Theorem subrgunit 19289

Description: An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (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)
subrgunit.4	⊢ 𝐼 = (invr‘𝑅)

Assertion

Ref	Expression
subrgunit	⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)))

Proof of Theorem subrgunit

Step	Hyp	Ref	Expression
1		subrgugrp.1	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		subrgugrp.2	. . . . 5 ⊢ 𝑈 = (Unit‘𝑅)
3		subrgugrp.3	. . . . 5 ⊢ 𝑉 = (Unit‘𝑆)
4	1, 2, 3	subrguss 19286	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5	4	sselda 3853	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑈)
6		eqid 2773	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	6, 3	unitcl 19145	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝑆))
8	7	adantl 474	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (Base‘𝑆))
9	1	subrgbas 19280	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
10	9	adantr 473	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
11	8, 10	eleqtrrd 2864	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐴)
12	1	subrgring 19274	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
13		eqid 2773	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
14	3, 13, 6	ringinvcl 19162	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆))
15	12, 14	sylan 572	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆))
16		subrgunit.4	. . . . 5 ⊢ 𝐼 = (invr‘𝑅)
17	1, 16, 3, 13	subrginv 19287	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) = ((invr‘𝑆)‘𝑋))
18	15, 17, 10	3eltr4d 2876	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝐴)
19	5, 11, 18	3jca 1109	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴))
20		simpr2 1176	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝐴)
21	9	adantr 473	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
22	20, 21	eleqtrd 2863	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
23		simpr3 1177	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ 𝐴)
24	23, 21	eleqtrd 2863	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ (Base‘𝑆))
25		eqid 2773	. . . . . 6 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
26		eqid 2773	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
27	6, 25, 26	dvdsrmul 19134	. . . . 5 ⊢ ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼‘𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘𝑆)((𝐼‘𝑋)(.r‘𝑆)𝑋))
28	22, 24, 27	syl2anc 576	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)((𝐼‘𝑋)(.r‘𝑆)𝑋))
29		subrgrcl 19276	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
30	29	adantr 473	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑅 ∈ Ring)
31		simpr1 1175	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑈)
32		eqid 2773	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
33		eqid 2773	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
34	2, 16, 32, 33	unitlinv 19163	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
35	30, 31, 34	syl2anc 576	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
36	1, 32	ressmulr 16480	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
37	36	adantr 473	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘𝑅) = (.r‘𝑆))
38	37	oveqd 6992	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = ((𝐼‘𝑋)(.r‘𝑆)𝑋))
39	1, 33	subrg1 19281	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
40	39	adantr 473	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (1r‘𝑅) = (1r‘𝑆))
41	35, 38, 40	3eqtr3d 2817	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
42	28, 41	breqtrd 4952	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)(1r‘𝑆))
43		eqid 2773	. . . . . . 7 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
44	43, 6	opprbas 19115	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘(oppr‘𝑆))
45		eqid 2773	. . . . . 6 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
46		eqid 2773	. . . . . 6 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
47	44, 45, 46	dvdsrmul 19134	. . . . 5 ⊢ ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼‘𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘(oppr‘𝑆))((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋))
48	22, 24, 47	syl2anc 576	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋))
49	6, 26, 43, 46	opprmul 19112	. . . . 5 ⊢ ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (𝑋(.r‘𝑆)(𝐼‘𝑋))
50	2, 16, 32, 33	unitrinv 19164	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
51	30, 31, 50	syl2anc 576	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
52	37	oveqd 6992	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (𝑋(.r‘𝑆)(𝐼‘𝑋)))
53	51, 52, 40	3eqtr3d 2817	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑆)(𝐼‘𝑋)) = (1r‘𝑆))
54	49, 53	syl5eq 2821	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (1r‘𝑆))
55	48, 54	breqtrd 4952	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆))
56		eqid 2773	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
57	3, 56, 25, 43, 45	isunit 19143	. . 3 ⊢ (𝑋 ∈ 𝑉 ↔ (𝑋(∥r‘𝑆)(1r‘𝑆) ∧ 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆)))
58	42, 55, 57	sylanbrc 575	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑉)
59	19, 58	impbida 789	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   class class class wbr 4926  ‘cfv 6186  (class class class)co 6975  Basecbs 16338   ↾_s cress 16339  ._rcmulr 16421  1_rcur 18987  Ringcrg 19033  opp_rcoppr 19108  ∥_rcdsr 19124  Unitcui 19125  inv_rcinvr 19157  SubRingcsubrg 19267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-tpos 7694  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-nn 11439  df-2 11502  df-3 11503  df-ndx 16341  df-slot 16342  df-base 16344  df-sets 16345  df-ress 16346  df-plusg 16433  df-mulr 16434  df-0g 16570  df-mgm 17723  df-sgrp 17765  df-mnd 17776  df-grp 17907  df-minusg 17908  df-subg 18073  df-mgp 18976  df-ur 18988  df-ring 19035  df-oppr 19109  df-dvdsr 19127  df-unit 19128  df-invr 19158  df-subrg 19269

This theorem is referenced by:  issubdrg  19296  gzrngunit  20329  zringunit  20353  cphreccllem  23501