Definition df-srg 14207

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 14206	. 2 class SRing
2		vf	. . . . . . 7 setvar f
3	2	cv 1397	. . . . . 6 class f
4		cmgp 14159	. . . . . 6 class mulGrp
5	3, 4	cfv 5357	. . . . 5 class (mulGrp ` f )
6		cmnd 13677	. . . . 5 class Mnd
7	5, 6	wcel 2205	. . . 4 wff (mulGrp ` f ) e. Mnd
8		vx	. . . . . . . . . . . . . . . 16 setvar x
9	8	cv 1397	. . . . . . . . . . . . . . 15 class x
10		vy	. . . . . . . . . . . . . . . . 17 setvar y
11	10	cv 1397	. . . . . . . . . . . . . . . 16 class y
12		vz	. . . . . . . . . . . . . . . . 17 setvar z
13	12	cv 1397	. . . . . . . . . . . . . . . 16 class z
14		vp	. . . . . . . . . . . . . . . . 17 setvar p
15	14	cv 1397	. . . . . . . . . . . . . . . 16 class p
16	11, 13, 15	co 6058	. . . . . . . . . . . . . . 15 class ( y p z )
17		vt	. . . . . . . . . . . . . . . 16 setvar t
18	17	cv 1397	. . . . . . . . . . . . . . 15 class t
19	9, 16, 18	co 6058	. . . . . . . . . . . . . 14 class ( x t ( y p z ) )
20	9, 11, 18	co 6058	. . . . . . . . . . . . . . 15 class ( x t y )
21	9, 13, 18	co 6058	. . . . . . . . . . . . . . 15 class ( x t z )
22	20, 21, 15	co 6058	. . . . . . . . . . . . . 14 class ( ( x t y ) p ( x t z ) )
23	19, 22	wceq 1398	. . . . . . . . . . . . 13 wff ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) )
24	9, 11, 15	co 6058	. . . . . . . . . . . . . . 15 class ( x p y )
25	24, 13, 18	co 6058	. . . . . . . . . . . . . 14 class ( ( x p y ) t z )
26	11, 13, 18	co 6058	. . . . . . . . . . . . . . 15 class ( y t z )
27	21, 26, 15	co 6058	. . . . . . . . . . . . . 14 class ( ( x t z ) p ( y t z ) )
28	25, 27	wceq 1398	. . . . . . . . . . . . 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 1397	. . . . . . . . . . . 12 class r
32	29, 12, 31	wral 2522	. . . . . . . . . . 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 2522	. . . . . . . . . 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 1397	. . . . . . . . . . . . 13 class n
36	35, 9, 18	co 6058	. . . . . . . . . . . 12 class ( n t x )
37	36, 35	wceq 1398	. . . . . . . . . . 11 wff ( n t x ) = n
38	9, 35, 18	co 6058	. . . . . . . . . . . 12 class ( x t n )
39	38, 35	wceq 1398	. . . . . . . . . . 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 2522	. . . . . . . 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 13553	. . . . . . . . 9 class 0g
44	3, 43	cfv 5357	. . . . . . . 8 class ( 0g ` f )
45	42, 34, 44	wsbc 3045	. . . . . . 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 13375	. . . . . . . 8 class .r
47	3, 46	cfv 5357	. . . . . . 7 class ( .r ` f )
48	45, 17, 47	wsbc 3045	. . . . . 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 13374	. . . . . . 7 class +g
50	3, 49	cfv 5357	. . . . . 6 class ( +g ` f )
51	48, 14, 50	wsbc 3045	. . . . 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 13296	. . . . . 6 class Base
53	3, 52	cfv 5357	. . . . 5 class ( Base ` f )
54	51, 30, 53	wsbc 3045	. . . 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 14037	. . 3 class CMnd
57	55, 2, 56	crab 2526	. 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 1398	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  14208