Theorem 1unit 13281

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 2177	. . . 4 |- ( Base ` R ) = ( Base ` R )
2		unit.2	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	ringidcl 13208	. . 3 |- ( R e. Ring -> .1. e. ( Base ` R ) )
4		eqid 2177	. . . 4 |- ( ||r ` R ) = ( ||r ` R )
5	1, 4	dvdsrid 13274	. . 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 2177	. . . 4 |- (oppr ` R ) = (oppr ` R )
8	7	opprring 13254	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
9	7, 1	opprbasg 13252	. . . 4 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
10	3, 9	eleqtrd 2256	. . 3 |- ( R e. Ring -> .1. e. ( Base ` (oppr ` R ) ) )
11		eqid 2177	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
12		eqid 2177	. . . 4 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
13	11, 12	dvdsrid 13274	. . 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 2178	. . 3 |- ( R e. Ring -> ( ||r ` R ) = ( ||r ` R ) )
19		eqidd 2178	. . 3 |- ( R e. Ring -> (oppr ` R ) = (oppr ` R ) )
20		eqidd 2178	. . 3 |- ( R e. Ring -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
21		ringsrg 13229	. . 3 |- ( R e. Ring -> R e. SRing )
22	16, 17, 18, 19, 20, 21	isunitd 13280	. 2 |- ( R e. Ring -> ( .1. e. U <-> ( .1. ( ||r ` R ) .1. /\ .1. ( ||r ` (oppr ` R ) ) .1. ) ) )
23	6, 14, 22	mpbir2and 944	1 |- ( R e. Ring -> .1. e. U )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218   Basecbs 12464   1rcur 13147   Ringcrg 13184  opp_rcoppr 13244   ||_rcdsr 13260  Unitcui 13261

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-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

This theorem is referenced by:  unitgrp  13290  unitgrpid  13292  unitsubm  13293  1rinv  13302  0unit  13303  dvr1  13312  subrgugrp  13366