Theorem 1unit 14120

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 2231	. . . 4 |- ( Base ` R ) = ( Base ` R )
2		unit.2	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	ringidcl 14032	. . 3 |- ( R e. Ring -> .1. e. ( Base ` R ) )
4		eqid 2231	. . . 4 |- ( ||r ` R ) = ( ||r ` R )
5	1, 4	dvdsrid 14113	. . 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 2231	. . . 4 |- (oppr ` R ) = (oppr ` R )
8	7	opprring 14091	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
9	7, 1	opprbasg 14087	. . . 4 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
10	3, 9	eleqtrd 2310	. . 3 |- ( R e. Ring -> .1. e. ( Base ` (oppr ` R ) ) )
11		eqid 2231	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
12		eqid 2231	. . . 4 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
13	11, 12	dvdsrid 14113	. . 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 2232	. . 3 |- ( R e. Ring -> ( ||r ` R ) = ( ||r ` R ) )
19		eqidd 2232	. . 3 |- ( R e. Ring -> (oppr ` R ) = (oppr ` R ) )
20		eqidd 2232	. . 3 |- ( R e. Ring -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
21		ringsrg 14059	. . 3 |- ( R e. Ring -> R e. SRing )
22	16, 17, 18, 19, 20, 21	isunitd 14119	. 2 |- ( R e. Ring -> ( .1. e. U <-> ( .1. ( ||r ` R ) .1. /\ .1. ( ||r ` (oppr ` R ) ) .1. ) ) )
23	6, 14, 22	mpbir2and 952	1 |- ( R e. Ring -> .1. e. U )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326   Basecbs 13081   1rcur 13971   Ringcrg 14008  opp_rcoppr 14079   ||_rcdsr 14098  Unitcui 14099

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-oppr 14080  df-dvdsr 14101  df-unit 14102

This theorem is referenced by:  unitgrp  14129  unitgrpid  14131  unitsubm  14132  1rinv  14141  0unit  14142  dvr1  14151  subrgugrp  14253