Theorem invrpropdg 13645

Description: The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
unitpropdg.1	|- ( ph -> B = ( Base ` K ) )
unitpropdg.2	|- ( ph -> B = ( Base ` L ) )
unitpropdg.3	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
unitpropdg.k	|- ( ph -> K e. Ring )
unitpropdg.l	|- ( ph -> L e. Ring )

Assertion

Ref	Expression
invrpropdg	|- ( ph -> ( invr ` K ) = ( invr ` L ) )

Distinct variable groups:   x, y, B   x, K, y   x, L, y   ph, x, y

Proof of Theorem invrpropdg

Step	Hyp	Ref	Expression
1		eqidd 2194	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
2		eqidd 2194	. . . 4 |- ( ph -> ( (mulGrp ` K ) ↾s (Unit ` K ) ) = ( (mulGrp ` K ) ↾s (Unit ` K ) ) )
3		unitpropdg.k	. . . . 5 |- ( ph -> K e. Ring )
4		ringsrg 13543	. . . . 5 |- ( K e. Ring -> K e. SRing )
5	3, 4	syl 14	. . . 4 |- ( ph -> K e. SRing )
6	1, 2, 5	unitgrpbasd 13611	. . 3 |- ( ph -> (Unit ` K ) = ( Base ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
7		unitpropdg.1	. . . . 5 |- ( ph -> B = ( Base ` K ) )
8		unitpropdg.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
9		unitpropdg.3	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
10		unitpropdg.l	. . . . 5 |- ( ph -> L e. Ring )
11	7, 8, 9, 3, 10	unitpropdg 13644	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` L ) )
12		eqidd 2194	. . . . 5 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
13		eqidd 2194	. . . . 5 |- ( ph -> ( (mulGrp ` L ) ↾s (Unit ` L ) ) = ( (mulGrp ` L ) ↾s (Unit ` L ) ) )
14		ringsrg 13543	. . . . . 6 |- ( L e. Ring -> L e. SRing )
15	10, 14	syl 14	. . . . 5 |- ( ph -> L e. SRing )
16	12, 13, 15	unitgrpbasd 13611	. . . 4 |- ( ph -> (Unit ` L ) = ( Base ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
17	11, 16	eqtrd 2226	. . 3 |- ( ph -> (Unit ` K ) = ( Base ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
18		eqid 2193	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
19	18	ringmgp 13498	. . . . 5 |- ( K e. Ring -> (mulGrp ` K ) e. Mnd )
20	3, 19	syl 14	. . . 4 |- ( ph -> (mulGrp ` K ) e. Mnd )
21		basfn 12676	. . . . . . 7 |- Base Fn _V
22	3	elexd 2773	. . . . . . 7 |- ( ph -> K e. _V )
23		funfvex 5571	. . . . . . . 8 |- ( ( Fun Base /\ K e. dom Base ) -> ( Base ` K ) e. _V )
24	23	funfni 5354	. . . . . . 7 |- ( ( Base Fn _V /\ K e. _V ) -> ( Base ` K ) e. _V )
25	21, 22, 24	sylancr 414	. . . . . 6 |- ( ph -> ( Base ` K ) e. _V )
26	7, 25	eqeltrd 2270	. . . . 5 |- ( ph -> B e. _V )
27	7, 1, 5	unitssd 13605	. . . . 5 |- ( ph -> (Unit ` K ) C_ B )
28	26, 27	ssexd 4169	. . . 4 |- ( ph -> (Unit ` K ) e. _V )
29		ressex 12683	. . . 4 |- ( ( (mulGrp ` K ) e. Mnd /\ (Unit ` K ) e. _V ) -> ( (mulGrp ` K ) ↾s (Unit ` K ) ) e. _V )
30	20, 28, 29	syl2anc 411	. . 3 |- ( ph -> ( (mulGrp ` K ) ↾s (Unit ` K ) ) e. _V )
31		eqid 2193	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
32	31	ringmgp 13498	. . . . 5 |- ( L e. Ring -> (mulGrp ` L ) e. Mnd )
