Theorem 1unit 14387

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 2238	. . . 4 |- ( Base ` R ) = ( Base ` R )
2		unit.2	. . . 4 |- .1. = ( 1r ` R )
3	1, 2	ringidcl 14298	. . 3 |- ( R e. Ring -> .1. e. ( Base ` R ) )
4		eqid 2238	. . . 4 |- ( ||r ` R ) = ( ||r ` R )
5	1, 4	dvdsrid 14380	. . 3 |- ( ( R e. Ring /\ .1. e. ( Base ` R ) ) -> .1. ( ||r ` R ) .1. )
6	3, 5	mpdan 425	. 2 |- ( R e. Ring -> .1. ( ||r ` R ) .1. )
7		eqid 2238	. . . 4 |- (oppr ` R ) = (oppr ` R )
8	7	opprring 14357	. . 3 |- ( R e. Ring -> (oppr ` R ) e. Ring )
9	7, 1	opprbasg 14353	. . . 4 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
10	3, 9	eleqtrd 2317	. . 3 |- ( R e. Ring -> .1. e. ( Base ` (oppr ` R ) ) )
11		eqid 2238	. . . 4 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
12		eqid 2238	. . . 4 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
13	11, 12	dvdsrid 14380	. . 3 |- ( ( (oppr ` R ) e. Ring /\ .1. e. ( Base ` (oppr ` R ) ) ) -> .1. ( ||r ` (oppr ` R ) ) .1. )
14	8, 10, 13	syl2anc 415	. 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 2239	. . 3 |- ( R e. Ring -> ( ||r ` R ) = ( ||r ` R ) )
19		eqidd 2239	. . 3 |- ( R e. Ring -> (oppr ` R ) = (oppr ` R ) )
20		eqidd 2239	. . 3 |- ( R e. Ring -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
21		ringsrg 14325	. . 3 |- ( R e. Ring -> R e. SRing )
22	16, 17, 18, 19, 20, 21	isunitd 14386	. 2 |- ( R e. Ring -> ( .1. e. U <-> ( .1. ( ||r ` R ) .1. /\ .1. ( ||r ` (oppr ` R ) ) .1. ) ) )
23	6, 14, 22	mpbir2and 957	1 |- ( R e. Ring -> .1. e. U )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372   Basecbs 13330   1rcur 14237   Ringcrg 14274  opp_rcoppr 14345   ||_rcdsr 14365  Unitcui 14366

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-tpos 6506  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-cmn 14066  df-abl 14067  df-mgp 14195  df-ur 14238  df-srg 14242  df-ring 14276  df-oppr 14346  df-dvdsr 14368  df-unit 14369

This theorem is referenced by:  unitgrp  14396  unitgrpid  14398  unitsubm  14399  1rinv  14408  0unit  14409  dvr1  14418  subrgugrp  14521  aprnzr  14572  aprlring  14573