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Theorem resrhm 17009
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 16925 . . 3  |-  ( F  e.  ( S RingHom  T
)  ->  T  e.  Ring )
2 resrhm.u . . . 4  |-  U  =  ( Ss  X )
32subrgrng 16983 . . 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 16930 . . . 4  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
6 subrgsubg 16986 . . . 4  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubGrp `  S ) )
72resghm 15874 . . . 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 2451 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
10 eqid 2451 . . . . . 6  |-  (mulGrp `  T )  =  (mulGrp `  T )
119, 10rhmmhm 16927 . . . . 5  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
129subrgsubm 16993 . . . . 5  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubMnd `  (mulGrp `  S
) ) )
13 eqid 2451 . . . . . 6  |-  ( (mulGrp `  S )s  X )  =  ( (mulGrp `  S )s  X
)
1413resmhm 15598 . . . . 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 16924 . . . . . 6  |-  ( F  e.  ( S RingHom  T
)  ->  S  e.  Ring )
172, 9mgpress 16716 . . . . . 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 6208 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (
(mulGrp `  S )s  X
) MndHom  (mulGrp `  T )
)  =  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
2015, 19eleqtrd 2541 . . 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 2451 . . 3  |-  (mulGrp `  U )  =  (mulGrp `  U )
2322, 10isrhm 16926 . 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 1370    e. wcel 1758    |` cres 4943   ` cfv 5519  (class class class)co 6193   ↾s cress 14286   MndHom cmhm 15573  SubMndcsubmnd 15574  SubGrpcsubg 15786    GrpHom cghm 15855  mulGrpcmgp 16705   Ringcrg 16760   RingHom crh 16919  SubRingcsubrg 16976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-0g 14491  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-subg 15789  df-ghm 15856  df-mgp 16706  df-ur 16718  df-rng 16762  df-rnghom 16921  df-subrg 16978
This theorem is referenced by:  evlsval2  17722
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