Theorem srgideu 14250

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 2238	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	srgmgp 14246	. . . 4 |- ( R e. SRing -> (mulGrp ` R ) e. Mnd )
3		eqid 2238	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
4		eqid 2238	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
5	3, 4	mndideu 13716	. . . 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 14198	. . . . . . . 8 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
9	8	oveqd 6092	. . . . . . 7 |- ( R e. SRing -> ( u .x. x ) = ( u ( +g ` (mulGrp ` R ) ) x ) )
10	9	eqeq1d 2247	. . . . . 6 |- ( R e. SRing -> ( ( u .x. x ) = x <-> ( u ( +g ` (mulGrp ` R ) ) x ) = x ) )
11	8	oveqd 6092	. . . . . . 7 |- ( R e. SRing -> ( x .x. u ) = ( x ( +g ` (mulGrp ` R ) ) u ) )
12	11	eqeq1d 2247	. . . . . 6 |- ( R e. SRing -> ( ( x .x. u ) = x <-> ( x ( +g ` (mulGrp ` R ) ) u ) = x ) )
13	10, 12	anbi12d 477	. . . . 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 2550	. . . 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 2737	. . 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 14200	. . 3 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
19		raleq 2749	. . . 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 2763	. . 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 1402   e. wcel 2209   A.wral 2528   E!wreu 2530   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   .rcmulr 13409   Mndcmnd 13706  mulGrpcmgp 14194  SRingcsrg 14241

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-sep 4244  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-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-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-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  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-mgp 14195  df-srg 14242

This theorem is referenced by:  issrgid  14259