Theorem invrpropdg 14162

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 2232	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
2		eqidd 2232	. . . 4 |- ( ph -> ( (mulGrp ` K ) ↾s (Unit ` K ) ) = ( (mulGrp ` K ) ↾s (Unit ` K ) ) )
3		unitpropdg.k	. . . . 5 |- ( ph -> K e. Ring )
4		ringsrg 14059	. . . . 5 |- ( K e. Ring -> K e. SRing )
5	3, 4	syl 14	. . . 4 |- ( ph -> K e. SRing )
6	1, 2, 5	unitgrpbasd 14128	. . 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 14161	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` L ) )
12		eqidd 2232	. . . . 5 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
13		eqidd 2232	. . . . 5 |- ( ph -> ( (mulGrp ` L ) ↾s (Unit ` L ) ) = ( (mulGrp ` L ) ↾s (Unit ` L ) ) )
14		ringsrg 14059	. . . . . 6 |- ( L e. Ring -> L e. SRing )
15	10, 14	syl 14	. . . . 5 |- ( ph -> L e. SRing )
16	12, 13, 15	unitgrpbasd 14128	. . . 4 |- ( ph -> (Unit ` L ) = ( Base ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
17	11, 16	eqtrd 2264	. . 3 |- ( ph -> (Unit ` K ) = ( Base ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
18		eqid 2231	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
19	18	ringmgp 14014	. . . . 5 |- ( K e. Ring -> (mulGrp ` K ) e. Mnd )
20	3, 19	syl 14	. . . 4 |- ( ph -> (mulGrp ` K ) e. Mnd )
21		basfn 13140	. . . . . . 7 |- Base Fn _V
22	3	elexd 2816	. . . . . . 7 |- ( ph -> K e. _V )
23		funfvex 5656	. . . . . . . 8 |- ( ( Fun Base /\ K e. dom Base ) -> ( Base ` K ) e. _V )
24	23	funfni 5432	. . . . . . 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 2308	. . . . 5 |- ( ph -> B e. _V )
27	7, 1, 5	unitssd 14122	. . . . 5 |- ( ph -> (Unit ` K ) C_ B )
28	26, 27	ssexd 4229	. . . 4 |- ( ph -> (Unit ` K ) e. _V )
29		ressex 13147	. . . 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 2231	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
32	31	ringmgp 14014	. . . . 5 |- ( L e. Ring -> (mulGrp ` L ) e. Mnd )
33	10, 32	syl 14	. . . 4 |- ( ph -> (mulGrp ` L ) e. Mnd )
34	11, 28	eqeltrrd 2309	. . . 4 |- ( ph -> (Unit ` L ) e. _V )
35		ressex 13147	. . . 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 3227	. . . . . 6 |- ( ( ph /\ x e. (Unit ` K ) ) -> x e. B )
38	27	sselda 3227	. . . . . 6 |- ( ( ph /\ y e. (Unit ` K ) ) -> y e. B )
39	37, 38	anim12dan 604	. . . . 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 2231	. . . . . . . 8 |- ( .r ` K ) = ( .r ` K )
42	18, 41	mgpplusgg 13936	. . . . . . 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 13211	. . . . 5 |- ( ph -> ( .r ` K ) = ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
45	44	oveqdr 6045	. . . 4 |- ( ( ph /\ ( x e. (Unit ` K ) /\ y e. (Unit ` K ) ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) y ) )
46		eqid 2231	. . . . . . . 8 |- ( .r ` L ) = ( .r ` L )
47	31, 46	mgpplusgg 13936	. . . . . . 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 13211	. . . . 5 |- ( ph -> ( .r ` L ) = ( +g ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
50	49	oveqdr 6045	. . . 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 2272	. . 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 13657	. 2 |- ( ph -> ( invg ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) = ( invg ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
53		eqidd 2232	. . 3 |- ( ph -> ( invr ` K ) = ( invr ` K ) )
54	1, 2, 53, 3	invrfvald 14135	. 2 |- ( ph -> ( invr ` K ) = ( invg ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
55		eqidd 2232	. . 3 |- ( ph -> ( invr ` L ) = ( invr ` L ) )
56	12, 13, 55, 10	invrfvald 14135	. 2 |- ( ph -> ( invr ` L ) = ( invg ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
57	52, 54, 56	3eqtr4d 2274	1 |- ( ph -> ( invr ` K ) = ( invr ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   +g cplusg 13159   .rcmulr 13160   Mndcmnd 13498   invgcminusg 13583  mulGrpcmgp 13932  SRingcsrg 13975   Ringcrg 14008  Unitcui 14099   invrcinvr 14133

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-oppr 14080  df-dvdsr 14101  df-unit 14102  df-invr 14134

This theorem is referenced by: (None)