Theorem invrfvald 13999

Description: Multiplicative inverse function for a ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)

Hypotheses

Ref	Expression
invrfvald.u	|- ( ph -> U = (Unit ` R ) )
invrfvald.g	|- ( ph -> G = ( (mulGrp ` R ) ↾s U ) )
invrfvald.i	|- ( ph -> I = ( invr ` R ) )
invrfvald.r	|- ( ph -> R e. Ring )

Assertion

Ref	Expression
invrfvald	|- ( ph -> I = ( invg ` G ) )

Proof of Theorem invrfvald

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		invrfvald.u	. . . 4 |- ( ph -> U = (Unit ` R ) )
2	1	oveq2d 5983	. . 3 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
3	2	fveq2d 5603	. 2 |- ( ph -> ( invg ` ( (mulGrp ` R ) ↾s U ) ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
4		invrfvald.g	. . 3 |- ( ph -> G = ( (mulGrp ` R ) ↾s U ) )
5	4	fveq2d 5603	. 2 |- ( ph -> ( invg ` G ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
6		invrfvald.i	. . 3 |- ( ph -> I = ( invr ` R ) )
7		df-invr 13998	. . . 4 |- invr = ( r e. _V |-> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) )
8		fveq2 5599	. . . . . 6 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
9		fveq2 5599	. . . . . 6 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
10	8, 9	oveq12d 5985	. . . . 5 |- ( r = R -> ( (mulGrp ` r ) ↾s (Unit ` r ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
11	10	fveq2d 5603	. . . 4 |- ( r = R -> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
12		invrfvald.r	. . . . 5 |- ( ph -> R e. Ring )
13	12	elexd 2790	. . . 4 |- ( ph -> R e. _V )
14		eqid 2207	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
15		eqid 2207	. . . . . . . 8 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
16	14, 15	unitgrp 13993	. . . . . . 7 |- ( R e. Ring -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
17	12, 16	syl 14	. . . . . 6 |- ( ph -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp )
18		eqid 2207	. . . . . . 7 |- ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
19		eqid 2207	. . . . . . 7 |- ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
20	18, 19	grpinvfng 13491	. . . . . 6 |- ( ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. Grp -> ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) Fn ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
21	17, 20	syl 14	. . . . 5 |- ( ph -> ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) Fn ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
22		basfn 13005	. . . . . 6 |- Base Fn _V
23	17	elexd 2790	. . . . . 6 |- ( ph -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. _V )
24		funfvex 5616	. . . . . . 7 |- ( ( Fun Base /\ ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. dom Base ) -> ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
25	24	funfni 5395	. . . . . 6 |- ( ( Base Fn _V /\ ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. _V ) -> ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
26	22, 23, 25	sylancr 414	. . . . 5 |- ( ph -> ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
27		fnex 5829	. . . . 5 |- ( ( ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) Fn ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) /\ ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V ) -> ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
28	21, 26, 27	syl2anc 411	. . . 4 |- ( ph -> ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
29	7, 11, 13, 28	fvmptd3 5696	. . 3 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
30	6, 29	eqtrd 2240	. 2 |- ( ph -> I = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
31	3, 5, 30	3eqtr4rd 2251	1 |- ( ph -> I = ( invg ` G ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   _Vcvv 2776   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   Grpcgrp 13447   invgcminusg 13448  mulGrpcmgp 13797   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:  unitinvcl  14000  unitinvinv  14001  unitlinv  14003  unitrinv  14004  rdivmuldivd  14021  invrpropdg  14026  subrgugrp  14117