Theorem resrhm2b 14411

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 13996 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 14389	. . . . . 6 |- ( X e. (SubRing ` T ) -> X e. (SubGrp ` T ) )
2		resrhm2b.u	. . . . . . 7 |- U = ( T ↾s X )
3	2	resghm2b 13996	. . . . . 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 2234	. . . . . . . 8 |- (mulGrp ` T ) = (mulGrp ` T )
6	5	subrgsubm 14396	. . . . . . 7 |- ( X e. (SubRing ` T ) -> X e. (SubMnd ` (mulGrp ` T ) ) )
7		eqid 2234	. . . . . . . 8 |- ( (mulGrp ` T ) ↾s X ) = ( (mulGrp ` T ) ↾s X )
8	7	resmhm2b 13719	. . . . . . 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 14388	. . . . . . . . . 10 |- ( X e. (SubRing ` T ) -> T e. Ring )
11	2, 5	mgpress 14092	. . . . . . . . . 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 6068	. . . . . . 7 |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( (mulGrp ` S ) MndHom ( (mulGrp ` T ) ↾s X ) ) = ( (mulGrp ` S ) MndHom (mulGrp ` U ) ) )
15	14	eleq2d 2304	. . . . . 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 14386	. . . . . 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 2234	. . 3 |- (mulGrp ` S ) = (mulGrp ` S )
28	27, 5	isrhm 14320	. 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 2234	. . 3 |- (mulGrp ` U ) = (mulGrp ` U )
30	27, 29	isrhm 14320	. 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 1398   e. wcel 2205   C_ wss 3213   ran crn 4752   ` cfv 5354  (class class class)co 6052   ↾_s cress 13230   MndHom cmhm 13687  SubMndcsubmnd 13688  SubGrpcsubg 13901   GrpHom cghm 13974  mulGrpcmgp 14081   Ringcrg 14157   RingHom crh 14312  SubRingcsubrg 14379

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-mhm 13689  df-submnd 13690  df-grp 13733  df-minusg 13734  df-subg 13904  df-ghm 13975  df-mgp 14082  df-ur 14121  df-ring 14159  df-rhm 14314  df-subrg 14381

This theorem is referenced by: (None)