Definition df-srg 13078

Description: Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Like with rings, the additive identity is an absorbing element of the multiplicative law, but in the case of semirings, this has to be part of the definition, as it cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.)

Assertion

Ref	Expression
df-srg	|- SRing = { f e. CMnd | ( (mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) ) ) }

Distinct variable group:   f, n, p, r, t, x, y, z

Detailed syntax breakdown of Definition df-srg

Step	Hyp	Ref	Expression
1		csrg 13077	. 2 class SRing
2		vf	. . . . . . 7 setvar f
3	2	cv 1352	. . . . . 6 class f
4		cmgp 13061	. . . . . 6 class mulGrp
5	3, 4	cfv 5215	. . . . 5 class (mulGrp ` f )
6		cmnd 12749	. . . . 5 class Mnd
7	5, 6	wcel 2148	. . . 4 wff (mulGrp ` f ) e. Mnd
8		vx	. . . . . . . . . . . . . . . 16 setvar x
9	8	cv 1352	. . . . . . . . . . . . . . 15 class x
10		vy	. . . . . . . . . . . . . . . . 17 setvar y
11	10	cv 1352	. . . . . . . . . . . . . . . 16 class y
12		vz	. . . . . . . . . . . . . . . . 17 setvar z
13	12	cv 1352	. . . . . . . . . . . . . . . 16 class z
14		vp	. . . . . . . . . . . . . . . . 17 setvar p
15	14	cv 1352	. . . . . . . . . . . . . . . 16 class p
16	11, 13, 15	co 5872	. . . . . . . . . . . . . . 15 class ( y p z )
17		vt	. . . . . . . . . . . . . . . 16 setvar t
18	17	cv 1352	. . . . . . . . . . . . . . 15 class t
19	9, 16, 18	co 5872	. . . . . . . . . . . . . 14 class ( x t ( y p z ) )
20	9, 11, 18	co 5872	. . . . . . . . . . . . . . 15 class ( x t y )
21	9, 13, 18	co 5872	. . . . . . . . . . . . . . 15 class ( x t z )
22	20, 21, 15	co 5872	. . . . . . . . . . . . . 14 class ( ( x t y ) p ( x t z ) )
23	19, 22	wceq 1353	. . . . . . . . . . . . 13 wff ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) )
24	9, 11, 15	co 5872	. . . . . . . . . . . . . . 15 class ( x p y )
25	24, 13, 18	co 5872	. . . . . . . . . . . . . 14 class ( ( x p y ) t z )
26	11, 13, 18	co 5872	. . . . . . . . . . . . . . 15 class ( y t z )
27	21, 26, 15	co 5872	. . . . . . . . . . . . . 14 class ( ( x t z ) p ( y t z ) )
28	25, 27	wceq 1353	. . . . . . . . . . . . 13 wff ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) )
29	23, 28	wa 104	. . . . . . . . . . . 12 wff ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) )
30		vr	. . . . . . . . . . . . 13 setvar r
31	30	cv 1352	. . . . . . . . . . . 12 class r
32	29, 12, 31	wral 2455	. . . . . . . . . . 11 wff A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) )
33	32, 10, 31	wral 2455	. . . . . . . . . 10 wff A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) )
34		vn	. . . . . . . . . . . . . 14 setvar n
35	34	cv 1352	. . . . . . . . . . . . 13 class n
36	35, 9, 18	co 5872	. . . . . . . . . . . 12 class ( n t x )
37	36, 35	wceq 1353	. . . . . . . . . . 11 wff ( n t x ) = n
38	9, 35, 18	co 5872	. . . . . . . . . . . 12 class ( x t n )
39	38, 35	wceq 1353	. . . . . . . . . . 11 wff ( x t n ) = n
40	37, 39	wa 104	. . . . . . . . . 10 wff ( ( n t x ) = n /\ ( x t n ) = n )
41	33, 40	wa 104	. . . . . . . . 9 wff ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) )
42	41, 8, 31	wral 2455	. . . . . . . 8 wff A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) )
43		c0g 12693	. . . . . . . . 9 class 0g
44	3, 43	cfv 5215	. . . . . . . 8 class ( 0g ` f )
45	42, 34, 44	wsbc 2962	. . . . . . 7 wff [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) )
46		cmulr 12529	. . . . . . . 8 class .r
47	3, 46	cfv 5215	. . . . . . 7 class ( .r ` f )
48	45, 17, 47	wsbc 2962	. . . . . 6 wff [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) )
49		cplusg 12528	. . . . . . 7 class +g
50	3, 49	cfv 5215	. . . . . 6 class ( +g ` f )
51	48, 14, 50	wsbc 2962	. . . . 5 wff [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) )
52		cbs 12454	. . . . . 6 class Base
53	3, 52	cfv 5215	. . . . 5 class ( Base ` f )
54	51, 30, 53	wsbc 2962	. . . 4 wff [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) )
55	7, 54	wa 104	. . 3 wff ( (mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) ) )
56		ccmn 13019	. . 3 class CMnd
57	55, 2, 56	crab 2459	. 2 class { f e. CMnd | ( (mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) ) ) }
58	1, 57	wceq 1353	1 wff SRing = { f e. CMnd | ( (mulGrp ` f ) e. Mnd /\ [. ( Base ` f ) / r ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. [. ( 0g ` f ) / n ]. A. x e. r ( A. y e. r A. z e. r ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) /\ ( ( n t x ) = n /\ ( x t n ) = n ) ) ) }

This definition is referenced by:  issrg  13079