Theorem 1unit 14086

Description: The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
unit.1	|- U = (Unit ` R )
unit.2	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
1unit	|- ( R e. Ring -> .1. e. U )

Proof of Theorem 1unit

Step	Hyp	Ref	Expression
1		eqid 2229	. . . 4 |- ( Base ` R ) = ( Base ` R )
2		unit.2	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	ringidcl 13998	. . 3 |- ( R e. Ring -> .1. e. ( Base ` R ) )
4		eqid 2229	. . . 4 |- ( ||r ` R ) = ( ||r ` R )
5	1, 4	dvdsrid 14079	. . 3 |- ( ( R e. Ring /\ .1. e. ( Base ` R ) ) -> .1. ( ||r ` R ) .1. )
6	3, 5	mpdan 421	. 2 |- ( R e. Ring -> .1. ( ||r ` R ) .1. )
7		eqid 2229	. . . 4 |- (oppr ` R ) = (oppr ` R )
8	7	opprring 14057	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
9	7, 1	opprbasg 14053	. . . 4 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
10	3, 9	eleqtrd 2308	. . 3 |- ( R e. Ring -> .1. e. ( Base ` (oppr ` R ) ) )
11		eqid 2229	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
12		eqid 2229	. . . 4 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
13	11, 12	dvdsrid 14079	. . 3 |- ( ( (oppr ` R ) e. Ring /\ .1. e. ( Base ` (oppr ` R ) ) ) -> .1. ( ||r ` (oppr ` R ) ) .1. )
14	8, 10, 13	syl2anc 411	. 2 |- ( R e. Ring -> .1. ( ||r ` (oppr ` R ) ) .1. )
15		unit.1	. . . 4 |- U = (Unit ` R )
16	15	a1i 9	. . 3 |- ( R e. Ring -> U = (Unit ` R ) )
17	2	a1i 9	. . 3 |- ( R e. Ring -> .1. = ( 1r ` R ) )
18		eqidd 2230	. . 3 |- ( R e. Ring -> ( ||r ` R ) = ( ||r ` R ) )
19		eqidd 2230	. . 3 |- ( R e. Ring -> (oppr ` R ) = (oppr ` R ) )
20		eqidd 2230	. . 3 |- ( R e. Ring -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
21		ringsrg 14025	. . 3 |- ( R e. Ring -> R e. SRing )
22	16, 17, 18, 19, 20, 21	isunitd 14085	. 2 |- ( R e. Ring -> ( .1. e. U <-> ( .1. ( ||r ` R ) .1. /\ .1. ( ||r ` (oppr ` R ) ) .1. ) ) )
23	6, 14, 22	mpbir2and 950	1 |- ( R e. Ring -> .1. e. U )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318   Basecbs 13047   1rcur 13937   Ringcrg 13974  opp_rcoppr 14045   ||_rcdsr 14064  Unitcui 14065

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 13050  df-slot 13051  df-base 13053  df-sets 13054  df-plusg 13138  df-mulr 13139  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-minusg 13552  df-cmn 13838  df-abl 13839  df-mgp 13899  df-ur 13938  df-srg 13942  df-ring 13976  df-oppr 14046  df-dvdsr 14067  df-unit 14068

This theorem is referenced by:  unitgrp  14095  unitgrpid  14097  unitsubm  14098  1rinv  14107  0unit  14108  dvr1  14117  subrgugrp  14219