Theorem invrfvald 13296

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 5893	. . 3 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
3	2	fveq2d 5521	. 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 5521	. 2 |- ( ph -> ( invg ` G ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
6		invrfvald.i	. . 3 |- ( ph -> I = ( invr ` R ) )
7		df-invr 13295	. . . 4 |- invr = ( r e. _V |-> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) )
8		fveq2 5517	. . . . . 6 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
9		fveq2 5517	. . . . . 6 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
10	8, 9	oveq12d 5895	. . . . 5 |- ( r = R -> ( (mulGrp ` r ) ↾s (Unit ` r ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
11	10	fveq2d 5521	. . . 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 2752	. . . 4 |- ( ph -> R e. _V )
14		eqid 2177	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
15		eqid 2177	. . . . . . . 8 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
16	14, 15	unitgrp 13290	. . . . . . 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 2177	. . . . . . 7 |- ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
19		eqid 2177	. . . . . . 7 |- ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
20	18, 19	grpinvfng 12922	. . . . . 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 12522	. . . . . 6 |- Base Fn _V
23	17	elexd 2752	. . . . . 6 |- ( ph -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. _V )
24		funfvex 5534	. . . . . . 7 |- ( ( Fun Base /\ ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. dom Base ) -> ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
25	24	funfni 5318	. . . . . 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 5740	. . . . 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 5611	. . 3 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
30	6, 29	eqtrd 2210	. 2 |- ( ph -> I = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
31	3, 5, 30	3eqtr4rd 2221	1 |- ( ph -> I = ( invg ` G ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2739   Fn wfn 5213   ` cfv 5218  (class class class)co 5877   Basecbs 12464   ↾_s cress 12465   Grpcgrp 12882   invgcminusg 12883  mulGrpcmgp 13135   Ringcrg 13184  Unitcui 13261   invrcinvr 13294

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-tpos 6248  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-cmn 13095  df-abl 13096  df-mgp 13136  df-ur 13148  df-srg 13152  df-ring 13186  df-oppr 13245  df-dvdsr 13263  df-unit 13264  df-invr 13295

This theorem is referenced by:  unitinvcl  13297  unitinvinv  13298  unitlinv  13300  unitrinv  13301  rdivmuldivd  13318  invrpropdg  13323  subrgugrp  13366