Theorem srgideu 13849

Description: The unity element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
srgcl.b	|- B = ( Base ` R )
srgcl.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
srgideu	|- ( R e. SRing -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )

Distinct variable groups:   x, u, B   u, R, x   u, .x. , x

Proof of Theorem srgideu

Step	Hyp	Ref	Expression
1		eqid 2207	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13845	. . . 4 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3		eqid 2207	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
4		eqid 2207	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
5	3, 4	mndideu 13373	. . . 4 |- ( (mulGrp ` R ) e. Mnd -> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) )
6	2, 5	syl 14	. . 3 |- ( R e. SRing -> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) )
7		srgcl.t	. . . . . . . . 9 |- .x. = ( .r ` R )
8	1, 7	mgpplusgg 13801	. . . . . . . 8 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
9	8	oveqd 5984	. . . . . . 7 |- ( R e. SRing -> ( u .x. x ) = ( u ( +g ` (mulGrp ` R ) ) x ) )
10	9	eqeq1d 2216	. . . . . 6 |- ( R e. SRing -> ( ( u .x. x ) = x <-> ( u ( +g ` (mulGrp ` R ) ) x ) = x ) )
11	8	oveqd 5984	. . . . . . 7 |- ( R e. SRing -> ( x .x. u ) = ( x ( +g ` (mulGrp ` R ) ) u ) )
12	11	eqeq1d 2216	. . . . . 6 |- ( R e. SRing -> ( ( x .x. u ) = x <-> ( x ( +g ` (mulGrp ` R ) ) u ) = x ) )
13	10, 12	anbi12d 473	. . . . 5 |- ( R e. SRing -> ( ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
14	13	ralbidv 2508	. . . 4 |- ( R e. SRing -> ( A. x e. ( Base ` (mulGrp ` R ) ) ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
15	14	reubidv 2693	. . 3 |- ( R e. SRing -> ( E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
16	6, 15	mpbird 167	. 2 |- ( R e. SRing -> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )
17		srgcl.b	. . . 4 |- B = ( Base ` R )
18	1, 17	mgpbasg 13803	. . 3 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
19		raleq 2705	. . . 4 |- ( B = ( Base ` (mulGrp ` R ) ) -> ( A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( u .x. x ) = x /\ ( x .x. u ) = x ) ) )
20	19	reueqd 2719	. . 3 |- ( B = ( Base ` (mulGrp ` R ) ) -> ( E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u .x. x ) = x /\ ( x .x. u ) = x ) ) )
21	18, 20	syl 14	. 2 |- ( R e. SRing -> ( E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u .x. x ) = x /\ ( x .x. u ) = x ) ) )
22	16, 21	mpbird 167	1 |- ( R e. SRing -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   E!wreu 2488   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   Mndcmnd 13363  mulGrpcmgp 13797  SRingcsrg 13840

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-mgp 13798  df-srg 13841

This theorem is referenced by:  issrgid  13858