Theorem issrgid 13480

Description: Properties showing that an element I is the unity element of a semiring. (Contributed by NM, 7-Aug-2013.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
srgidm.b	|- B = ( Base ` R )
srgidm.t	|- .x. = ( .r ` R )
srgidm.u	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
issrgid	|- ( R e. SRing -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> .1. = I ) )

Distinct variable groups:   x, B   x, I   x, R   x, .x.   x, .1.

Proof of Theorem issrgid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . 3 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
2		eqid 2193	. . 3 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
3		eqid 2193	. . 3 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
4		srgidm.b	. . . . . 6 |- B = ( Base ` R )
5		srgidm.t	. . . . . 6 |- .x. = ( .r ` R )
6	4, 5	srgideu 13471	. . . . 5 |- ( R e. SRing -> E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
7		reurex 2712	. . . . 5 |- ( E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) -> E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
8	6, 7	syl 14	. . . 4 |- ( R e. SRing -> E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
9		eqid 2193	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
10	9, 4	mgpbasg 13425	. . . . 5 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
11	9, 5	mgpplusgg 13423	. . . . . . . . 9 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
12	11	oveqd 5936	. . . . . . . 8 |- ( R e. SRing -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
13	12	eqeq1d 2202	. . . . . . 7 |- ( R e. SRing -> ( ( y .x. x ) = x <-> ( y ( +g ` (mulGrp ` R ) ) x ) = x ) )
14	11	oveqd 5936	. . . . . . . 8 |- ( R e. SRing -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	14	eqeq1d 2202	. . . . . . 7 |- ( R e. SRing -> ( ( x .x. y ) = x <-> ( x ( +g ` (mulGrp ` R ) ) y ) = x ) )
16	13, 15	anbi12d 473	. . . . . 6 |- ( R e. SRing -> ( ( ( y .x. x ) = x /\ ( x .x. y ) = x ) <-> ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) ) )
17	10, 16	raleqbidv 2706	. . . . 5 |- ( R e. SRing -> ( A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) ) )
18	10, 17	rexeqbidv 2707	. . . 4 |- ( R e. SRing -> ( E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) <-> E. y e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) ) )
19	8, 18	mpbid 147	. . 3 |- ( R e. SRing -> E. y e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) )
20	1, 2, 3, 19	ismgmid 12963	. 2 |- ( R e. SRing -> ( ( I e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) <-> ( 0g ` (mulGrp ` R ) ) = I ) )
21	10	eleq2d 2263	. . 3 |- ( R e. SRing -> ( I e. B <-> I e. ( Base ` (mulGrp ` R ) ) ) )
22	11	oveqd 5936	. . . . . 6 |- ( R e. SRing -> ( I .x. x ) = ( I ( +g ` (mulGrp ` R ) ) x ) )
23	22	eqeq1d 2202	. . . . 5 |- ( R e. SRing -> ( ( I .x. x ) = x <-> ( I ( +g ` (mulGrp ` R ) ) x ) = x ) )
24	11	oveqd 5936	. . . . . 6 |- ( R e. SRing -> ( x .x. I ) = ( x ( +g ` (mulGrp ` R ) ) I ) )
25	24	eqeq1d 2202	. . . . 5 |- ( R e. SRing -> ( ( x .x. I ) = x <-> ( x ( +g ` (mulGrp ` R ) ) I ) = x ) )
26	23, 25	anbi12d 473	. . . 4 |- ( R e. SRing -> ( ( ( I .x. x ) = x /\ ( x .x. I ) = x ) <-> ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) )
27	10, 26	raleqbidv 2706	. . 3 |- ( R e. SRing -> ( A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) )
28	21, 27	anbi12d 473	. 2 |- ( R e. SRing -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> ( I e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) ) )
29		srgidm.u	. . . 4 |- .1. = ( 1r ` R )
30	9, 29	ringidvalg 13460	. . 3 |- ( R e. SRing -> .1. = ( 0g ` (mulGrp ` R ) ) )
31	30	eqeq1d 2202	. 2 |- ( R e. SRing -> ( .1. = I <-> ( 0g ` (mulGrp ` R ) ) = I ) )
32	20, 28, 31	3bitr4d 220	1 |- ( R e. SRing -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> .1. = I ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   E!wreu 2474   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   .rcmulr 12699   0gc0g 12870  mulGrpcmgp 13419   1rcur 13458  SRingcsrg 13462

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mgp 13420  df-ur 13459  df-srg 13463

This theorem is referenced by: (None)