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Theorem resrhm 16872
Description: Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
Hypothesis
Ref Expression
resrhm.u  |-  U  =  ( Ss  X )
Assertion
Ref Expression
resrhm  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( U RingHom  T ) )

Proof of Theorem resrhm
StepHypRef Expression
1 rhmrcl2 16798 . . 3  |-  ( F  e.  ( S RingHom  T
)  ->  T  e.  Ring )
2 resrhm.u . . . 4  |-  U  =  ( Ss  X )
32subrgrng 16846 . . 3  |-  ( X  e.  (SubRing `  S
)  ->  U  e.  Ring )
41, 3anim12ci 567 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( U  e.  Ring  /\  T  e.  Ring ) )
5 rhmghm 16801 . . . 4  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
6 subrgsubg 16849 . . . 4  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubGrp `  S ) )
72resghm 15754 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
85, 6, 7syl2an 477 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
9 eqid 2438 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
10 eqid 2438 . . . . . 6  |-  (mulGrp `  T )  =  (mulGrp `  T )
119, 10rhmmhm 16800 . . . . 5  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
129subrgsubm 16856 . . . . 5  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubMnd `  (mulGrp `  S
) ) )
13 eqid 2438 . . . . . 6  |-  ( (mulGrp `  S )s  X )  =  ( (mulGrp `  S )s  X
)
1413resmhm 15478 . . . . 5  |-  ( ( F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) )  /\  X  e.  (SubMnd `  (mulGrp `  S ) ) )  ->  ( F  |`  X )  e.  ( ( (mulGrp `  S
)s 
X ) MndHom  (mulGrp `  T
) ) )
1511, 12, 14syl2an 477 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( ( (mulGrp `  S
)s 
X ) MndHom  (mulGrp `  T
) ) )
16 rhmrcl1 16797 . . . . . 6  |-  ( F  e.  ( S RingHom  T
)  ->  S  e.  Ring )
172, 9mgpress 16590 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1816, 17sylan 471 . . . . 5  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1918oveq1d 6101 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (
(mulGrp `  S )s  X
) MndHom  (mulGrp `  T )
)  =  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
2015, 19eleqtrd 2514 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
218, 20jca 532 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( ( F  |`  X )  e.  ( U  GrpHom  T )  /\  ( F  |`  X )  e.  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) ) )
22 eqid 2438 . . 3  |-  (mulGrp `  U )  =  (mulGrp `  U )
2322, 10isrhm 16799 . 2  |-  ( ( F  |`  X )  e.  ( U RingHom  T )  <->  ( ( U  e.  Ring  /\  T  e.  Ring )  /\  ( ( F  |`  X )  e.  ( U  GrpHom  T )  /\  ( F  |`  X )  e.  ( (mulGrp `  U ) MndHom  (mulGrp `  T
) ) ) ) )
244, 21, 23sylanbrc 664 1  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( U RingHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    |` cres 4837   ` cfv 5413  (class class class)co 6086   ↾s cress 14167   MndHom cmhm 15454  SubMndcsubmnd 15455  SubGrpcsubg 15666    GrpHom cghm 15735  mulGrpcmgp 16579   Ringcrg 16633   RingHom crh 16792  SubRingcsubrg 16839
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-subg 15669  df-ghm 15736  df-mgp 16580  df-ur 16592  df-rng 16635  df-rnghom 16794  df-subrg 16841
This theorem is referenced by:  evlsval2  17581
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