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Theorem subrgugrp 20493

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 20489	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5	1	subrgring 20476	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
6		eqid 2726	. . . 4 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
7	3, 6	1unit 20276	. . 3 ⊢ (𝑆 ∈ Ring → (1_r‘𝑆) ∈ 𝑉)
8		ne0i 4329	. . 3 ⊢ ((1_r‘𝑆) ∈ 𝑉 → 𝑉 ≠ ∅)
9	5, 7, 8	3syl 18	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ≠ ∅)
10		eqid 2726	. . . . . . . . . 10 ⊢ (._r‘𝑅) = (._r‘𝑅)
11	1, 10	ressmulr 17261	. . . . . . . . 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 7422	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥(._r‘𝑅)𝑦) = (𝑥(._r‘𝑆)𝑦))
14		eqid 2726	. . . . . . . . 9 ⊢ (._r‘𝑆) = (._r‘𝑆)
15	3, 14	unitmulcl 20282	. . . . . . . 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 2827	. . . . . 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 3140	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉)
20		eqid 2726	. . . . . 6 ⊢ (inv_r‘𝑅) = (inv_r‘𝑅)
21		eqid 2726	. . . . . 6 ⊢ (inv_r‘𝑆) = (inv_r‘𝑆)
22	1, 20, 3, 21	subrginv 20490	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑅)‘𝑥) = ((inv_r‘𝑆)‘𝑥))
23	3, 21	unitinvcl 20292	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑆)‘𝑥) ∈ 𝑉)
24	5, 23	sylan 579	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑆)‘𝑥) ∈ 𝑉)
25	22, 24	eqeltrd 2827	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → ((inv_r‘𝑅)‘𝑥) ∈ 𝑉)
26	19, 25	jca 511	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉 ∧ ((inv_r‘𝑅)‘𝑥) ∈ 𝑉))
27	26	ralrimiva 3140	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝑉 (∀𝑦 ∈ 𝑉 (𝑥(._r‘𝑅)𝑦) ∈ 𝑉 ∧ ((inv_r‘𝑅)‘𝑥) ∈ 𝑉))
28		subrgrcl 20478	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
29		subrgugrp.4	. . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾_s 𝑈)
30	2, 29	unitgrp 20285	. . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
31	2, 29	unitgrpbas 20284	. . . 4 ⊢ 𝑈 = (Base‘𝐺)
32	2	fvexi 6899	. . . . 5 ⊢ 𝑈 ∈ V
33		eqid 2726	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
34	33, 10	mgpplusg 20043	. . . . . 6 ⊢ (._r‘𝑅) = (+_g‘(mulGrp‘𝑅))
35	29, 34	ressplusg 17244	. . . . 5 ⊢ (𝑈 ∈ V → (._r‘𝑅) = (+_g‘𝐺))
36	32, 35	ax-mp 5	. . . 4 ⊢ (._r‘𝑅) = (+_g‘𝐺)
37	2, 29, 20	invrfval 20291	. . . 4 ⊢ (inv_r‘𝑅) = (inv_g‘𝐺)
38	31, 36, 37	issubg2 19068	. . 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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 Vcvv 3468 ⊆ wss 3943 ∅c0 4317 ‘cfv 6537 (class class class)co 7405 ↾_s cress 17182 +_gcplusg 17206 ._rcmulr 17207 Grpcgrp 18863 SubGrpcsubg 19047 mulGrpcmgp 20039 1_rcur 20086 Ringcrg 20138 Unitcui 20257 inv_rcinvr 20289 SubRingcsubrg 20469

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-subrg 20471

This theorem is referenced by: (None)

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