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Theorem resrhm 17653
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 17564 . . 3  |-  ( F  e.  ( S RingHom  T
)  ->  T  e.  Ring )
2 resrhm.u . . . 4  |-  U  =  ( Ss  X )
32subrgring 17627 . . 3  |-  ( X  e.  (SubRing `  S
)  ->  U  e.  Ring )
41, 3anim12ci 565 . 2  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( U  e.  Ring  /\  T  e.  Ring ) )
5 rhmghm 17569 . . . 4  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
6 subrgsubg 17630 . . . 4  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubGrp `  S ) )
72resghm 16482 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
85, 6, 7syl2an 475 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
9 eqid 2454 . . . . . 6  |-  (mulGrp `  S )  =  (mulGrp `  S )
10 eqid 2454 . . . . . 6  |-  (mulGrp `  T )  =  (mulGrp `  T )
119, 10rhmmhm 17566 . . . . 5  |-  ( F  e.  ( S RingHom  T
)  ->  F  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
129subrgsubm 17637 . . . . 5  |-  ( X  e.  (SubRing `  S
)  ->  X  e.  (SubMnd `  (mulGrp `  S
) ) )
13 eqid 2454 . . . . . 6  |-  ( (mulGrp `  S )s  X )  =  ( (mulGrp `  S )s  X
)
1413resmhm 16189 . . . . 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 475 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( ( (mulGrp `  S
)s 
X ) MndHom  (mulGrp `  T
) ) )
16 rhmrcl1 17563 . . . . . 6  |-  ( F  e.  ( S RingHom  T
)  ->  S  e.  Ring )
172, 9mgpress 17347 . . . . . 6  |-  ( ( S  e.  Ring  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1816, 17sylan 469 . . . . 5  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (mulGrp `  S )s  X )  =  (mulGrp `  U ) )
1918oveq1d 6285 . . . 4  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( (
(mulGrp `  S )s  X
) MndHom  (mulGrp `  T )
)  =  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
2015, 19eleqtrd 2544 . . 3  |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S )
)  ->  ( F  |`  X )  e.  ( (mulGrp `  U ) MndHom  (mulGrp `  T ) ) )
218, 20jca 530 . 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 2454 . . 3  |-  (mulGrp `  U )  =  (mulGrp `  U )
2322, 10isrhm 17565 . 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 662 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 367    = wceq 1398    e. wcel 1823    |` cres 4990   ` cfv 5570  (class class class)co 6270   ↾s cress 14717   MndHom cmhm 16163  SubMndcsubmnd 16164  SubGrpcsubg 16394    GrpHom cghm 16463  mulGrpcmgp 17336   Ringcrg 17393   RingHom crh 17556  SubRingcsubrg 17620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-subg 16397  df-ghm 16464  df-mgp 17337  df-ur 17349  df-ring 17395  df-rnghom 17559  df-subrg 17622
This theorem is referenced by:  evlsval2  18384
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