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Theorem resrhm 18037
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 17948 . . 3  |-  ( F  e.  ( S RingHom  T
)  ->  T  e.  Ring )
2 resrhm.u . . . 4  |-  U  =  ( Ss  X )
32subrgring 18011 . . 3  |-  ( X  e.  (SubRing `  S
)  ->  U  e.  Ring )
41, 3anim12ci 571 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( U  e.  Ring  /\  T  e.  Ring ) )
5 rhmghm 17953 . . . 4  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
6 subrgsubg 18014 . . . 4  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubGrp `  S ) )
72resghm 16899 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
85, 6, 7syl2an 480 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
9 eqid 2451 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
10 eqid 2451 . . . . . 6  |-  (mulGrp `  T )  =  (mulGrp `  T )
119, 10rhmmhm 17950 . . . . 5  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
129subrgsubm 18021 . . . . 5  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubMnd `  (mulGrp `  S
) ) )
13 eqid 2451 . . . . . 6  |-  ( (mulGrp `  S )s  X )  =  ( (mulGrp `  S )s  X
)
1413resmhm 16606 . . . . 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 480 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( ( (mulGrp `  S
)s 
X ) MndHom  (mulGrp `  T
) ) )
16 rhmrcl1 17947 . . . . . 6  |-  ( F  e.  ( S RingHom  T
)  ->  S  e.  Ring )
172, 9mgpress 17734 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1816, 17sylan 474 . . . . 5  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1918oveq1d 6305 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (
(mulGrp `  S )s  X
) MndHom  (mulGrp `  T )
)  =  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
2015, 19eleqtrd 2531 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
218, 20jca 535 . 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 2451 . . 3  |-  (mulGrp `  U )  =  (mulGrp `  U )
2322, 10isrhm 17949 . 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 670 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 371    = wceq 1444    e. wcel 1887    |` cres 4836   ` cfv 5582  (class class class)co 6290   ↾s cress 15122   MndHom cmhm 16580  SubMndcsubmnd 16581  SubGrpcsubg 16811    GrpHom cghm 16880  mulGrpcmgp 17723   Ringcrg 17780   RingHom crh 17940  SubRingcsubrg 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-subg 16814  df-ghm 16881  df-mgp 17724  df-ur 17736  df-ring 17782  df-rnghom 17943  df-subrg 18006
This theorem is referenced by:  evlsval2  18743
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