Definition df-srg 13598

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 13597	. 2 class SRing
2		vf	. . . . . . 7 setvar f
3	2	cv 1363	. . . . . 6 class f
4		cmgp 13554	. . . . . 6 class mulGrp
5	3, 4	cfv 5259	. . . . 5 class (mulGrp ` f )
6		cmnd 13120	. . . . 5 class Mnd
7	5, 6	wcel 2167	. . . 4 wff (mulGrp ` f ) e. Mnd
8		vx	. . . . . . . . . . . . . . . 16 setvar x
9	8	cv 1363	. . . . . . . . . . . . . . 15 class x
10		vy	. . . . . . . . . . . . . . . . 17 setvar y
11	10	cv 1363	. . . . . . . . . . . . . . . 16 class y
12		vz	. . . . . . . . . . . . . . . . 17 setvar z
13	12	cv 1363	. . . . . . . . . . . . . . . 16 class z
14		vp	. . . . . . . . . . . . . . . . 17 setvar p
15	14	cv 1363	. . . . . . . . . . . . . . . 16 class p
16	11, 13, 15	co 5925	. . . . . . . . . . . . . . 15 class ( y p z )
17		vt	. . . . . . . . . . . . . . . 16 setvar t
18	17	cv 1363	. . . . . . . . . . . . . . 15 class t
19	9, 16, 18	co 5925	. . . . . . . . . . . . . 14 class ( x t ( y p z ) )
20	9, 11, 18	co 5925	. . . . . . . . . . . . . . 15 class ( x t y )
21	9, 13, 18	co 5925	. . . . . . . . . . . . . . 15 class ( x t z )
22	20, 21, 15	co 5925	. . . . . . . . . . . . . 14 class ( ( x t y ) p ( x t z ) )
23	19, 22	wceq 1364	. . . . . . . . . . . . 13 wff ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) )
24	9, 11, 15	co 5925	. . . . . . . . . . . . . . 15 class ( x p y )
25	24, 13, 18	co 5925	. . . . . . . . . . . . . 14 class ( ( x p y ) t z )
26	11, 13, 18	co 5925	. . . . . . . . . . . . . . 15 class ( y t z )
27	21, 26, 15	co 5925	. . . . . . . . . . . . . 14 class ( ( x t z ) p ( y t z ) )
28	25, 27	wceq 1364	. . . . . . . . . . . . 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 1363	. . . . . . . . . . . 12 class r
32	29, 12, 31	wral 2475	. . . . . . . . . . 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 2475	. . . . . . . . . 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 1363	. . . . . . . . . . . . 13 class n
36	35, 9, 18	co 5925	. . . . . . . . . . . 12 class ( n t x )
37	36, 35	wceq 1364	. . . . . . . . . . 11 wff ( n t x ) = n
38	9, 35, 18	co 5925	. . . . . . . . . . . 12 class ( x t n )
39	38, 35	wceq 1364	. . . . . . . . . . 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 2475	. . . . . . . 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 12960	. . . . . . . . 9 class 0g
44	3, 43	cfv 5259	. . . . . . . 8 class ( 0g ` f )
45	42, 34, 44	wsbc 2989	. . . . . . 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 12783	. . . . . . . 8 class .r
47	3, 46	cfv 5259	. . . . . . 7 class ( .r ` f )
48	45, 17, 47	wsbc 2989	. . . . . 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 12782	. . . . . . 7 class +g
50	3, 49	cfv 5259	. . . . . 6 class ( +g ` f )
51	48, 14, 50	wsbc 2989	. . . . 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 12705	. . . . . 6 class Base
53	3, 52	cfv 5259	. . . . 5 class ( Base ` f )
54	51, 30, 53	wsbc 2989	. . . 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 13492	. . 3 class CMnd
57	55, 2, 56	crab 2479	. 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 1364	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  13599