Theorem invrpropdg 14026

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 2208	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` K ) )
2		eqidd 2208	. . . 4 |- ( ph -> ( (mulGrp ` K ) ↾s (Unit ` K ) ) = ( (mulGrp ` K ) ↾s (Unit ` K ) ) )
3		unitpropdg.k	. . . . 5 |- ( ph -> K e. Ring )
4		ringsrg 13924	. . . . 5 |- ( K e. Ring -> K e. SRing )
5	3, 4	syl 14	. . . 4 |- ( ph -> K e. SRing )
6	1, 2, 5	unitgrpbasd 13992	. . 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 14025	. . . 4 |- ( ph -> (Unit ` K ) = (Unit ` L ) )
12		eqidd 2208	. . . . 5 |- ( ph -> (Unit ` L ) = (Unit ` L ) )
13		eqidd 2208	. . . . 5 |- ( ph -> ( (mulGrp ` L ) ↾s (Unit ` L ) ) = ( (mulGrp ` L ) ↾s (Unit ` L ) ) )
14		ringsrg 13924	. . . . . 6 |- ( L e. Ring -> L e. SRing )
15	10, 14	syl 14	. . . . 5 |- ( ph -> L e. SRing )
16	12, 13, 15	unitgrpbasd 13992	. . . 4 |- ( ph -> (Unit ` L ) = ( Base ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
17	11, 16	eqtrd 2240	. . 3 |- ( ph -> (Unit ` K ) = ( Base ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
18		eqid 2207	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
19	18	ringmgp 13879	. . . . 5 |- ( K e. Ring -> (mulGrp ` K ) e. Mnd )
20	3, 19	syl 14	. . . 4 |- ( ph -> (mulGrp ` K ) e. Mnd )
21		basfn 13005	. . . . . . 7 |- Base Fn _V
22	3	elexd 2790	. . . . . . 7 |- ( ph -> K e. _V )
23		funfvex 5616	. . . . . . . 8 |- ( ( Fun Base /\ K e. dom Base ) -> ( Base ` K ) e. _V )
24	23	funfni 5395	. . . . . . 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 2284	. . . . 5 |- ( ph -> B e. _V )
27	7, 1, 5	unitssd 13986	. . . . 5 |- ( ph -> (Unit ` K ) C_ B )
28	26, 27	ssexd 4200	. . . 4 |- ( ph -> (Unit ` K ) e. _V )
29		ressex 13012	. . . 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 2207	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
32	31	ringmgp 13879	. . . . 5 |- ( L e. Ring -> (mulGrp ` L ) e. Mnd )
33	10, 32	syl 14	. . . 4 |- ( ph -> (mulGrp ` L ) e. Mnd )
34	11, 28	eqeltrrd 2285	. . . 4 |- ( ph -> (Unit ` L ) e. _V )
35		ressex 13012	. . . 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 3201	. . . . . 6 |- ( ( ph /\ x e. (Unit ` K ) ) -> x e. B )
38	27	sselda 3201	. . . . . 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 2207	. . . . . . . 8 |- ( .r ` K ) = ( .r ` K )
42	18, 41	mgpplusgg 13801	. . . . . . 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 13076	. . . . 5 |- ( ph -> ( .r ` K ) = ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
45	44	oveqdr 5995	. . . 4 |- ( ( ph /\ ( x e. (Unit ` K ) /\ y e. (Unit ` K ) ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) y ) )
46		eqid 2207	. . . . . . . 8 |- ( .r ` L ) = ( .r ` L )
47	31, 46	mgpplusgg 13801	. . . . . . 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 13076	. . . . 5 |- ( ph -> ( .r ` L ) = ( +g ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
50	49	oveqdr 5995	. . . 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 2248	. . 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 13522	. 2 |- ( ph -> ( invg ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) = ( invg ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
53		eqidd 2208	. . 3 |- ( ph -> ( invr ` K ) = ( invr ` K ) )
54	1, 2, 53, 3	invrfvald 13999	. 2 |- ( ph -> ( invr ` K ) = ( invg ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
55		eqidd 2208	. . 3 |- ( ph -> ( invr ` L ) = ( invr ` L ) )
56	12, 13, 55, 10	invrfvald 13999	. 2 |- ( ph -> ( invr ` L ) = ( invg ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
57	52, 54, 56	3eqtr4d 2250	1 |- ( ph -> ( invr ` K ) = ( invr ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   +g cplusg 13024   .rcmulr 13025   Mndcmnd 13363   invgcminusg 13448  mulGrpcmgp 13797  SRingcsrg 13840   Ringcrg 13873  Unitcui 13964   invrcinvr 13997

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-tpos 6354  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-cmn 13737  df-abl 13738  df-mgp 13798  df-ur 13837  df-srg 13841  df-ring 13875  df-oppr 13945  df-dvdsr 13966  df-unit 13967  df-invr 13998

This theorem is referenced by: (None)