Theorem subrginv 20604

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 20592	. . . . 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 20597	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	subrgss 20588	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
7	4, 6	eqsstrrd 4034	. . . . . 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 20590	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10		subrginv.3	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑆)
11		subrginv.4	. . . . . . 7 ⊢ 𝐽 = (invr‘𝑆)
12		eqid 2734	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
13	10, 11, 12	ringinvcl 20408	. . . . . 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 3995	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑅))
16	12, 10	unitcl 20391	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑆))
17	16	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆))
18	8, 17	sseldd 3995	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅))
19		eqid 2734	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
20	3, 19, 10	subrguss 20603	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
21	20	sselda 3994	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Unit‘𝑅))
22		subrginv.2	. . . . . 6 ⊢ 𝐼 = (invr‘𝑅)
23	19, 22, 5	ringinvcl 20408	. . . . 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 2734	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
26	5, 25	ringass 20270	. . . 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 1371	. . 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 2734	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
29		eqid 2734	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	10, 11, 28, 29	unitlinv 20409	. . . . . 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 17352	. . . . . . 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 7447	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = ((𝐽‘𝑋)(.r‘𝑆)𝑋))
35		eqid 2734	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	3, 35	subrg1 20598	. . . . . 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 2784	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
39	38	oveq1d 7445	. . 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 20410	. . . . 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 7446	. . 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 2782	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
44	5, 25, 35	ringlidm 20282	. . 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 20283	. . 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 2782	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ⊆ wss 3962  ‘cfv 6562  (class class class)co 7430  Basecbs 17244   ↾_s cress 17273  ._rcmulr 17298  1_rcur 20198  Ringcrg 20250  Unitcui 20371  inv_rcinvr 20403  SubRingcsubrg 20585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-subg 19153  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-subrg 20586

This theorem is referenced by:  subrgdv  20605  subrgunit  20606  subrgugrp  20607  issubdrg  20797  gzrngunit  21468  sdrginvcl  33281