Theorem resrhm2b 13745

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 13332 analog. (Contributed by SN, 7-Feb-2025.)

Hypothesis

Ref	Expression
resrhm2b.u	|- U = ( T ↾s X )

Assertion

Ref	Expression
resrhm2b	|- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( S RingHom T ) <-> F e. ( S RingHom U ) ) )

Proof of Theorem resrhm2b

Step	Hyp	Ref	Expression
1		subrgsubg 13723	. . . . . 6 |- ( X e. (SubRing ` T ) -> X e. (SubGrp ` T ) )
2		resrhm2b.u	. . . . . . 7 |- U = ( T ↾s X )
3	2	resghm2b 13332	. . . . . 6 |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )
4	1, 3	sylan 283	. . . . 5 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )
5		eqid 2193	. . . . . . . 8 |- (mulGrp ` T ) = (mulGrp ` T )
6	5	subrgsubm 13730	. . . . . . 7 |- ( X e. (SubRing ` T ) -> X e. (SubMnd ` (mulGrp ` T ) ) )
7		eqid 2193	. . . . . . . 8 |- ( (mulGrp ` T ) ↾s X ) = ( (mulGrp ` T ) ↾s X )
8	7	resmhm2b 13061	. . . . . . 7 |- ( ( X e. (SubMnd ` (mulGrp ` T ) ) /\ ran F C_ X ) -> ( F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) <-> F e. ( (mulGrp ` S ) MndHom ( (mulGrp ` T ) ↾s X ) ) ) )
9	6, 8	sylan 283	. . . . . 6 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) <-> F e. ( (mulGrp ` S ) MndHom ( (mulGrp ` T ) ↾s X ) ) ) )
10		subrgrcl 13722	. . . . . . . . . 10 |- ( X e. (SubRing ` T ) -> T e. Ring )
11	2, 5	mgpress 13427	. . . . . . . . . 10 |- ( ( T e. Ring /\ X e. (SubRing ` T ) ) -> ( (mulGrp ` T ) ↾s X ) = (mulGrp ` U ) )
12	10, 11	mpancom 422	. . . . . . . . 9 |- ( X e. (SubRing ` T ) -> ( (mulGrp ` T ) ↾s X ) = (mulGrp ` U ) )
13	12	adantr 276	. . . . . . . 8 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( (mulGrp ` T ) ↾s X ) = (mulGrp ` U ) )
14	13	oveq2d 5934	. . . . . . 7 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( (mulGrp ` S ) MndHom ( (mulGrp ` T ) ↾s X ) ) = ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) )
15	14	eleq2d 2263	. . . . . 6 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( (mulGrp ` S ) MndHom ( (mulGrp ` T ) ↾s X ) ) <-> F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) )
16	9, 15	bitrd 188	. . . . 5 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) <-> F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) )
17	4, 16	anbi12d 473	. . . 4 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( ( F e. ( S GrpHom T ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) <-> ( F e. ( S GrpHom U ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) ) )
18	17	anbi2d 464	. . 3 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( ( S e. Ring /\ ( F e. ( S GrpHom T ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) ) <-> ( S e. Ring /\ ( F e. ( S GrpHom U ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) ) ) )
19	10	adantr 276	. . . . 5 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> T e. Ring )
20	19	biantrud 304	. . . 4 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( S e. Ring <-> ( S e. Ring /\ T e. Ring ) ) )
21	20	anbi1d 465	. . 3 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( ( S e. Ring /\ ( F e. ( S GrpHom T ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) ) <-> ( ( S e. Ring /\ T e. Ring ) /\ ( F e. ( S GrpHom T ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) ) ) )
22	2	subrgring 13720	. . . . . 6 |- ( X e. (SubRing ` T ) -> U e. Ring )
23	22	adantr 276	. . . . 5 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> U e. Ring )
24	23	biantrud 304	. . . 4 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( S e. Ring <-> ( S e. Ring /\ U e. Ring ) ) )
25	24	anbi1d 465	. . 3 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( ( S e. Ring /\ ( F e. ( S GrpHom U ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) ) <-> ( ( S e. Ring /\ U e. Ring ) /\ ( F e. ( S GrpHom U ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) ) ) )
26	18, 21, 25	3bitr3d 218	. 2 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( ( ( S e. Ring /\ T e. Ring ) /\ ( F e. ( S GrpHom T ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) ) <-> ( ( S e. Ring /\ U e. Ring ) /\ ( F e. ( S GrpHom U ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) ) ) )
27		eqid 2193	. . 3 |- (mulGrp ` S ) = (mulGrp ` S )
28	27, 5	isrhm 13654	. 2 |- ( F e. ( S RingHom T ) <-> ( ( S e. Ring /\ T e. Ring ) /\ ( F e. ( S GrpHom T ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` T ) ) ) ) )
29		eqid 2193	. . 3 |- (mulGrp ` U ) = (mulGrp ` U )
30	27, 29	isrhm 13654	. 2 |- ( F e. ( S RingHom U ) <-> ( ( S e. Ring /\ U e. Ring ) /\ ( F e. ( S GrpHom U ) /\ F e. ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) ) ) )
31	26, 28, 30	3bitr4g 223	1 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( S RingHom T ) <-> F e. ( S RingHom U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   C_ wss 3153   ran crn 4660   ` cfv 5254  (class class class)co 5918   ↾_s cress 12619   MndHom cmhm 13029  SubMndcsubmnd 13030  SubGrpcsubg 13237   GrpHom cghm 13310  mulGrpcmgp 13416   Ringcrg 13492   RingHom crh 13646  SubRingcsubrg 13713

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mhm 13031  df-submnd 13032  df-grp 13075  df-minusg 13076  df-subg 13240  df-ghm 13311  df-mgp 13417  df-ur 13456  df-ring 13494  df-rhm 13648  df-subrg 13715

This theorem is referenced by: (None)