Theorem subrgugrp 14521

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 14517	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5		subrgrcl 14507	. . . 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 14325	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
10	6, 8, 9	unitgrpbasd 14395	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑈 = (Base‘𝐺))
11	5, 10	syl 14	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 = (Base‘𝐺))
12	4, 11	sseqtrd 3286	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ (Base‘𝐺))
13	1	subrgring 14505	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
14		eqid 2238	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
15	3, 14	1unit 14387	. . 3 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ 𝑉)
16		elex2 2838	. . 3 ⊢ ((1r‘𝑆) ∈ 𝑉 → ∃𝑤 𝑤 ∈ 𝑉)
17	13, 15, 16	3syl 17	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∃𝑤 𝑤 ∈ 𝑉)
18		eqid 2238	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	1, 18	ressmulrg 13476	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
20	5, 19	mpdan 425	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
21	20	3ad2ant1 1049	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 6092	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑆)𝑦))
23		eqid 2238	. . . . . . . . . 10 ⊢ (.r‘𝑆) = (.r‘𝑆)
24	3, 23	unitmulcl 14393	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
25	13, 24	syl3an1 1311	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑉)
26	22, 25	eqeltrd 2315	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
27	26	3expa 1234	. . . . . 6 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
28	27	ralrimiva 2623	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉)
29		eqid 2238	. . . . . . 7 ⊢ (invr‘𝑅) = (invr‘𝑅)
30		eqid 2238	. . . . . . 7 ⊢ (invr‘𝑆) = (invr‘𝑆)
31	1, 29, 3, 30	subrginv 14518	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((invr‘𝑅)‘𝑥) = ((invr‘𝑆)‘𝑥))
32	3, 30	unitinvcl 14403	. . . . . . 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 2315	. . . . 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 2623	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ∧ ((invr‘𝑅)‘𝑥) ∈ 𝑉))
37		eqid 2238	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
38	37, 18	mgpplusgg 14198	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
39		basfn 13389	. . . . . . . . . . . 12 ⊢ Base Fn V
40		elex 2833	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
41		funfvex 5707	. . . . . . . . . . . . 13 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
42	41	funfni 5478	. . . . . . . . . . . 12 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
43	39, 40, 42	sylancr 418	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ V)
44		eqidd 2239	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑅))
45	44, 6, 9	unitssd 14389	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (Base‘𝑅))
46	43, 45	ssexd 4268	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ V)
47	37	ringmgp 14280	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
48	8, 38, 46, 47	ressplusgd 13460	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘𝐺))
49	5, 48	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (+g‘𝐺))
50	49	oveqd 6092	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘𝐺)𝑦))
51	50	eleq1d 2307	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑉 ↔ (𝑥(+g‘𝐺)𝑦) ∈ 𝑉))
52	51	ralbidv 2550	. . . . 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 2239	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invr‘𝑅))
56	53, 54, 55, 5	invrfvald 14402	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (invr‘𝑅) = (invg‘𝐺))
57	56	fveq1d 5692	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((invr‘𝑅)‘𝑥) = ((invg‘𝐺)‘𝑥))
58	57	eleq1d 2307	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (((invr‘𝑅)‘𝑥) ∈ 𝑉 ↔ ((invg‘𝐺)‘𝑥) ∈ 𝑉))
59	52, 58	anbi12d 477	. . . 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 2550	. . 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 14396	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
63		eqid 2238	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
64		eqid 2238	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
65		eqid 2238	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
66	63, 64, 65	issubg2m 13969	. . 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 1211	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075  Basecbs 13330   ↾_s cress 13331  +_gcplusg 13408  ._rcmulr 13409  Mndcmnd 13706  Grpcgrp 13782  inv_gcminusg 13783  SubGrpcsubg 13947  mulGrpcmgp 14194  1_rcur 14237  Ringcrg 14274  Unitcui 14366  inv_rcinvr 14400  SubRingcsubrg 14498

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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-tpos 6506  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-cmn 14066  df-abl 14067  df-mgp 14195  df-ur 14238  df-srg 14242  df-ring 14276  df-oppr 14346  df-dvdsr 14368  df-unit 14369  df-invr 14401  df-subrg 14500

This theorem is referenced by: (None)