghmima 15766 . . 3  |-  ( ( F  e.  ( M 
GrpHom  N )  /\  X  e.  (SubGrp `  M )
)  ->  ( F " X )  e.  (SubGrp `  N ) )
41, 2, 3syl2an 477 . 2  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubGrp `  N ) )
5 eqid 2442 . . . 4  |-  (mulGrp `  M )  =  (mulGrp `  M )
6 eqid 2442 . . . 4  |-  (mulGrp `  N )  =  (mulGrp `  N )
75, 6rhmmhm 16811 . . 3  |-  ( F  e.  ( M RingHom  N
)  ->  F  e.  ( (mulGrp `  M ) MndHom  (mulGrp `  N ) ) )
85subrgsubm 16877 . . 3  |-  ( X  e.  (SubRing `  M
)  ->  X  e.  (SubMnd `  (mulGrp `  M
) ) )
9 mhmima 15490 . . 3  |-  ( ( F  e.  ( (mulGrp `  M ) MndHom  (mulGrp `  N ) )  /\  X  e.  (SubMnd `  (mulGrp `  M ) ) )  ->  ( F " X )  e.  (SubMnd `  (mulGrp `  N )
) )
107, 8, 9syl2an 477 . 2  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubMnd `  (mulGrp `  N )
) )
11 rhmrcl2 16809 . . . 4  |-  ( F  e.  ( M RingHom  N
)  ->  N  e.  Ring )
1211adantr 465 . . 3  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  N  e.  Ring )
136issubrg3 16892 . . 3  |-  ( N  e.  Ring  ->  ( ( F " X )  e.  (SubRing `  N
)  <->  ( ( F
" X )  e.  (SubGrp `  N )  /\  ( F " X
)  e.  (SubMnd `  (mulGrp `  N ) ) ) ) )
1412, 13syl 16 . 2  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( ( F " X )  e.  (SubRing `  N )  <->  ( ( F " X
)  e.  (SubGrp `  N )  /\  ( F " X )  e.  (SubMnd `  (mulGrp `  N
) ) ) ) )
154, 10, 14mpbir2and 913 1  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubRing `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   "cima 4842   ` cfv 5417  (class class class)co 6090   MndHom cmhm 15461  SubMndcsubmnd 15462  SubGrpcsubg 15674    GrpHom cghm 15743  mulGrpcmgp 16590   Ringcrg 16644   RingHom crh 16803  SubRingcsubrg 16860
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6831  df-rdg 6865  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-0g 14379  df-mnd 15414  df-mhm 15463  df-submnd 15464  df-grp 15544  df-minusg 15545  df-subg 15677  df-ghm 15744  df-mgp 16591  df-ur 16603  df-rng 16646  df-rnghom 16805  df-subrg 16862
This theorem is referenced by:  mpfsubrg  17617  pf1subrg  17781  plypf1  21679
  Copyright terms: Public domain W3C validator