Theorem rhmco 14138

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 14120	. . 3 |- ( F e. ( T RingHom U ) -> U e. Ring )
2		rhmrcl1 14119	. . 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 14126	. . . 4 |- ( F e. ( T RingHom U ) -> F e. ( T GrpHom U ) )
5		rhmghm 14126	. . . 4 |- ( G e. ( S RingHom T ) -> G e. ( S GrpHom T ) )
6		ghmco 13801	. . . 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 2229	. . . . 5 |- (mulGrp ` T ) = (mulGrp ` T )
9		eqid 2229	. . . . 5 |- (mulGrp ` U ) = (mulGrp ` U )
10	8, 9	rhmmhm 14123	. . . 4 |- ( F e. ( T RingHom U ) -> F e. ( (mulGrp ` T ) MndHom (mulGrp ` U ) ) )
11		eqid 2229	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
12	11, 8	rhmmhm 14123	. . . 4 |- ( G e. ( S RingHom T ) -> G e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
13		mhmco 13523	. . . 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 14122	. 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 417	1 |- ( ( F e. ( T RingHom U ) /\ G e. ( S RingHom T ) ) -> ( F o. G ) e. ( S RingHom U ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   o. ccom 4723   ` cfv 5318  (class class class)co 6001   MndHom cmhm 13490   GrpHom cghm 13777  mulGrpcmgp 13883   Ringcrg 13959   RingHom crh 14114

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-mhm 13492  df-grp 13536  df-ghm 13778  df-mgp 13884  df-ur 13923  df-ring 13961  df-rhm 14116

This theorem is referenced by: (None)