Theorem issrg 18868

Description: The predicate "is a semiring." (Contributed by Thierry Arnoux, 21-Mar-2018.)

Hypotheses

Ref	Expression
issrg.b	⊢ 𝐵 = (Base‘𝑅)
issrg.g	⊢ 𝐺 = (mulGrp‘𝑅)
issrg.p	⊢ + = (+g‘𝑅)
issrg.t	⊢ · = (.r‘𝑅)
issrg.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
issrg	⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))

Distinct variable groups:   𝑥,𝑦,𝑧, +   𝑥, 0 ,𝑦,𝑧   𝑥, · ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem issrg

Dummy variables 𝑛 𝑏 𝑝 𝑟 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issrg.g	. . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅)
2	1	eleq1i 2897	. . . . 5 ⊢ (𝐺 ∈ Mnd ↔ (mulGrp‘𝑅) ∈ Mnd)
3	2	bicomi 216	. . . 4 ⊢ ((mulGrp‘𝑅) ∈ Mnd ↔ 𝐺 ∈ Mnd)
4		issrg.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5	4	fvexi 6451	. . . . 5 ⊢ 𝐵 ∈ V
6		issrg.p	. . . . . 6 ⊢ + = (+g‘𝑅)
7	6	fvexi 6451	. . . . 5 ⊢ + ∈ V
8		issrg.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
9	8	fvexi 6451	. . . . . . 7 ⊢ · ∈ V
10	9	a1i 11	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → · ∈ V)
11		issrg.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
12	11	fvexi 6451	. . . . . . . 8 ⊢ 0 ∈ V
13	12	a1i 11	. . . . . . 7 ⊢ (((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 0 ∈ V)
14		simplll 791	. . . . . . . 8 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑏 = 𝐵)
15		simplr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑡 = · )
16		eqidd 2826	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑥 = 𝑥)
17		simpllr 793	. . . . . . . . . . . . . . 15 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑝 = + )
18	17	oveqd 6927	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑦𝑝𝑧) = (𝑦 + 𝑧))
19	15, 16, 18	oveq123d 6931	. . . . . . . . . . . . 13 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡(𝑦𝑝𝑧)) = (𝑥 · (𝑦 + 𝑧)))
20	15	oveqd 6927	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑦) = (𝑥 · 𝑦))
21	15	oveqd 6927	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑧) = (𝑥 · 𝑧))
22	17, 20, 21	oveq123d 6931	. . . . . . . . . . . . 13 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
23	19, 22	eqeq12d 2840	. . . . . . . . . . . 12 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ↔ (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))))
24	17	oveqd 6927	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
25		eqidd 2826	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑧 = 𝑧)
26	15, 24, 25	oveq123d 6931	. . . . . . . . . . . . 13 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥 + 𝑦) · 𝑧))
27	15	oveqd 6927	. . . . . . . . . . . . . 14 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑦𝑡𝑧) = (𝑦 · 𝑧))
28	17, 21, 27	oveq123d 6931	. . . . . . . . . . . . 13 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
29	26, 28	eqeq12d 2840	. . . . . . . . . . . 12 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) ↔ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
30	23, 29	anbi12d 624	. . . . . . . . . . 11 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
31	14, 30	raleqbidv 3364	. . . . . . . . . 10 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
32	14, 31	raleqbidv 3364	. . . . . . . . 9 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
33		simpr 479	. . . . . . . . . . . 12 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → 𝑛 = 0 )
34	15, 33, 16	oveq123d 6931	. . . . . . . . . . 11 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑛𝑡𝑥) = ( 0 · 𝑥))
35	34, 33	eqeq12d 2840	. . . . . . . . . 10 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑛𝑡𝑥) = 𝑛 ↔ ( 0 · 𝑥) = 0 ))
36	15, 16, 33	oveq123d 6931	. . . . . . . . . . 11 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (𝑥𝑡𝑛) = (𝑥 · 0 ))
37	36, 33	eqeq12d 2840	. . . . . . . . . 10 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((𝑥𝑡𝑛) = 𝑛 ↔ (𝑥 · 0 ) = 0 ))
38	35, 37	anbi12d 624	. . . . . . . . 9 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛) ↔ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
39	32, 38	anbi12d 624	. . . . . . . 8 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → ((∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
40	14, 39	raleqbidv 3364	. . . . . . 7 ⊢ ((((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) ∧ 𝑛 = 0 ) → (∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
41	13, 40	sbcied 3699	. . . . . 6 ⊢ (((𝑏 = 𝐵 ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ([ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
42	10, 41	sbcied 3699	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → ([ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
43	5, 7, 42	sbc2ie 3730	. . . 4 ⊢ ([𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
44	3, 43	anbi12i 620	. . 3 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
45	44	anbi2i 616	. 2 ⊢ ((𝑅 ∈ CMnd ∧ ((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))))
46		fveq2 6437	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
47	46	eleq1d 2891	. . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ Mnd ↔ (mulGrp‘𝑅) ∈ Mnd))
48		fveq2 6437	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
49	48, 4	syl6eqr 2879	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
50		fveq2 6437	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
51	50, 6	syl6eqr 2879	. . . . . 6 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
52		fveq2 6437	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
53	52, 8	syl6eqr 2879	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
54		fveq2 6437	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
55	54, 11	syl6eqr 2879	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
56	55	sbceq1d 3667	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ([(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
57	53, 56	sbceqbid 3669	. . . . . 6 ⊢ (𝑟 = 𝑅 → ([(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
58	51, 57	sbceqbid 3669	. . . . 5 ⊢ (𝑟 = 𝑅 → ([(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
59	49, 58	sbceqbid 3669	. . . 4 ⊢ (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)) ↔ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))))
60	47, 59	anbi12d 624	. . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛))) ↔ ((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))))
61		df-srg 18867	. . 3 ⊢ SRing = {𝑟 ∈ CMnd ∣ ((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡][(0g‘𝑟) / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
62	60, 61	elrab2 3589	. 2 ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ ((mulGrp‘𝑅) ∈ Mnd ∧ [𝐵 / 𝑏][ + / 𝑝][ · / 𝑡][ 0 / 𝑛]∀𝑥 ∈ 𝑏 (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))))
63		3anass 1120	. 2 ⊢ ((𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))) ↔ (𝑅 ∈ CMnd ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))))
64	45, 62, 63	3bitr4i 295	1 ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))

Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414  [wsbc 3662  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  +_gcplusg 16312  ._rcmulr 16313  0_gc0g 16460  Mndcmnd 17654  CMndccmn 18553  mulGrpcmgp 18850  SRingcsrg 18866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-srg 18867

This theorem is referenced by:  srgcmn  18869  srgmgp  18871  srgi  18872  srgrz  18887  srglz  18888  ringsrg  18950  nn0srg  20183  rge0srg  20184