Theorem subrginv 20548

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 20536	. . . . 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 20541	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	subrgss 20532	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
7	4, 6	eqsstrrd 3994	. . . . . 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 20534	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10		subrginv.3	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑆)
11		subrginv.4	. . . . . . 7 ⊢ 𝐽 = (invr‘𝑆)
12		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
13	10, 11, 12	ringinvcl 20352	. . . . . 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 3959	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑅))
16	12, 10	unitcl 20335	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑆))
17	16	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆))
18	8, 17	sseldd 3959	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅))
19		eqid 2735	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
20	3, 19, 10	subrguss 20547	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
21	20	sselda 3958	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Unit‘𝑅))
22		subrginv.2	. . . . . 6 ⊢ 𝐼 = (invr‘𝑅)
23	19, 22, 5	ringinvcl 20352	. . . . 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 2735	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
26	5, 25	ringass 20213	. . . 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 2735	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
29		eqid 2735	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	10, 11, 28, 29	unitlinv 20353	. . . . . 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 17321	. . . . . . 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 7422	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = ((𝐽‘𝑋)(.r‘𝑆)𝑋))
35		eqid 2735	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	3, 35	subrg1 20542	. . . . . 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 2780	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
39	38	oveq1d 7420	. . 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 20354	. . . . 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 7421	. . 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 2778	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
44	5, 25, 35	ringlidm 20229	. . 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 20230	. . 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 2778	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  ‘cfv 6531  (class class class)co 7405  Basecbs 17228   ↾_s cress 17251  ._rcmulr 17272  1_rcur 20141  Ringcrg 20193  Unitcui 20315  inv_rcinvr 20347  SubRingcsubrg 20529

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-subg 19106  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-subrg 20530

This theorem is referenced by:  subrgdv  20549  subrgunit  20550  subrgugrp  20551  issubdrg  20740  gzrngunit  21401  sdrginvcl  33294