Theorem srgideu 13676

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 2204	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 13672	. . . 4 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3		eqid 2204	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
4		eqid 2204	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
5	3, 4	mndideu 13200	. . . 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 13628	. . . . . . . 8 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
9	8	oveqd 5960	. . . . . . 7 |- ( R e. SRing -> ( u .x. x ) = ( u ( +g ` (mulGrp ` R ) ) x ) )
10	9	eqeq1d 2213	. . . . . 6 |- ( R e. SRing -> ( ( u .x. x ) = x <-> ( u ( +g ` (mulGrp ` R ) ) x ) = x ) )
11	8	oveqd 5960	. . . . . . 7 |- ( R e. SRing -> ( x .x. u ) = ( x ( +g ` (mulGrp ` R ) ) u ) )
12	11	eqeq1d 2213	. . . . . 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 2505	. . . 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 2689	. . 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 13630	. . 3 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
19		raleq 2701	. . . 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 2715	. . 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 1372   e. wcel 2175   A.wral 2483   E!wreu 2485   ` cfv 5270  (class class class)co 5943   Basecbs 12774   +g cplusg 12851   .rcmulr 12852   Mndcmnd 13190  mulGrpcmgp 13624  SRingcsrg 13667

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-plusg 12864  df-mulr 12865  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-mgp 13625  df-srg 13668

This theorem is referenced by:  issrgid  13685