Theorem subrgunit 19555

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 19552	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
5	4	sselda 3969	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑈)
6		eqid 2823	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	6, 3	unitcl 19411	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝑆))
8	7	adantl 484	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (Base‘𝑆))
9	1	subrgbas 19546	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
10	9	adantr 483	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝐴 = (Base‘𝑆))
11	8, 10	eleqtrrd 2918	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐴)
12	1	subrgring 19540	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
13		eqid 2823	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
14	3, 13, 6	ringinvcl 19428	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆))
15	12, 14	sylan 582	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → ((invr‘𝑆)‘𝑋) ∈ (Base‘𝑆))
16		subrgunit.4	. . . . 5 ⊢ 𝐼 = (invr‘𝑅)
17	1, 16, 3, 13	subrginv 19553	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) = ((invr‘𝑆)‘𝑋))
18	15, 17, 10	3eltr4d 2930	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝐼‘𝑋) ∈ 𝐴)
19	5, 11, 18	3jca 1124	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴))
20		simpr2 1191	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝐴)
21	9	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝐴 = (Base‘𝑆))
22	20, 21	eleqtrd 2917	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ (Base‘𝑆))
23		simpr3 1192	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ 𝐴)
24	23, 21	eleqtrd 2917	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝐼‘𝑋) ∈ (Base‘𝑆))
25		eqid 2823	. . . . . 6 ⊢ (∥r‘𝑆) = (∥r‘𝑆)
26		eqid 2823	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
27	6, 25, 26	dvdsrmul 19400	. . . . 5 ⊢ ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼‘𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘𝑆)((𝐼‘𝑋)(.r‘𝑆)𝑋))
28	22, 24, 27	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)((𝐼‘𝑋)(.r‘𝑆)𝑋))
29		subrgrcl 19542	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
30	29	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑅 ∈ Ring)
31		simpr1 1190	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑈)
32		eqid 2823	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
33		eqid 2823	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
34	2, 16, 32, 33	unitlinv 19429	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
35	30, 31, 34	syl2anc 586	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
36	1, 32	ressmulr 16627	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
37	36	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (.r‘𝑅) = (.r‘𝑆))
38	37	oveqd 7175	. . . . 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 19547	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
40	39	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (1r‘𝑅) = (1r‘𝑆))
41	35, 38, 40	3eqtr3d 2866	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
42	28, 41	breqtrd 5094	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘𝑆)(1r‘𝑆))
43		eqid 2823	. . . . . . 7 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
44	43, 6	opprbas 19381	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘(oppr‘𝑆))
45		eqid 2823	. . . . . 6 ⊢ (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆))
46		eqid 2823	. . . . . 6 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
47	44, 45, 46	dvdsrmul 19400	. . . . 5 ⊢ ((𝑋 ∈ (Base‘𝑆) ∧ (𝐼‘𝑋) ∈ (Base‘𝑆)) → 𝑋(∥r‘(oppr‘𝑆))((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋))
48	22, 24, 47	syl2anc 586	. . . 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 19378	. . . . 5 ⊢ ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (𝑋(.r‘𝑆)(𝐼‘𝑋))
50	2, 16, 32, 33	unitrinv 19430	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
51	30, 31, 50	syl2anc 586	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
52	37	oveqd 7175	. . . . . 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 2866	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → (𝑋(.r‘𝑆)(𝐼‘𝑋)) = (1r‘𝑆))
54	49, 53	syl5eq 2870	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → ((𝐼‘𝑋)(.r‘(oppr‘𝑆))𝑋) = (1r‘𝑆))
55	48, 54	breqtrd 5094	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆))
56		eqid 2823	. . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆)
57	3, 56, 25, 43, 45	isunit 19409	. . 3 ⊢ (𝑋 ∈ 𝑉 ↔ (𝑋(∥r‘𝑆)(1r‘𝑆) ∧ 𝑋(∥r‘(oppr‘𝑆))(1r‘𝑆)))
58	42, 55, 57	sylanbrc 585	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)) → 𝑋 ∈ 𝑉)
59	19, 58	impbida 799	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485   ↾_s cress 16486  ._rcmulr 16568  1_rcur 19253  Ringcrg 19299  opp_rcoppr 19374  ∥_rcdsr 19390  Unitcui 19391  inv_rcinvr 19423  SubRingcsubrg 19533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-tpos 7894  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-subg 18278  df-mgp 19242  df-ur 19254  df-ring 19301  df-oppr 19375  df-dvdsr 19393  df-unit 19394  df-invr 19424  df-subrg 19535

This theorem is referenced by:  issubdrg  19562  gzrngunit  20613  zringunit  20637  cphreccllem  23784