Theorem rhmimasubrng 20481

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 20401	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		subrngsubg 20467	. . 3 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3		ghmima 19149	. . 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 2731	. . . 4 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
6		eqid 2731	. . . 4 ⊢ (mulGrp‘𝑁) = (mulGrp‘𝑁)
7	5, 6	rhmmhm 20397	. . 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 2731	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
10	5, 9	mgpbas 20063	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
11	10	eqcomi 2740	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
12	11	subrngss 20463	. . . . . 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 2732	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15		eqidd 2732	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16		eqid 2731	. . . . . . . . 9 ⊢ (.r‘𝑀) = (.r‘𝑀)
17	5, 16	mgpplusg 20062	. . . . . . . 8 ⊢ (.r‘𝑀) = (+g‘(mulGrp‘𝑀))
18	17	eqcomi 2740	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑀)) = (.r‘𝑀)
19	18	subrngmcl 20472	. . . . . 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 18732	. . . 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 2731	. . . . . . . . 9 ⊢ (.r‘𝑁) = (.r‘𝑁)
23	6, 22	mgpplusg 20062	. . . . . . . 8 ⊢ (.r‘𝑁) = (+g‘(mulGrp‘𝑁))
24	23	eqcomi 2740	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑁)) = (.r‘𝑁)
25	24	oveqi 7359	. . . . . 6 ⊢ (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r‘𝑁)𝑦)
26	25	eleq1i 2822	. . . . 5 ⊢ ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋) ↔ (𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
27	26	2ralbii 3107	. . . 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 20395	. . . . 5 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31		ringrng 20203	. . . . 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 2731	. . . 4 ⊢ (Base‘𝑁) = (Base‘𝑁)
35	34, 22	issubrng2 20473	. . 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 2111  ∀wral 3047   ⊆ wss 3897   “ cima 5617  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162   MndHom cmhm 18689  SubGrpcsubg 19033   GrpHom cghm 19124  mulGrpcmgp 20058  Rngcrng 20070  Ringcrg 20151   RingHom crh 20387  SubRngcsubrng 20460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-0g 17345  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-grp 18849  df-minusg 18850  df-subg 19036  df-ghm 19125  df-cmn 19694  df-abl 19695  df-mgp 20059  df-rng 20071  df-ur 20100  df-ring 20153  df-rhm 20390  df-subrng 20461

This theorem is referenced by: (None)