Theorem resrhm2b 14269

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 13854 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 14247	. . . . . 6 |- ( X e. (SubRing ` T ) -> X e. (SubGrp ` T ) )
2		resrhm2b.u	. . . . . . 7 |- U = ( T ↾s X )
3	2	resghm2b 13854	. . . . . 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 2231	. . . . . . . 8 |- (mulGrp ` T ) = (mulGrp ` T )
6	5	subrgsubm 14254	. . . . . . 7 |- ( X e. (SubRing ` T ) -> X e. (SubMnd ` (mulGrp ` T ) ) )
7		eqid 2231	. . . . . . . 8 |- ( (mulGrp ` T ) ↾s X ) = ( (mulGrp ` T ) ↾s X )
8	7	resmhm2b 13577	. . . . . . 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 14246	. . . . . . . . . 10 |- ( X e. (SubRing ` T ) -> T e. Ring )
11	2, 5	mgpress 13950	. . . . . . . . . 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 6034	. . . . . . 7 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( (mulGrp ` S ) MndHom ( (mulGrp ` T ) ↾s X ) ) = ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) )
15	14	eleq2d 2301	. . . . . 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 14244	. . . . . 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 2231	. . 3 |- (mulGrp ` S ) = (mulGrp ` S )
28	27, 5	isrhm 14178	. 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 2231	. . 3 |- (mulGrp ` U ) = (mulGrp ` U )
30	27, 29	isrhm 14178	. 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 1397   e. wcel 2202   C_ wss 3200   ran crn 4726   ` cfv 5326  (class class class)co 6018   ↾_s cress 13088   MndHom cmhm 13545  SubMndcsubmnd 13546  SubGrpcsubg 13759   GrpHom cghm 13832  mulGrpcmgp 13939   Ringcrg 14015   RingHom crh 14170  SubRingcsubrg 14237

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-iress 13095  df-plusg 13178  df-mulr 13179  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-mhm 13547  df-submnd 13548  df-grp 13591  df-minusg 13592  df-subg 13762  df-ghm 13833  df-mgp 13940  df-ur 13979  df-ring 14017  df-rhm 14172  df-subrg 14239

This theorem is referenced by: (None)