33	10, 32	syl 14	. . . 4 |- ( ph -> (mulGrp ` L ) e. Mnd )
34	11, 28	eqeltrrd 2271	. . . 4 |- ( ph -> (Unit ` L ) e. _V )
35		ressex 12683	. . . 4 |- ( ( (mulGrp ` L ) e. Mnd /\ (Unit ` L ) e. _V ) -> ( (mulGrp ` L ) ↾s (Unit ` L ) ) e. _V )
36	33, 34, 35	syl2anc 411	. . 3 |- ( ph -> ( (mulGrp ` L ) ↾s (Unit ` L ) ) e. _V )
37	27	sselda 3179	. . . . . 6 |- ( ( ph /\ x e. (Unit ` K ) ) -> x e. B )
38	27	sselda 3179	. . . . . 6 |- ( ( ph /\ y e. (Unit ` K ) ) -> y e. B )
39	37, 38	anim12dan 600	. . . . 5 |- ( ( ph /\ ( x e. (Unit ` K ) /\ y e. (Unit ` K ) ) ) -> ( x e. B /\ y e. B ) )
40	39, 9	syldan 282	. . . 4 |- ( ( ph /\ ( x e. (Unit ` K ) /\ y e. (Unit ` K ) ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
41		eqid 2193	. . . . . . . 8 |- ( .r ` K ) = ( .r ` K )
42	18, 41	mgpplusgg 13420	. . . . . . 7 |- ( K e. Ring -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
43	3, 42	syl 14	. . . . . 6 |- ( ph -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
44	2, 43, 28, 20	ressplusgd 12746	. . . . 5 |- ( ph -> ( .r ` K ) = ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
45	44	oveqdr 5946	. . . 4 |- ( ( ph /\ ( x e. (Unit ` K ) /\ y e. (Unit ` K ) ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) y ) )
46		eqid 2193	. . . . . . . 8 |- ( .r ` L ) = ( .r ` L )
47	31, 46	mgpplusgg 13420	. . . . . . 7 |- ( L e. Ring -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
48	10, 47	syl 14	. . . . . 6 |- ( ph -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
49	13, 48, 34, 33	ressplusgd 12746	. . . . 5 |- ( ph -> ( .r ` L ) = ( +g ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
50	49	oveqdr 5946	. . . 4 |- ( ( ph /\ ( x e. (Unit ` K ) /\ y e. (Unit ` K ) ) ) -> ( x ( .r ` L ) y ) = ( x ( +g ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) y ) )
51	40, 45, 50	3eqtr3d 2234	. . 3 |- ( ( ph /\ ( x e. (Unit ` K ) /\ y e. (Unit ` K ) ) ) -> ( x ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) y ) = ( x ( +g ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) y ) )
52	6, 17, 30, 36, 51	grpinvpropdg 13147	. 2 |- ( ph -> ( invg ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) = ( invg ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
53		eqidd 2194	. . 3 |- ( ph -> ( invr ` K ) = ( invr ` K ) )
54	1, 2, 53, 3	invrfvald 13618	. 2 |- ( ph -> ( invr ` K ) = ( invg ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
55		eqidd 2194	. . 3 |- ( ph -> ( invr ` L ) = ( invr ` L ) )
56	12, 13, 55, 10	invrfvald 13618	. 2 |- ( ph -> ( invr ` L ) = ( invg ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
57	52, 54, 56	3eqtr4d 2236	1 |- ( ph -> ( invr ` K ) = ( invr ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   .rcmulr 12696   Mndcmnd 12997   invgcminusg 13073  mulGrpcmgp 13416  SRingcsrg 13459   Ringcrg 13492  Unitcui 13583   invrcinvr 13616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586  df-invr 13617

This theorem is referenced by: (None)