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Theorem subrgugrp 20544

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgugrp.1	. . 3 ⊢ 𝑆 = (𝑅 ↾_s 𝐴)
2		subrgugrp.2	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
3		subrgugrp.3	. . 3 ⊢ 𝑉 = (Unit‘𝑆)
4	1, 2, 3	subrguss 20540	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5	1	subrgring 20527	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
6		eqid 2728	. . . 4 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
7	3, 6	1unit 20327	. . 3 ⊢ (𝑆 ∈ Ring → (1_r‘𝑆) ∈ 𝑉)
8		ne0i 4338	. . 3 ⊢ ((1_r‘𝑆) ∈ 𝑉 → 𝑉 ≠ ∅)
9	5, 7, 8	3syl 18	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ≠ ∅)
10		eqid 2728	. . . . . . . . . 10 ⊢ (._r‘𝑅) = (._r‘𝑅)
11	1, 10	ressmulr 17297	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (._r‘𝑅) = (._r‘𝑆))
12	11	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (._r‘𝑅) = (._r‘𝑆))
13	12	oveqd 7443	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(._r‘𝑅)𝑦) = (𝑥(._r‘𝑆)𝑦))
14		eqid 2728	. . . . . . . . 9 ⊢ (._r‘𝑆) = (._r‘𝑆)
15	3, 14	unitmulcl 20333	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(._r‘𝑆)𝑦) ∈ 𝑉)
16	5, 15	syl3an1 1160	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(._r‘𝑆)𝑦) ∈ 𝑉)
17	13, 16	eqeltrd 2829	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(._r‘𝑅)𝑦) ∈ 𝑉)
18	17	3expa 1115	. . . . 5 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑥(._r‘𝑅)𝑦) ∈ 𝑉)
19	18	ralrimiva 3143	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉)
20		eqid 2728	. . . . . 6 ⊢ (inv_r‘𝑅) = (inv_r‘𝑅)
21		eqid 2728	. . . . . 6 ⊢ (inv_r‘𝑆) = (inv_r‘𝑆)
22	1, 20, 3, 21	subrginv 20541	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑅)‘𝑥) = ((inv_r‘𝑆)‘𝑥))
23	3, 21	unitinvcl 20343	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑆)‘𝑥) ∈ 𝑉)
24	5, 23	sylan 578	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑆)‘𝑥) ∈ 𝑉)
25	22, 24	eqeltrd 2829	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑅)‘𝑥) ∈ 𝑉)
26	19, 25	jca 510	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉 ∧ ((inv_r‘𝑅)‘𝑥) ∈ 𝑉))
27	26	ralrimiva 3143	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉 ∧ ((inv_r‘𝑅)‘𝑥) ∈ 𝑉))
28		subrgrcl 20529	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
29		subrgugrp.4	. . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾_s 𝑈)
30	2, 29	unitgrp 20336	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
31	2, 29	unitgrpbas 20335	. . . 4 ⊢ 𝑈 = (Base‘𝐺)
32	2	fvexi 6916	. . . . 5 ⊢ 𝑈 ∈ V
33		eqid 2728	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
34	33, 10	mgpplusg 20092	. . . . . 6 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
35	29, 34	ressplusg 17280	. . . . 5 ⊢ (𝑈 ∈ V → (._r‘𝑅) = (+_g‘𝐺))
36	32, 35	ax-mp 5	. . . 4 ⊢ (._r‘𝑅) = (+_g‘𝐺)
37	2, 29, 20	invrfval 20342	. . . 4 ⊢ (inv_r‘𝑅) = (inv_g‘𝐺)
38	31, 36, 37	issubg2 19110	. . 3 ⊢ (𝐺 ∈ Grp → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ 𝑈 ∧ 𝑉 ≠ ∅ ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉 ∧ ((inv_r‘𝑅)‘𝑥) ∈ 𝑉))))
39	28, 30, 38	3syl 18	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑉 ∈ (SubGrp‘𝐺) ↔ (𝑉 ⊆ 𝑈 ∧ 𝑉 ≠ ∅ ∧ ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉 ∧ ((inv_r‘𝑅)‘𝑥) ∈ 𝑉))))
40	4, 9, 27, 39	mpbir3and 1339	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 Vcvv 3473 ⊆ wss 3949 ∅c0 4326 ‘cfv 6553 (class class class)co 7426 ↾_s cress 17218 +_gcplusg 17242 ._rcmulr 17243 Grpcgrp 18904 SubGrpcsubg 19089 mulGrpcmgp 20088 1_rcur 20135 Ringcrg 20187 Unitcui 20308 inv_rcinvr 20340 SubRingcsubrg 20520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-subg 19092 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-subrg 20522

This theorem is referenced by: (None)

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