Theorem resrhm2b 14266

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 13851 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 14244	. . . . . 6 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubGrp‘𝑇))
2		resrhm2b.u	. . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
3	2	resghm2b 13851	. . . . . 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 2231	. . . . . . . 8 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇)
6	5	subrgsubm 14251	. . . . . . 7 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑇)))
7		eqid 2231	. . . . . . . 8 ⊢ ((mulGrp‘𝑇) ↾s 𝑋) = ((mulGrp‘𝑇) ↾s 𝑋)
8	7	resmhm2b 13574	. . . . . . 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 14243	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑇 ∈ Ring)
11	2, 5	mgpress 13947	. . . . . . . . . 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 6034	. . . . . . 7 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋)) = ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))
15	14	eleq2d 2301	. . . . . 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 14241	. . . . . 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 2231	. . 3 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
28	27, 5	isrhm 14175	. 2 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))))
29		eqid 2231	. . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
30	27, 29	isrhm 14175	. 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 1397   ∈ wcel 2202   ⊆ wss 3200  ran crn 4726  ‘cfv 5326  (class class class)co 6018   ↾_s cress 13085   MndHom cmhm 13542  SubMndcsubmnd 13543  SubGrpcsubg 13756   GrpHom cghm 13829  mulGrpcmgp 13936  Ringcrg 14012   RingHom crh 14167  SubRingcsubrg 14234

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 13087  df-slot 13088  df-base 13090  df-sets 13091  df-iress 13092  df-plusg 13175  df-mulr 13176  df-0g 13343  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-mhm 13544  df-submnd 13545  df-grp 13588  df-minusg 13589  df-subg 13759  df-ghm 13830  df-mgp 13937  df-ur 13976  df-ring 14014  df-rhm 14169  df-subrg 14236

This theorem is referenced by: (None)