Theorem rhmco 14481

Description: The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)

Assertion

Ref	Expression
rhmco	|- ( ( F e. ( T RingHom U ) /\ G e. ( S RingHom T ) ) -> ( F o. G ) e. ( S RingHom U ) )

Proof of Theorem rhmco

Step	Hyp	Ref	Expression
1		rhmrcl2 14463	. . 3 |- ( F e. ( T RingHom U ) -> U e. Ring )
2		rhmrcl1 14462	. . 3 |- ( G e. ( S RingHom T ) -> S e. Ring )
3	1, 2	anim12ci 339	. 2 |- ( ( F e. ( T RingHom U ) /\ G e. ( S RingHom T ) ) -> ( S e. Ring /\ U e. Ring ) )
4		rhmghm 14469	. . . 4 |- ( F e. ( T RingHom U ) -> F e. ( T GrpHom U ) )
5		rhmghm 14469	. . . 4 |- ( G e. ( S RingHom T ) -> G e. ( S GrpHom T ) )
6		ghmco 14067	. . . 4 |- ( ( F e. ( T GrpHom U ) /\ G e. ( S GrpHom T ) ) -> ( F o. G ) e. ( S GrpHom U ) )
7	4, 5, 6	syl2an 289	. . 3 |- ( ( F e. ( T RingHom U ) /\ G e. ( S RingHom T ) ) -> ( F o. G ) e. ( S GrpHom U ) )
8		eqid 2238	. . . . 5 |- (mulGrp ` T ) = (mulGrp ` T )
9		eqid 2238	. . . . 5 |- (mulGrp ` U ) = (mulGrp ` U )
10	8, 9	rhmmhm 14466	. . . 4 |- ( F e. ( T RingHom U ) -> F e. ( (mulGrp ` T ) MndHom (mulGrp ` U ) ) )
11		eqid 2238	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
12	11, 8	rhmmhm 14466	. . . 4 |- ( G e. ( S RingHom T ) -> G e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
13		mhmco 13797	. . . 4 |- ( ( F e. ( (mulGrp ` T ) MndHom (mulGrp ` U ) ) /\ G e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) -> ( F o. G ) e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) )
14	10, 12, 13	syl2an 289	. . 3 |- ( ( F e. ( T RingHom U ) /\ G e. ( S RingHom T ) ) -> ( F o. G ) e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) )
15	7, 14	jca 306	. 2 |- ( ( F e. ( T RingHom U ) /\ G e. ( S RingHom T ) ) -> ( ( F o. G ) e. ( S GrpHom U ) /\ ( F o. G ) e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) )
16	11, 9	isrhm 14465	. 2 |- ( ( F o. G ) e. ( S RingHom U ) <-> ( ( S e. Ring /\ U e. Ring ) /\ ( ( F o. G ) e. ( S GrpHom U ) /\ ( F o. G ) e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) ) )
17	3, 15, 16	sylanbrc 421	1 |- ( ( F e. ( T RingHom U ) /\ G e. ( S RingHom T ) ) -> ( F o. G ) e. ( S RingHom U ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   o. ccom 4778   ` cfv 5377  (class class class)co 6085   MndHom cmhm 13764   GrpHom cghm 14043  mulGrpcmgp 14217   Ringcrg 14300   RingHom crh 14457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-mhm 13766  df-grp 13808  df-ghm 14044  df-mgp 14218  df-ur 14263  df-ring 14302  df-rhm 14459

This theorem is used by: (None)