Theorem rhmco 14051

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 14033	. . 3 |- ( F e. ( T RingHom U ) -> U e. Ring )
2		rhmrcl1 14032	. . 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 14039	. . . 4 |- ( F e. ( T RingHom U ) -> F e. ( T GrpHom U ) )
5		rhmghm 14039	. . . 4 |- ( G e. ( S RingHom T ) -> G e. ( S GrpHom T ) )
6		ghmco 13715	. . . 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 2207	. . . . 5 |- (mulGrp ` T ) = (mulGrp ` T )
9		eqid 2207	. . . . 5 |- (mulGrp ` U ) = (mulGrp ` U )
10	8, 9	rhmmhm 14036	. . . 4 |- ( F e. ( T RingHom U ) -> F e. ( (mulGrp ` T ) MndHom (mulGrp ` U ) ) )
11		eqid 2207	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
12	11, 8	rhmmhm 14036	. . . 4 |- ( G e. ( S RingHom T ) -> G e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) )
13		mhmco 13437	. . . 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 14035	. 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 2178   o. ccom 4697   ` cfv 5290  (class class class)co 5967   MndHom cmhm 13404   GrpHom cghm 13691  mulGrpcmgp 13797   Ringcrg 13873   RingHom crh 14027

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-mhm 13406  df-grp 13450  df-ghm 13692  df-mgp 13798  df-ur 13837  df-ring 13875  df-rhm 14029

This theorem is referenced by: (None)