Theorem subrgugrp 13299

Description: The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrgugrp.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrgugrp.2	⊢ 𝑈 = (Unit‘𝑅)
subrgugrp.3	⊢ 𝑉 = (Unit‘𝑆)
subrgugrp.4	⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)

Assertion

Ref	Expression
subrgugrp	⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Proof of Theorem subrgugrp

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

Step	Hyp	Ref	Expression
1		subrgugrp.1	. . . 4 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		subrgugrp.2	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3		subrgugrp.3	. . . 4 ⊢ 𝑉 = (Unit‘𝑆)
4	1, 2, 3	subrguss 13295	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5		subrgrcl 13285	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
6	2	a1i 9	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑈 = (Unit‘𝑅))
7		subrgugrp.4	. . . . . 6 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
8	7	a1i 9	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))
9		ringsrg 13155	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
10	6, 8, 9	unitgrpbasd 13215	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
11	5, 10	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Base‘𝐺))
12	4, 11	sseqtrd 3193	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ (Base‘𝐺))
13	1	subrgring 13283	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
14		eqid 2177	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
15	3, 14	1unit 13207	. . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉)
16		elex2 2753	. . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → ∃𝑤 𝑤 ∈ 𝑉)
17	13, 15, 16	3syl 17	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∃𝑤 𝑤 ∈ 𝑉)
18		eqid 2177	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	1, 18	ressmulrg 12595	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
20	5, 19	mpdan 421	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
21	20	3ad2ant1 1018	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 5889	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦))
23		eqid 2177	. . . . . . . . . 10 ⊢ (.r‘𝑆) = (.r‘𝑆)
24	3, 23	unitmulcl 13213	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
25	13, 24	syl3an1 1271	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
26	22, 25	eqeltrd 2254	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
27	26	3expa 1203	. . . . . 6 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
28	27	ralrimiva 2550	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
29		eqid 2177	. . . . . . 7 ⊢ (invr‘𝑅) = (invr‘𝑅)
30		eqid 2177	. . . . . . 7 ⊢ (invr‘𝑆) = (invr‘𝑆)
31	1, 29, 3, 30	subrginv 13296	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥))
32	3, 30	unitinvcl 13223	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉)
33	13, 32	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑆)‘𝑥) ∈ 𝑉)
34	31, 33	eqeltrd 2254	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) ∈ 𝑉)
35	28, 34	jca 306	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉))
36	35	ralrimiva 2550	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉))
37		eqid 2177	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
38	37, 18	mgpplusgg 13065	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
39		basfn 12512	. . . . . . . . . . . 12 ⊢ Base Fn V
40		elex 2748	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
41		funfvex 5531	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
42	41	funfni 5315	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
43	39, 40, 42	sylancr 414	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
44		eqidd 2178	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
45	44, 6, 9	unitssd 13209	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
46	43, 45	ssexd 4142	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
47	37	ringmgp 13116	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
48	8, 38, 46, 47	ressplusgd 12579	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
49	5, 48	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (+g‘𝐺))
50	49	oveqd 5889	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘𝐺)𝑦))
51	50	eleq1d 2246	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ↔ (𝑥(+g‘𝐺)𝑦) ∈ 𝑉))
52	51	ralbidv 2477	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ↔ ∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉))
53	2	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Unit‘𝑅))
54	7	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))
55		eqidd 2178	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invr‘𝑅))
56	53, 54, 55, 5	invrfvald 13222	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invg‘𝐺))
57	56	fveq1d 5516	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((invr‘𝑅)‘𝑥) = ((invg‘𝐺)‘𝑥))
58	57	eleq1d 2246	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (((invr‘𝑅)‘𝑥) ∈ 𝑉 ↔ ((invg‘𝐺)‘𝑥) ∈ 𝑉))
59	52, 58	anbi12d 473	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉) ↔ (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉)))
60	59	ralbidv 2477	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉) ↔ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉)))
61	36, 60	mpbid 147	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉))
62	2, 7	unitgrp 13216	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
63		eqid 2177	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
64		eqid 2177	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
65		eqid 2177	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
66	63, 64, 65	issubg2m 12980	. . 3 ⊢ (𝐺 ∈ Grp → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ (Base‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉))))
67	5, 62, 66	3syl 17	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ (Base‘𝐺) ∧ ∃𝑤 𝑤 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(+g‘𝐺)𝑦) ∈ 𝑉 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝑉))))
68	12, 17, 61, 67	mpbir3and 1180	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   ⊆ wss 3129   Fn wfn 5210  ‘cfv 5215  (class class class)co 5872  Basecbs 12454   ↾_s cress 12455  +_gcplusg 12528  ._rcmulr 12529  Mndcmnd 12749  Grpcgrp 12809  inv_gcminusg 12810  SubGrpcsubg 12958  mulGrpcmgp 13061  1_rcur 13073  Ringcrg 13110  Unitcui 13187  inv_rcinvr 13220  SubRingcsubrg 13276

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-tpos 6243  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-oppr 13171  df-dvdsr 13189  df-unit 13190  df-invr 13221  df-subrg 13278

This theorem is referenced by: (None)