Theorem cznrng 41955

Description: The ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is a non-unital ring. (Contributed by AV, 17-Feb-2020.)

Hypotheses

Ref	Expression
cznrng.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
cznrng.b	⊢ 𝐵 = (Base‘𝑌)
cznrng.x	⊢ 𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)
cznrng.0	⊢ 0 = (0g‘𝑌)

Assertion

Ref	Expression
cznrng	⊢ ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑋 ∈ Rng)

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑁,𝑦   𝑥,𝑋   𝑥,𝑌,𝑦   𝑥, 0 ,𝑦

Allowed substitution hint:   𝑋(𝑦)

Proof of Theorem cznrng

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 11299	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2		cznrng.y	. . . . . . 7 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
3	2	zncrng 19893	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
4	1, 3	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑌 ∈ CRing)
5		crngring 18558	. . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
6		cznrng.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌)
7		cznrng.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
8	6, 7	ring0cl 18569	. . . . . . 7 ⊢ (𝑌 ∈ Ring → 0 ∈ 𝐵)
9		eleq1a 2696	. . . . . . 7 ⊢ ( 0 ∈ 𝐵 → (𝐶 = 0 → 𝐶 ∈ 𝐵))
10	8, 9	syl 17	. . . . . 6 ⊢ (𝑌 ∈ Ring → (𝐶 = 0 → 𝐶 ∈ 𝐵))
11	5, 10	syl 17	. . . . 5 ⊢ (𝑌 ∈ CRing → (𝐶 = 0 → 𝐶 ∈ 𝐵))
12	4, 11	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐶 = 0 → 𝐶 ∈ 𝐵))
13	12	imp 445	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝐶 ∈ 𝐵)
14		cznrng.x	. . . . . 6 ⊢ 𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)
15	2, 6, 14	cznabel 41954	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑋 ∈ Abel)
16	15	adantlr 751	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) → 𝑋 ∈ Abel)
17		eqid 2622	. . . . . 6 ⊢ (mulGrp‘𝑋) = (mulGrp‘𝑋)
18	2, 6, 14	cznrnglem 41953	. . . . . 6 ⊢ 𝐵 = (Base‘𝑋)
19	17, 18	mgpbas 18495	. . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑋))
20	14	fveq2i 6194	. . . . . . 7 ⊢ (mulGrp‘𝑋) = (mulGrp‘(𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩))
21		fvex 6201	. . . . . . . . 9 ⊢ (ℤ/nℤ‘𝑁) ∈ V
22	2, 21	eqeltri 2697	. . . . . . . 8 ⊢ 𝑌 ∈ V
23		fvex 6201	. . . . . . . . . 10 ⊢ (Base‘𝑌) ∈ V
24	6, 23	eqeltri 2697	. . . . . . . . 9 ⊢ 𝐵 ∈ V
25	24, 24	mpt2ex 7247	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
26		mulrid 15997	. . . . . . . . 9 ⊢ .r = Slot (.r‘ndx)
27	26	setsid 15914	. . . . . . . 8 ⊢ ((𝑌 ∈ V ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘(𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)))
28	22, 25, 27	mp2an 708	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘(𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩))
29	20, 28	mgpplusg 18493	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘(mulGrp‘𝑋))
30	29	eqcomi 2631	. . . . 5 ⊢ (+g‘(mulGrp‘𝑋)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)
31		ne0i 3921	. . . . . 6 ⊢ (𝐶 ∈ 𝐵 → 𝐵 ≠ ∅)
32	31	adantl 482	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) → 𝐵 ≠ ∅)
33		simpr 477	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ 𝐵)
34	19, 30, 32, 33	copissgrp 41808	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) → (mulGrp‘𝑋) ∈ SGrp)
35		oveq1 6657	. . . . . . . . 9 ⊢ (𝐶 = 0 → (𝐶(+g‘𝑌)𝐶) = ( 0 (+g‘𝑌)𝐶))
36	35	ad3antlr 767	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐶(+g‘𝑌)𝐶) = ( 0 (+g‘𝑌)𝐶))
37	4, 5	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑌 ∈ Ring)
38		ringmnd 18556	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
39	37, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑌 ∈ Mnd)
40	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑌 ∈ Mnd)
41	40	anim1i 592	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) → (𝑌 ∈ Mnd ∧ 𝐶 ∈ 𝐵))
42	41	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑌 ∈ Mnd ∧ 𝐶 ∈ 𝐵))
43		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝑌) = (+g‘𝑌)
44	6, 43, 7	mndlid 17311	. . . . . . . . 9 ⊢ ((𝑌 ∈ Mnd ∧ 𝐶 ∈ 𝐵) → ( 0 (+g‘𝑌)𝐶) = 𝐶)
45	42, 44	syl 17	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ( 0 (+g‘𝑌)𝐶) = 𝐶)
46	36, 45	eqtrd 2656	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐶(+g‘𝑌)𝐶) = 𝐶)
47		eqidd 2623	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶))
48		eqidd 2623	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → 𝐶 = 𝐶)
49		simpr1 1067	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
50		simpr2 1068	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
51	33	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
52	47, 48, 49, 50, 51	ovmpt2d 6788	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
53		eqidd 2623	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑐)) → 𝐶 = 𝐶)
54		simpr3 1069	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ 𝐵)
55	47, 53, 49, 54, 51	ovmpt2d 6788	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = 𝐶)
56	52, 55	oveq12d 6668	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(+g‘𝑌)(𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) = (𝐶(+g‘𝑌)𝐶))
