Theorem resrhm2b 13805

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 13392 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 13783	. . . . . 6 |- ( X e. (SubRing ` T ) -> X e. (SubGrp ` T ) )
2		resrhm2b.u	. . . . . . 7 |- U = ( T ↾s X )
3	2	resghm2b 13392	. . . . . 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 2196	. . . . . . . 8 |- (mulGrp ` T ) = (mulGrp ` T )
6	5	subrgsubm 13790	. . . . . . 7 |- ( X e. (SubRing ` T ) -> X e. (SubMnd ` (mulGrp ` T ) ) )
7		eqid 2196	. . . . . . . 8 |- ( (mulGrp ` T ) ↾s X ) = ( (mulGrp ` T ) ↾s X )
8	7	resmhm2b 13121	. . . . . . 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 13782	. . . . . . . . . 10 |- ( X e. (SubRing ` T ) -> T e. Ring )
11	2, 5	mgpress 13487	. . . . . . . . . 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 5938	. . . . . . 7 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( (mulGrp ` S ) MndHom ( (mulGrp ` T ) ↾s X ) ) = ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) )
15	14	eleq2d 2266	. . . . . 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 13780	. . . . . 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 2196	. . 3 |- (mulGrp ` S ) = (mulGrp ` S )
28	27, 5	isrhm 13714	. 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 2196	. . 3 |- (mulGrp ` U ) = (mulGrp ` U )
30	27, 29	isrhm 13714	. 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 2167   C_ wss 3157   ran crn 4664   ` cfv 5258  (class class class)co 5922   ↾_s cress 12679   MndHom cmhm 13089  SubMndcsubmnd 13090  SubGrpcsubg 13297   GrpHom cghm 13370  mulGrpcmgp 13476   Ringcrg 13552   RingHom crh 13706  SubRingcsubrg 13773

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-submnd 13092  df-grp 13135  df-minusg 13136  df-subg 13300  df-ghm 13371  df-mgp 13477  df-ur 13516  df-ring 13554  df-rhm 13708  df-subrg 13775

This theorem is referenced by: (None)