Theorem subrginv 20588

Description: A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrginv.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrginv.2	⊢ 𝐼 = (invr‘𝑅)
subrginv.3	⊢ 𝑈 = (Unit‘𝑆)
subrginv.4	⊢ 𝐽 = (invr‘𝑆)

Assertion

Ref	Expression
subrginv	⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))

Proof of Theorem subrginv

Step	Hyp	Ref	Expression
1		subrgrcl 20576	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring)
3		subrginv.1	. . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
4	3	subrgbas 20581	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	subrgss 20572	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
7	4, 6	eqsstrrd 4019	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
8	7	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
9	3	subrgring 20574	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10		subrginv.3	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑆)
11		subrginv.4	. . . . . . 7 ⊢ 𝐽 = (invr‘𝑆)
12		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
13	10, 11, 12	ringinvcl 20392	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑆))
14	9, 13	sylan 580	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑆))
15	8, 14	sseldd 3984	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑅))
16	12, 10	unitcl 20375	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑆))
17	16	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆))
18	8, 17	sseldd 3984	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅))
19		eqid 2737	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
20	3, 19, 10	subrguss 20587	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
21	20	sselda 3983	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Unit‘𝑅))
22		subrginv.2	. . . . . 6 ⊢ 𝐼 = (invr‘𝑅)
23	19, 22, 5	ringinvcl 20392	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ (Base‘𝑅))
24	1, 21, 23	syl2an2r 685	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ (Base‘𝑅))
25		eqid 2737	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
26	5, 25	ringass 20250	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((𝐽‘𝑋) ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝐼‘𝑋) ∈ (Base‘𝑅))) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))))
27	2, 15, 18, 24, 26	syl13anc 1374	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))))
28		eqid 2737	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
29		eqid 2737	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	10, 11, 28, 29	unitlinv 20393	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
31	9, 30	sylan 580	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
32	3, 25	ressmulr 17351	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
33	32	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑆))
34	33	oveqd 7448	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = ((𝐽‘𝑋)(.r‘𝑆)𝑋))
35		eqid 2737	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	3, 35	subrg1 20582	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
37	36	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (1r‘𝑅) = (1r‘𝑆))
38	31, 34, 37	3eqtr4d 2787	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
39	38	oveq1d 7446	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)))
40	19, 22, 25, 35	unitrinv 20394	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
41	1, 21, 40	syl2an2r 685	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
42	41	oveq2d 7447	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
43	27, 39, 42	3eqtr3d 2785	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
44	5, 25, 35	ringlidm 20266	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑋) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = (𝐼‘𝑋))
45	1, 24, 44	syl2an2r 685	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = (𝐼‘𝑋))
46	5, 25, 35	ringridm 20267	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐽‘𝑋) ∈ (Base‘𝑅)) → ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)) = (𝐽‘𝑋))
47	1, 15, 46	syl2an2r 685	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)) = (𝐽‘𝑋))
48	43, 45, 47	3eqtr3d 2785	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ‘cfv 6561  (class class class)co 7431  Basecbs 17247   ↾_s cress 17274  ._rcmulr 17298  1_rcur 20178  Ringcrg 20230  Unitcui 20355  inv_rcinvr 20387  SubRingcsubrg 20569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-subrg 20570

This theorem is referenced by:  subrgdv  20589  subrgunit  20590  subrgugrp  20591  issubdrg  20781  gzrngunit  21451  sdrginvcl  33302