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Theorem resrhm 15667
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 15593 . . 3  |-  ( F  e.  ( S RingHom  T
)  ->  T  e.  Ring )
2 resrhm.u . . . 4  |-  U  =  ( Ss  X )
32subrgrng 15641 . . 3  |-  ( X  e.  (SubRing `  S
)  ->  U  e.  Ring )
41, 3anim12ci 550 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( U  e.  Ring  /\  T  e.  Ring ) )
5 rhmghm 15596 . . . 4  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
6 subrgsubg 15644 . . . 4  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubGrp `  S ) )
72resghm 14792 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
85, 6, 7syl2an 463 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
9 eqid 2358 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
10 eqid 2358 . . . . . 6  |-  (mulGrp `  T )  =  (mulGrp `  T )
119, 10rhmmhm 15595 . . . . 5  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
129subrgsubm 15651 . . . . 5  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubMnd `  (mulGrp `  S
) ) )
13 eqid 2358 . . . . . 6  |-  ( (mulGrp `  S )s  X )  =  ( (mulGrp `  S )s  X
)
1413resmhm 14529 . . . . 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 463 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( ( (mulGrp `  S
)s 
X ) MndHom  (mulGrp `  T
) ) )
16 rhmrcl1 15592 . . . . . 6  |-  ( F  e.  ( S RingHom  T
)  ->  S  e.  Ring )
172, 9mgpress 15429 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1816, 17sylan 457 . . . . 5  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1918oveq1d 5957 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (
(mulGrp `  S )s  X
) MndHom  (mulGrp `  T )
)  =  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
2015, 19eleqtrd 2434 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
218, 20jca 518 . 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 2358 . . 3  |-  (mulGrp `  U )  =  (mulGrp `  U )
2322, 10isrhm 15594 . 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 645 1  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( U RingHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    |` cres 4770   ` cfv 5334  (class class class)co 5942   ↾s cress 13240   MndHom cmhm 14506  SubMndcsubmnd 14507  SubGrpcsubg 14708    GrpHom cghm 14773  mulGrpcmgp 15418   Ringcrg 15430   RingHom crh 15587  SubRingcsubrg 15634
This theorem is referenced by:  evlsval2  19502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-0g 13497  df-mnd 14460  df-mhm 14508  df-submnd 14509  df-grp 14582  df-subg 14711  df-ghm 14774  df-mgp 15419  df-rng 15433  df-ur 15435  df-rnghom 15589  df-subrg 15636
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