57		eqidd 2623	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = (𝑏(+g‘𝑌)𝑐))) → 𝐶 = 𝐶)
58	37	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑌 ∈ Ring)
59	6, 43	ringacl 18578	. . . . . . . . 9 ⊢ ((𝑌 ∈ Ring ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) → (𝑏(+g‘𝑌)𝑐) ∈ 𝐵)
60	58, 50, 54, 59	syl3anc 1326	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏(+g‘𝑌)𝑐) ∈ 𝐵)
61	47, 57, 49, 60, 51	ovmpt2d 6788	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(+g‘𝑌)𝑐)) = 𝐶)
62	46, 56, 61	3eqtr4rd 2667	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(+g‘𝑌)𝑐)) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(+g‘𝑌)(𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)))
63		eqidd 2623	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑐)) → 𝐶 = 𝐶)
64	47, 63, 50, 54, 51	ovmpt2d 6788	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = 𝐶)
65	55, 64	oveq12d 6668	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)(+g‘𝑌)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) = (𝐶(+g‘𝑌)𝐶))
66		eqidd 2623	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = (𝑎(+g‘𝑌)𝑏) ∧ 𝑦 = 𝑐)) → 𝐶 = 𝐶)
67	6, 43	ringacl 18578	. . . . . . . . 9 ⊢ ((𝑌 ∈ Ring ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑌)𝑏) ∈ 𝐵)
68	58, 49, 50, 67	syl3anc 1326	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(+g‘𝑌)𝑏) ∈ 𝐵)
69	47, 66, 68, 54, 51	ovmpt2d 6788	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎(+g‘𝑌)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = 𝐶)
70	46, 65, 69	3eqtr4rd 2667	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎(+g‘𝑌)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)(+g‘𝑌)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)))
71	62, 70	jca 554	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(+g‘𝑌)𝑐)) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(+g‘𝑌)(𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) ∧ ((𝑎(+g‘𝑌)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)(+g‘𝑌)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐))))
72	71	ralrimivvva 2972	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(+g‘𝑌)𝑐)) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(+g‘𝑌)(𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) ∧ ((𝑎(+g‘𝑌)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)(+g‘𝑌)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐))))
73	16, 34, 72	3jca 1242	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ Abel ∧ (mulGrp‘𝑋) ∈ SGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(+g‘𝑌)𝑐)) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(+g‘𝑌)(𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) ∧ ((𝑎(+g‘𝑌)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)(+g‘𝑌)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)))))
74	13, 73	mpdan 702	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → (𝑋 ∈ Abel ∧ (mulGrp‘𝑋) ∈ SGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(+g‘𝑌)𝑐)) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(+g‘𝑌)(𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) ∧ ((𝑎(+g‘𝑌)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)(+g‘𝑌)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)))))
75		plusgid 15977	. . . . 5 ⊢ +g = Slot (+g‘ndx)
76		plusgndxnmulrndx 15998	. . . . 5 ⊢ (+g‘ndx) ≠ (.r‘ndx)
77	75, 76	setsnid 15915	. . . 4 ⊢ (+g‘𝑌) = (+g‘(𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩))
78	14	fveq2i 6194	. . . 4 ⊢ (+g‘𝑋) = (+g‘(𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩))
79	77, 78	eqtr4i 2647	. . 3 ⊢ (+g‘𝑌) = (+g‘𝑋)
80	14	eqcomi 2631	. . . . 5 ⊢ (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩) = 𝑋
81	80	fveq2i 6194	. . . 4 ⊢ (.r‘(𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)) = (.r‘𝑋)
82	28, 81	eqtri 2644	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (.r‘𝑋)
83	18, 17, 79, 82	isrng 41876	. 2 ⊢ (𝑋 ∈ Rng ↔ (𝑋 ∈ Abel ∧ (mulGrp‘𝑋) ∈ SGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(+g‘𝑌)𝑐)) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(+g‘𝑌)(𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) ∧ ((𝑎(+g‘𝑌)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)(+g‘𝑌)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)))))
84	74, 83	sylibr 224	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑋 ∈ Rng)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200  ∅c0 3915  ⟨cop 4183  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ℕcn 11020  ℕ₀cn0 11292  ndxcnx 15854   sSet csts 15855  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  0_gc0g 16100  SGrpcsgrp 17283  Mndcmnd 17294  Abelcabl 18194  mulGrpcmgp 18489  Ringcrg 18547  CRingccrg 18548  ℤ/nℤczn 19851  Rngcrng 41874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-nsg 17592  df-eqg 17593  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-cnfld 19747  df-zring 19819  df-zn 19855  df-rng0 41875

This theorem is referenced by: (None)