Theorem resrhm2b 14417

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 14000 analog. (Contributed by SN, 7-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
resrhm2b	⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))

Proof of Theorem resrhm2b

Step	Hyp	Ref	Expression
1		subrgsubg 14395	. . . . . 6 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubGrp‘𝑇))
2		resrhm2b.u	. . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
3	2	resghm2b 14000	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
4	1, 3	sylan 283	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
5		eqid 2234	. . . . . . . 8 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇)
6	5	subrgsubm 14402	. . . . . . 7 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑇)))
7		eqid 2234	. . . . . . . 8 ⊢ ((mulGrp‘𝑇) ↾s 𝑋) = ((mulGrp‘𝑇) ↾s 𝑋)
8	7	resmhm2b 13723	. . . . . . 7 ⊢ ((𝑋 ∈ (SubMnd‘(mulGrp‘𝑇)) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋))))
9	6, 8	sylan 283	. . . . . 6 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋))))
10		subrgrcl 14394	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑇 ∈ Ring)
11	2, 5	mgpress 14096	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Ring ∧ 𝑋 ∈ (SubRing‘𝑇)) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
12	10, 11	mpancom 422	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubRing‘𝑇) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
13	12	adantr 276	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
14	13	oveq2d 6068	. . . . . . 7 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋)) = ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))
15	14	eleq2d 2304	. . . . . 6 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))
16	9, 15	bitrd 188	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))
17	4, 16	anbi12d 473	. . . 4 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇))) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))))
18	17	anbi2d 464	. . 3 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))) ↔ (𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))))
19	10	adantr 276	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑇 ∈ Ring)
20	19	biantrud 304	. . . 4 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ 𝑇 ∈ Ring)))
21	20	anbi1d 465	. . 3 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇))))))
22	2	subrgring 14392	. . . . . 6 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑈 ∈ Ring)
23	22	adantr 276	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑈 ∈ Ring)
24	23	biantrud 304	. . . 4 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ 𝑈 ∈ Ring)))
25	24	anbi1d 465	. . 3 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))) ↔ ((𝑆 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))))
26	18, 21, 25	3bitr3d 218	. 2 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))) ↔ ((𝑆 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))))
27		eqid 2234	. . 3 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
28	27, 5	isrhm 14325	. 2 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))))
29		eqid 2234	. . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
30	27, 29	isrhm 14325	. 2 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑈) ↔ ((𝑆 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))))
31	26, 28, 30	3bitr4g 223	1 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   ⊆ wss 3213  ran crn 4752  ‘cfv 5354  (class class class)co 6052   ↾_s cress 13234   MndHom cmhm 13691  SubMndcsubmnd 13692  SubGrpcsubg 13905   GrpHom cghm 13978  mulGrpcmgp 14085  Ringcrg 14161   RingHom crh 14317  SubRingcsubrg 14385

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 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-lttrn 8246  ax-pre-ltadd 8248

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 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-iress 13241  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-mhm 13693  df-submnd 13694  df-grp 13737  df-minusg 13738  df-subg 13908  df-ghm 13979  df-mgp 14086  df-ur 14125  df-ring 14163  df-rhm 14319  df-subrg 14387

This theorem is referenced by: (None)