Theorem resrhm2b 20537

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 19167 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 20512	. . . . . 6 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubGrp‘𝑇))
2		resrhm2b.u	. . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
3	2	resghm2b 19167	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
4	1, 3	sylan 581	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
5		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇)
6	5	subrgsubm 20520	. . . . . . 7 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑇)))
7		eqid 2737	. . . . . . . 8 ⊢ ((mulGrp‘𝑇) ↾s 𝑋) = ((mulGrp‘𝑇) ↾s 𝑋)
8	7	resmhm2b 18748	. . . . . . 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 581	. . . . . 6 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋))))
10		subrgrcl 20511	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑇 ∈ Ring)
11	2, 5	mgpress 20089	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Ring ∧ 𝑋 ∈ (SubRing‘𝑇)) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
12	10, 11	mpancom 689	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubRing‘𝑇) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
14	13	oveq2d 7374	. . . . . . 7 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋)) = ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))
15	14	eleq2d 2823	. . . . . 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 279	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))
17	4, 16	anbi12d 633	. . . 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 631	. . 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 480	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑇 ∈ Ring)
20	19	biantrud 531	. . . 4 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ 𝑇 ∈ Ring)))
21	20	anbi1d 632	. . 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 20509	. . . . . 6 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑈 ∈ Ring)
23	22	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑈 ∈ Ring)
24	23	biantrud 531	. . . 4 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ 𝑈 ∈ Ring)))
25	24	anbi1d 632	. . 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 309	. 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 2737	. . 3 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
28	27, 5	isrhm 20416	. 2 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))))
29		eqid 2737	. . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
30	27, 29	isrhm 20416	. 2 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑈) ↔ ((𝑆 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))))
31	26, 28, 30	3bitr4g 314	1 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ran crn 5623  ‘cfv 6490  (class class class)co 7358   ↾_s cress 17158   MndHom cmhm 18707  SubMndcsubmnd 18708  SubGrpcsubg 19054   GrpHom cghm 19145  mulGrpcmgp 20079  Ringcrg 20172   RingHom crh 20407  SubRingcsubrg 20504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17138  df-ress 17159  df-plusg 17191  df-mulr 17192  df-0g 17362  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18709  df-submnd 18710  df-grp 18870  df-minusg 18871  df-subg 19057  df-ghm 19146  df-mgp 20080  df-ur 20121  df-ring 20174  df-rhm 20410  df-subrg 20505

This theorem is referenced by:  imadrhmcl  20732  idomsubr  33375  algextdeglem4  33870  selvcllem4  43013