Theorem rhmima 14501

Description: The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
rhmima	|- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubRing ` N ) )

Proof of Theorem rhmima

Step	Hyp	Ref	Expression
1		rhmghm 14411	. . 3 |- ( F e. ( M RingHom N ) -> F e. ( M GrpHom N ) )
2		subrgsubg 14477	. . 3 |- ( X e. (SubRing ` M ) -> X e. (SubGrp ` M ) )
3		ghmima 14022	. . 3 |- ( ( F e. ( M GrpHom N ) /\ X e. (SubGrp ` M ) ) -> ( F " X ) e. (SubGrp ` N ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubGrp ` N ) )
5		eqid 2234	. . . 4 |- (mulGrp ` M ) = (mulGrp ` M )
6		eqid 2234	. . . 4 |- (mulGrp ` N ) = (mulGrp ` N )
7	5, 6	rhmmhm 14408	. . 3 |- ( F e. ( M RingHom N ) -> F e. ( (mulGrp ` M ) MndHom (mulGrp ` N ) ) )
8	5	subrgsubm 14484	. . 3 |- ( X e. (SubRing ` M ) -> X e. (SubMnd ` (mulGrp ` M ) ) )
9		mhmima 13750	. . 3 |- ( ( F e. ( (mulGrp ` M ) MndHom (mulGrp ` N ) ) /\ X e. (SubMnd ` (mulGrp ` M ) ) ) -> ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) )
10	7, 8, 9	syl2an 289	. 2 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) )
11		rhmrcl2 14405	. . . 4 |- ( F e. ( M RingHom N ) -> N e. Ring )
12	11	adantr 276	. . 3 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> N e. Ring )
13	6	issubrg3 14497	. . 3 |- ( N e. Ring -> ( ( F " X ) e. (SubRing ` N ) <-> ( ( F " X ) e. (SubGrp ` N ) /\ ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) ) ) )
14	12, 13	syl 14	. 2 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( ( F " X ) e. (SubRing ` N ) <-> ( ( F " X ) e. (SubGrp ` N ) /\ ( F " X ) e. (SubMnd ` (mulGrp ` N ) ) ) ) )
15	4, 10, 14	mpbir2and 953	1 |- ( ( F e. ( M RingHom N ) /\ X e. (SubRing ` M ) ) -> ( F " X ) e. (SubRing ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2205   "cima 4757   ` cfv 5357  (class class class)co 6058   MndHom cmhm 13716  SubMndcsubmnd 13717  SubGrpcsubg 13924   GrpHom cghm 13997  mulGrpcmgp 14163   Ringcrg 14243   RingHom crh 14399  SubRingcsubrg 14467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-3 9317  df-ndx 13303  df-slot 13304  df-base 13306  df-sets 13307  df-iress 13308  df-plusg 13391  df-mulr 13392  df-0g 13559  df-mgm 13623  df-sgrp 13669  df-mnd 13682  df-mhm 13718  df-submnd 13719  df-grp 13762  df-minusg 13763  df-subg 13927  df-ghm 13998  df-mgp 14164  df-ur 14207  df-ring 14245  df-rhm 14401  df-subrg 14469

This theorem is referenced by:  rnrhmsubrg  14502