Theorem invrfvald 13678

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 5938	. . 3 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
3	2	fveq2d 5562	. 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 5562	. 2 |- ( ph -> ( invg ` G ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
6		invrfvald.i	. . 3 |- ( ph -> I = ( invr ` R ) )
7		df-invr 13677	. . . 4 |- invr = ( r e. _V |-> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) )
8		fveq2 5558	. . . . . 6 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
9		fveq2 5558	. . . . . 6 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
10	8, 9	oveq12d 5940	. . . . 5 |- ( r = R -> ( (mulGrp ` r ) ↾s (Unit ` r ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
11	10	fveq2d 5562	. . . 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 2776	. . . 4 |- ( ph -> R e. _V )
14		eqid 2196	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
15		eqid 2196	. . . . . . . 8 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
16	14, 15	unitgrp 13672	. . . . . . 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 2196	. . . . . . 7 |- ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
19		eqid 2196	. . . . . . 7 |- ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
20	18, 19	grpinvfng 13176	. . . . . 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 12736	. . . . . 6 |- Base Fn _V
23	17	elexd 2776	. . . . . 6 |- ( ph -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. _V )
24		funfvex 5575	. . . . . . 7 |- ( ( Fun Base /\ ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. dom Base ) -> ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
25	24	funfni 5358	. . . . . 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 5784	. . . . 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 5655	. . 3 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
30	6, 29	eqtrd 2229	. 2 |- ( ph -> I = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
31	3, 5, 30	3eqtr4rd 2240	1 |- ( ph -> I = ( invg ` G ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   Grpcgrp 13132   invgcminusg 13133  mulGrpcmgp 13476   Ringcrg 13552  Unitcui 13643   invrcinvr 13676

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-oppr 13624  df-dvdsr 13645  df-unit 13646  df-invr 13677

This theorem is referenced by:  unitinvcl  13679  unitinvinv  13680  unitlinv  13682  unitrinv  13683  rdivmuldivd  13700  invrpropdg  13705  subrgugrp  13796