Theorem rhmeql 14481

Description: The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
rhmeql	|- ( ( F e. ( S RingHom T ) /\ G e. ( S RingHom T ) ) -> dom ( F i^i G ) e. (SubRing ` S ) )

Proof of Theorem rhmeql

Step	Hyp	Ref	Expression
1		rhmghm 14392	. . 3 |- ( F e. ( S RingHom T ) -> F e. ( S GrpHom T ) )
2		rhmghm 14392	. . 3 |- ( G e. ( S RingHom T ) -> G e. ( S GrpHom T ) )
3		ghmeql 14068	. . 3 |- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> dom ( F i^i G ) e. (SubGrp ` S ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( F e. ( S RingHom T ) /\ G e. ( S RingHom T ) ) -> dom ( F i^i G ) e. (SubGrp ` S ) )
5		eqid 2234	. . . 4 |- (mulGrp ` S ) = (mulGrp ` S )
6		eqid 2234	. . . 4 |- (mulGrp ` T ) = (mulGrp ` T )
7	5, 6	rhmmhm 14389	. . 3 |- ( F e. ( S RingHom T ) -> F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
8	5, 6	rhmmhm 14389	. . 3 |- ( G e. ( S RingHom T ) -> G e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
9		mhmeql 13789	. . 3 |- ( ( F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) /\ G e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) -> dom ( F i^i G ) e. (SubMnd ` (mulGrp ` S ) ) )
10	7, 8, 9	syl2an 289	. 2 |- ( ( F e. ( S RingHom T ) /\ G e. ( S RingHom T ) ) -> dom ( F i^i G ) e. (SubMnd ` (mulGrp ` S ) ) )
11		rhmrcl1 14385	. . . 4 |- ( F e. ( S RingHom T ) -> S e. Ring )
12	11	adantr 276	. . 3 |- ( ( F e. ( S RingHom T ) /\ G e. ( S RingHom T ) ) -> S e. Ring )
13	5	issubrg3 14478	. . 3 |- ( S e. Ring -> ( dom ( F i^i G ) e. (SubRing ` S ) <-> ( dom ( F i^i G ) e. (SubGrp ` S ) /\ dom ( F i^i G ) e. (SubMnd ` (mulGrp ` S ) ) ) ) )
14	12, 13	syl 14	. 2 |- ( ( F e. ( S RingHom T ) /\ G e. ( S RingHom T ) ) -> ( dom ( F i^i G ) e. (SubRing ` S ) <-> ( dom ( F i^i G ) e. (SubGrp ` S ) /\ dom ( F i^i G ) e. (SubMnd ` (mulGrp ` S ) ) ) ) )
15	4, 10, 14	mpbir2and 953	1 |- ( ( F e. ( S RingHom T ) /\ G e. ( S RingHom T ) ) -> dom ( F i^i G ) e. (SubRing ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2205   i^i cin 3213   dom cdm 4754   ` cfv 5357  (class class class)co 6058   MndHom cmhm 13754  SubMndcsubmnd 13755  SubGrpcsubg 13968   GrpHom cghm 14041  mulGrpcmgp 14148   Ringcrg 14224   RingHom crh 14380  SubRingcsubrg 14448

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 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-mhm 13756  df-submnd 13757  df-grp 13800  df-minusg 13801  df-subg 13971  df-ghm 14042  df-mgp 14149  df-ur 14188  df-ring 14226  df-rhm 14382  df-subrg 14450

This theorem is referenced by: (None)