Theorem invrfvald 14102

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 6023	. . 3 |- ( ph -> ( (mulGrp ` R ) ↾s U ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
3	2	fveq2d 5633	. 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 5633	. 2 |- ( ph -> ( invg ` G ) = ( invg ` ( (mulGrp ` R ) ↾s U ) ) )
6		invrfvald.i	. . 3 |- ( ph -> I = ( invr ` R ) )
7		df-invr 14101	. . . 4 |- invr = ( r e. _V |-> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) )
8		fveq2 5629	. . . . . 6 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
9		fveq2 5629	. . . . . 6 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
10	8, 9	oveq12d 6025	. . . . 5 |- ( r = R -> ( (mulGrp ` r ) ↾s (Unit ` r ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
11	10	fveq2d 5633	. . . 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 2813	. . . 4 |- ( ph -> R e. _V )
14		eqid 2229	. . . . . . . 8 |- (Unit ` R ) = (Unit ` R )
15		eqid 2229	. . . . . . . 8 |- ( (mulGrp ` R ) ↾s (Unit ` R ) ) = ( (mulGrp ` R ) ↾s (Unit ` R ) )
16	14, 15	unitgrp 14096	. . . . . . 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 2229	. . . . . . 7 |- ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
19		eqid 2229	. . . . . . 7 |- ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) )
20	18, 19	grpinvfng 13593	. . . . . 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 13107	. . . . . 6 |- Base Fn _V
23	17	elexd 2813	. . . . . 6 |- ( ph -> ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. _V )
24		funfvex 5646	. . . . . . 7 |- ( ( Fun Base /\ ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. dom Base ) -> ( Base ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) e. _V )
25	24	funfni 5423	. . . . . 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 5865	. . . . 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 5730	. . 3 |- ( ph -> ( invr ` R ) = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
30	6, 29	eqtrd 2262	. 2 |- ( ph -> I = ( invg ` ( (mulGrp ` R ) ↾s (Unit ` R ) ) ) )
31	3, 5, 30	3eqtr4rd 2273	1 |- ( ph -> I = ( invg ` G ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   Basecbs 13048   ↾_s cress 13049   Grpcgrp 13549   invgcminusg 13550  mulGrpcmgp 13899   Ringcrg 13975  Unitcui 14066   invrcinvr 14100

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-iress 13056  df-plusg 13139  df-mulr 13140  df-0g 13307  df-mgm 13405  df-sgrp 13451  df-mnd 13466  df-grp 13552  df-minusg 13553  df-cmn 13839  df-abl 13840  df-mgp 13900  df-ur 13939  df-srg 13943  df-ring 13977  df-oppr 14047  df-dvdsr 14068  df-unit 14069  df-invr 14101

This theorem is referenced by:  unitinvcl  14103  unitinvinv  14104  unitlinv  14106  unitrinv  14107  rdivmuldivd  14124  invrpropdg  14129  subrgugrp  14220