Theorem issrgid 12957

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 2175	. . 3 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
2		eqid 2175	. . 3 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
3		eqid 2175	. . 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 12948	. . . . 5 |- ( R e. SRing -> E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
7		reurex 2688	. . . . 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 2175	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
10	9, 4	mgpbasg 12930	. . . . 5 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
11	9, 5	mgpplusgg 12929	. . . . . . . . 9 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
12	11	oveqd 5882	. . . . . . . 8 |- ( R e. SRing -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
13	12	eqeq1d 2184	. . . . . . 7 |- ( R e. SRing -> ( ( y .x. x ) = x <-> ( y ( +g ` (mulGrp ` R ) ) x ) = x ) )
14	11	oveqd 5882	. . . . . . . 8 |- ( R e. SRing -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	14	eqeq1d 2184	. . . . . . 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 2682	. . . . 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 2683	. . . 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 12661	. 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 2245	. . 3 |- ( R e. SRing -> ( I e. B <-> I e. ( Base ` (mulGrp ` R ) ) ) )
22	11	oveqd 5882	. . . . . 6 |- ( R e. SRing -> ( I .x. x ) = ( I ( +g ` (mulGrp ` R ) ) x ) )
23	22	eqeq1d 2184	. . . . 5 |- ( R e. SRing -> ( ( I .x. x ) = x <-> ( I ( +g ` (mulGrp ` R ) ) x ) = x ) )
24	11	oveqd 5882	. . . . . 6 |- ( R e. SRing -> ( x .x. I ) = ( x ( +g ` (mulGrp ` R ) ) I ) )
25	24	eqeq1d 2184	. . . . 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 2682	. . 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 12937	. . 3 |- ( R e. SRing -> .1. = ( 0g ` (mulGrp ` R ) ) )
31	30	eqeq1d 2184	. 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 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   E!wreu 2455   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   .rcmulr 12493   0gc0g 12626  mulGrpcmgp 12925   1rcur 12935  SRingcsrg 12939

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12431  df-slot 12432  df-base 12434  df-sets 12435  df-plusg 12505  df-mulr 12506  df-0g 12628  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-mgp 12926  df-ur 12936  df-srg 12940

This theorem is referenced by: (None)