Theorem rhmimasubrng 20566

Description: The homomorphic image of a subring is a subring. (Contributed by AV, 16-Feb-2025.)

Assertion

Ref	Expression
rhmimasubrng	⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))

Proof of Theorem rhmimasubrng

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmghm 20484	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		subrngsubg 20552	. . 3 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3		ghmima 19255	. . 3 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑋 ∈ (SubGrp‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
5		eqid 2737	. . . 4 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
6		eqid 2737	. . . 4 ⊢ (mulGrp‘𝑁) = (mulGrp‘𝑁)
7	5, 6	rhmmhm 20479	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
8		simpl 482	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
9		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
10	5, 9	mgpbas 20142	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
11	10	eqcomi 2746	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
12	11	subrngss 20548	. . . . . 6 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
13	12	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
14		eqidd 2738	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15		eqidd 2738	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑀) = (.r‘𝑀)
17	5, 16	mgpplusg 20141	. . . . . . . 8 ⊢ (.r‘𝑀) = (+g‘(mulGrp‘𝑀))
18	17	eqcomi 2746	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑀)) = (.r‘𝑀)
19	18	subrngmcl 20557	. . . . . 6 ⊢ ((𝑋 ∈ (SubRng‘𝑀) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
20	19	3adant1l 1177	. . . . 5 ⊢ (((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
21	8, 13, 14, 15, 20	mhmimalem 18837	. . . 4 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋))
22		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑁) = (.r‘𝑁)
23	6, 22	mgpplusg 20141	. . . . . . . 8 ⊢ (.r‘𝑁) = (+g‘(mulGrp‘𝑁))
24	23	eqcomi 2746	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑁)) = (.r‘𝑁)
25	24	oveqi 7444	. . . . . 6 ⊢ (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r‘𝑁)𝑦)
26	25	eleq1i 2832	. . . . 5 ⊢ ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋) ↔ (𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
27	26	2ralbii 3128	. . . 4 ⊢ (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
28	21, 27	sylib 218	. . 3 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
29	7, 28	sylan 580	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
30		rhmrcl2 20477	. . . . 5 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31		ringrng 20282	. . . . 5 ⊢ (𝑁 ∈ Ring → 𝑁 ∈ Rng)
32	30, 31	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Rng)
33	32	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑁 ∈ Rng)
34		eqid 2737	. . . 4 ⊢ (Base‘𝑁) = (Base‘𝑁)
35	34, 22	issubrng2 20558	. . 3 ⊢ (𝑁 ∈ Rng → ((𝐹 “ 𝑋) ∈ (SubRng‘𝑁) ↔ ((𝐹 “ 𝑋) ∈ (SubGrp‘𝑁) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
36	33, 35	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ((𝐹 “ 𝑋) ∈ (SubRng‘𝑁) ↔ ((𝐹 “ 𝑋) ∈ (SubGrp‘𝑁) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
37	4, 29, 36	mpbir2and 713	1 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951   “ cima 5688  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  ._rcmulr 17298   MndHom cmhm 18794  SubGrpcsubg 19138   GrpHom cghm 19230  mulGrpcmgp 20137  Rngcrng 20149  Ringcrg 20230   RingHom crh 20469  SubRngcsubrng 20545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-minusg 18955  df-subg 19141  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-rhm 20472  df-subrng 20546

This theorem is referenced by: (None)