Theorem subrginv 18619

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 18608	. . . . 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 18612	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
5		eqid 2610	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	subrgss 18604	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
7	4, 6	eqsstr3d 3603	. . . . . 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 18606	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
10		subrginv.3	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑆)
11		subrginv.4	. . . . . . 7 ⊢ 𝐽 = (invr‘𝑆)
12		eqid 2610	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
13	10, 11, 12	ringinvcl 18499	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑆))
14	9, 13	sylan 487	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑆))
15	8, 14	sseldd 3569	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐽‘𝑋) ∈ (Base‘𝑅))
16	12, 10	unitcl 18482	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑆))
17	16	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆))
18	8, 17	sseldd 3569	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅))
19		eqid 2610	. . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
20	3, 19, 10	subrguss 18618	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
21	20	sselda 3568	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (Unit‘𝑅))
22		subrginv.2	. . . . . . 7 ⊢ 𝐼 = (invr‘𝑅)
23	19, 22, 5	ringinvcl 18499	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ (Base‘𝑅))
24	1, 23	sylan 487	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ (Base‘𝑅))
25	21, 24	syldan 486	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ (Base‘𝑅))
26		eqid 2610	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
27	5, 26	ringass 18387	. . . 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))))
28	2, 15, 18, 25, 27	syl13anc 1320	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))))
29		eqid 2610	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
30		eqid 2610	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
31	10, 11, 29, 30	unitlinv 18500	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
32	9, 31	sylan 487	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑆)𝑋) = (1r‘𝑆))
33	3, 26	ressmulr 15829	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
34	33	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑆))
35	34	oveqd 6566	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = ((𝐽‘𝑋)(.r‘𝑆)𝑋))
36		eqid 2610	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
37	3, 36	subrg1 18613	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝑆))
38	37	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (1r‘𝑅) = (1r‘𝑆))
39	32, 35, 38	3eqtr4d 2654	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)𝑋) = (1r‘𝑅))
40	39	oveq1d 6564	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (((𝐽‘𝑋)(.r‘𝑅)𝑋)(.r‘𝑅)(𝐼‘𝑋)) = ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)))
41	19, 22, 26, 36	unitrinv 18501	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
42	1, 41	sylan 487	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
43	21, 42	syldan 486	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝑋(.r‘𝑅)(𝐼‘𝑋)) = (1r‘𝑅))
44	43	oveq2d 6565	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)(𝑋(.r‘𝑅)(𝐼‘𝑋))) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
45	28, 40, 44	3eqtr3d 2652	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)))
46	5, 26, 36	ringlidm 18394	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼‘𝑋) ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = (𝐼‘𝑋))
47	2, 25, 46	syl2anc 691	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((1r‘𝑅)(.r‘𝑅)(𝐼‘𝑋)) = (𝐼‘𝑋))
48	5, 26, 36	ringridm 18395	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐽‘𝑋) ∈ (Base‘𝑅)) → ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)) = (𝐽‘𝑋))
49	2, 15, 48	syl2anc 691	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → ((𝐽‘𝑋)(.r‘𝑅)(1r‘𝑅)) = (𝐽‘𝑋))
50	45, 47, 49	3eqtr3d 2652	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696  ._rcmulr 15769  1_rcur 18324  Ringcrg 18370  Unitcui 18462  inv_rcinvr 18494  SubRingcsubrg 18599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-subrg 18601

This theorem is referenced by:  subrgdv  18620  subrgunit  18621  subrgugrp  18622  issubdrg  18628  gzrngunit  19631