Theorem invrpropdg 14244

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 14141	. . . . 5 |- ( K e. Ring -> K e. SRing )
5	3, 4	syl 14	. . . 4 |- ( ph -> K e. SRing )
6	1, 2, 5	unitgrpbasd 14210	. . 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 14243	. . . 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 14141	. . . . . 6 |- ( L e. Ring -> L e. SRing )
15	10, 14	syl 14	. . . . 5 |- ( ph -> L e. SRing )
16	12, 13, 15	unitgrpbasd 14210	. . . 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 14096	. . . . 5 |- ( K e. Ring -> (mulGrp ` K ) e. Mnd )
20	3, 19	syl 14	. . . 4 |- ( ph -> (mulGrp ` K ) e. Mnd )
21		basfn 13221	. . . . . . 7 |- Base Fn _V
22	3	elexd 2817	. . . . . . 7 |- ( ph -> K e. _V )
23		funfvex 5665	. . . . . . . 8 |- ( ( Fun Base /\ K e. dom Base ) -> ( Base ` K ) e. _V )
24	23	funfni 5439	. . . . . . 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 14204	. . . . 5 |- ( ph -> (Unit ` K ) C_ B )
28	26, 27	ssexd 4234	. . . 4 |- ( ph -> (Unit ` K ) e. _V )
29		ressex 13228	. . . 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 14096	. . . . 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 13228	. . . 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 3228	. . . . . 6 |- ( ( ph /\ x e. (Unit ` K ) ) -> x e. B )
38	27	sselda 3228	. . . . . 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 14018	. . . . . . 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 13292	. . . . 5 |- ( ph -> ( .r ` K ) = ( +g ` ( (mulGrp ` K ) ↾s (Unit ` K ) ) ) )
45	44	oveqdr 6056	. . . 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 14018	. . . . . . 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 13292	. . . . 5 |- ( ph -> ( .r ` L ) = ( +g ` ( (mulGrp ` L ) ↾s (Unit ` L ) ) ) )
50	49	oveqdr 6056	. . . 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 13738	. 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 14217	. 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 14217	. 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 1398   e. wcel 2202   _Vcvv 2803   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   Basecbs 13162   ↾_s cress 13163   +g cplusg 13240   .rcmulr 13241   Mndcmnd 13579   invgcminusg 13664  mulGrpcmgp 14014  SRingcsrg 14057   Ringcrg 14090  Unitcui 14181   invrcinvr 14215

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-cmn 13953  df-abl 13954  df-mgp 14015  df-ur 14054  df-srg 14058  df-ring 14092  df-oppr 14162  df-dvdsr 14183  df-unit 14184  df-invr 14216

This theorem is referenced by: (None)