Theorem resrhm2b 14540

Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 14048 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 14518	. . . . . 6 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubGrp‘𝑇))
2		resrhm2b.u	. . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
3	2	resghm2b 14048	. . . . . 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 2238	. . . . . . . 8 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇)
6	5	subrgsubm 14525	. . . . . . 7 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑇)))
7		eqid 2238	. . . . . . . 8 ⊢ ((mulGrp‘𝑇) ↾s 𝑋) = ((mulGrp‘𝑇) ↾s 𝑋)
8	7	resmhm2b 13779	. . . . . . 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 14517	. . . . . . . . . 10 ⊢ (𝑋 ∈ (SubRing‘𝑇) → 𝑇 ∈ Ring)
11	2, 5	mgpress 14213	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Ring ∧ 𝑋 ∈ (SubRing‘𝑇)) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
12	10, 11	mpancom 426	. . . . . . . . 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 6095	. . . . . . 7 ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋)) = ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))
15	14	eleq2d 2308	. . . . . 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 477	. . . 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 468	. . 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 469	. . 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 14515	. . . . . 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 469	. . 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 2238	. . 3 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
28	27, 5	isrhm 14448	. 2 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))))
29		eqid 2238	. . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈)
30	27, 29	isrhm 14448	. 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 1402   ∈ wcel 2209   ⊆ wss 3220  ran crn 4773  ‘cfv 5375  (class class class)co 6079   ↾_s cress 13336   MndHom cmhm 13747  SubMndcsubmnd 13748  SubGrpcsubg 13953   GrpHom cghm 14026  mulGrpcmgp 14200  Ringcrg 14283   RingHom crh 14440  SubRingcsubrg 14508

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-map 6918  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-mhm 13749  df-submnd 13750  df-grp 13791  df-minusg 13792  df-subg 13956  df-ghm 14027  df-mgp 14201  df-ur 14246  df-ring 14285  df-rhm 14442  df-subrg 14510

This theorem is referenced by: (None)