Theorem issrgid 14099

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 2232	. . 3 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
2		eqid 2232	. . 3 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
3		eqid 2232	. . 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 14090	. . . . 5 |- ( R e. SRing -> E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
7		reurex 2762	. . . . 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 2232	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
10	9, 4	mgpbasg 14044	. . . . 5 |- ( R e. SRing -> B = ( Base ` (mulGrp ` R ) ) )
11	9, 5	mgpplusgg 14042	. . . . . . . . 9 |- ( R e. SRing -> .x. = ( +g ` (mulGrp ` R ) ) )
12	11	oveqd 6058	. . . . . . . 8 |- ( R e. SRing -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
13	12	eqeq1d 2241	. . . . . . 7 |- ( R e. SRing -> ( ( y .x. x ) = x <-> ( y ( +g ` (mulGrp ` R ) ) x ) = x ) )
14	11	oveqd 6058	. . . . . . . 8 |- ( R e. SRing -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	14	eqeq1d 2241	. . . . . . 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 2756	. . . . 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 2757	. . . 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 13564	. 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 2302	. . 3 |- ( R e. SRing -> ( I e. B <-> I e. ( Base ` (mulGrp ` R ) ) ) )
22	11	oveqd 6058	. . . . . 6 |- ( R e. SRing -> ( I .x. x ) = ( I ( +g ` (mulGrp ` R ) ) x ) )
23	22	eqeq1d 2241	. . . . 5 |- ( R e. SRing -> ( ( I .x. x ) = x <-> ( I ( +g ` (mulGrp ` R ) ) x ) = x ) )
24	11	oveqd 6058	. . . . . 6 |- ( R e. SRing -> ( x .x. I ) = ( x ( +g ` (mulGrp ` R ) ) I ) )
25	24	eqeq1d 2241	. . . . 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 2756	. . 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 14079	. . 3 |- ( R e. SRing -> .1. = ( 0g ` (mulGrp ` R ) ) )
31	30	eqeq1d 2241	. 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 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   E!wreu 2522   ` cfv 5343  (class class class)co 6041   Basecbs 13186   +g cplusg 13264   .rcmulr 13265   0gc0g 13443  mulGrpcmgp 14038   1rcur 14077  SRingcsrg 14081

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-addcom 8215  ax-addass 8217  ax-i2m1 8220  ax-0lt1 8221  ax-0id 8223  ax-rnegex 8224  ax-pre-ltirr 8227  ax-pre-ltadd 8231

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-pnf 8298  df-mnf 8299  df-ltxr 8301  df-inn 9226  df-2 9284  df-3 9285  df-ndx 13189  df-slot 13190  df-base 13192  df-sets 13193  df-plusg 13277  df-mulr 13278  df-0g 13445  df-mgm 13543  df-sgrp 13589  df-mnd 13604  df-mgp 14039  df-ur 14078  df-srg 14082

This theorem is referenced by: (None)