Theorem znval 14715

Description: The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)

Hypotheses

Ref	Expression
znval.s	⊢ 𝑆 = (RSpan‘ℤring)
znval.u	⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
znval.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
znval.f	⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊)
znval.w	⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
znval.l	⊢ ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)

Assertion

Ref	Expression
znval	⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))

Proof of Theorem znval

Dummy variables 𝑓 𝑛 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		znval.y	. 2 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
2		df-zn 14695	. . 3 ⊢ ℤ/nℤ = (𝑛 ∈ ℕ0 ↦ ⦋ℤring / 𝑧⦌⦋(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet ⟨(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)⟩))
3		zringring 14672	. . . . 5 ⊢ ℤring ∈ Ring
4	3	a1i 9	. . . 4 ⊢ (𝑛 = 𝑁 → ℤring ∈ Ring)
5		vex 2806	. . . . . . 7 ⊢ 𝑧 ∈ V
6		rspex 14553	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (RSpan‘𝑧) ∈ V)
7	6	elv 2807	. . . . . . . . 9 ⊢ (RSpan‘𝑧) ∈ V
8		vex 2806	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
9	8	snex 4281	. . . . . . . . 9 ⊢ {𝑛} ∈ V
10	7, 9	fvex 5668	. . . . . . . 8 ⊢ ((RSpan‘𝑧)‘{𝑛}) ∈ V
11		eqgex 13871	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ ((RSpan‘𝑧)‘{𝑛}) ∈ V) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V)
12	5, 10, 11	mp2an 426	. . . . . . 7 ⊢ (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V
13		qusex 13471	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ V)
14	5, 12, 13	mp2an 426	. . . . . 6 ⊢ (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ V
15	14	a1i 9	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ V)
16		id 19	. . . . . . 7 ⊢ (𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) → 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))))
17		simpr 110	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → 𝑧 = ℤring)
18	17	fveq2d 5652	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
19		znval.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (RSpan‘ℤring)
20	18, 19	eqtr4di 2282	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
21		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → 𝑛 = 𝑁)
22	21	sneqd 3686	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → {𝑛} = {𝑁})
23	20, 22	fveq12d 5655	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
24	17, 23	oveq12d 6046	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
25	17, 24	oveq12d 6046	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))))
26		znval.u	. . . . . . . 8 ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
27	25, 26	eqtr4di 2282	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
28	16, 27	sylan9eqr 2286	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
29		eqid 2231	. . . . . . . . . . . 12 ⊢ (ℤRHom‘𝑠) = (ℤRHom‘𝑠)
30	29	zrhex 14700	. . . . . . . . . . 11 ⊢ (𝑠 ∈ V → (ℤRHom‘𝑠) ∈ V)
31	30	elv 2807	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑠) ∈ V
32	31	resex 5060	. . . . . . . . 9 ⊢ ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) ∈ V
33	32	a1i 9	. . . . . . . 8 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) ∈ V)
34		id 19	. . . . . . . . . . . 12 ⊢ (𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) → 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))))
35	28	fveq2d 5652	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (ℤRHom‘𝑠) = (ℤRHom‘𝑈))
36		simpll 527	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑛 = 𝑁)
37	36	eqeq1d 2240	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
38	36	oveq2d 6044	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
39	37, 38	ifbieq2d 3634	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = if(𝑁 = 0, ℤ, (0..^𝑁)))
40		znval.w	. . . . . . . . . . . . . . 15 ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
41	39, 40	eqtr4di 2282	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
42	35, 41	reseq12d 5020	. . . . . . . . . . . . 13 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = ((ℤRHom‘𝑈) ↾ 𝑊))
43		znval.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊)
44	42, 43	eqtr4di 2282	. . . . . . . . . . . 12 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = 𝐹)
45	34, 44	sylan9eqr 2286	. . . . . . . . . . 11 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
46	45	coeq1d 4897	. . . . . . . . . 10 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → (𝑓 ∘ ≤ ) = (𝐹 ∘ ≤ ))
47	45	cnveqd 4912	. . . . . . . . . 10 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ◡𝑓 = ◡𝐹)
48	46, 47	coeq12d 4900	. . . . . . . . 9 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ((𝑓 ∘ ≤ ) ∘ ◡𝑓) = ((𝐹 ∘ ≤ ) ∘ ◡𝐹))
49		znval.l	. . . . . . . . 9 ⊢ ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)
50	48, 49	eqtr4di 2282	. . . . . . . 8 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ((𝑓 ∘ ≤ ) ∘ ◡𝑓) = ≤ )
51	33, 50	csbied 3175	. . . . . . 7 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓) = ≤ )
52	51	opeq2d 3874	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ⟨(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)⟩ = ⟨(le‘ndx), ≤ ⟩)
53	28, 52	oveq12d 6046	. . . . 5 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑠 sSet ⟨(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
54	15, 53	csbied 3175	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → ⦋(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet ⟨(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
55	4, 54	csbied 3175	. . 3 ⊢ (𝑛 = 𝑁 → ⦋ℤring / 𝑧⦌⦋(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet ⟨(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
56		id 19	. . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0)
57		rspex 14553	. . . . . . . . . 10 ⊢ (ℤring ∈ Ring → (RSpan‘ℤring) ∈ V)
58	3, 57	ax-mp 5	. . . . . . . . 9 ⊢ (RSpan‘ℤring) ∈ V
59	19, 58	eqeltri 2304	. . . . . . . 8 ⊢ 𝑆 ∈ V
60		snexg 4280	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → {𝑁} ∈ V)
61		fvexg 5667	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
62	59, 60, 61	sylancr 414	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘{𝑁}) ∈ V)
63		eqgex 13871	. . . . . . 7 ⊢ ((ℤring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ V) → (ℤring ~QG (𝑆‘{𝑁})) ∈ V)
64	3, 62, 63	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤring ~QG (𝑆‘{𝑁})) ∈ V)
65		qusex 13471	. . . . . 6 ⊢ ((ℤring ∈ Ring ∧ (ℤring ~QG (𝑆‘{𝑁})) ∈ V) → (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ∈ V)
66	3, 64, 65	sylancr 414	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ∈ V)
67	26, 66	eqeltrid 2318	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑈 ∈ V)
68		plendxnn 13349	. . . . 5 ⊢ (le‘ndx) ∈ ℕ
69	68	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (le‘ndx) ∈ ℕ)
70		eqid 2231	. . . . . . . . . . 11 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
71	70	zrhex 14700	. . . . . . . . . 10 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
72	67, 71	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) ∈ V)
73		resexg 5059	. . . . . . . . 9 ⊢ ((ℤRHom‘𝑈) ∈ V → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
74	72, 73	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
75	43, 74	eqeltrid 2318	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝐹 ∈ V)
76		xrex 10135	. . . . . . . . 9 ⊢ ℝ* ∈ V
77	76, 76	xpex 4848	. . . . . . . 8 ⊢ (ℝ* × ℝ*) ∈ V
78		lerelxr 8284	. . . . . . . 8 ⊢ ≤ ⊆ (ℝ* × ℝ*)
79	77, 78	ssexi 4232	. . . . . . 7 ⊢ ≤ ∈ V
80		coexg 5288	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
81	75, 79, 80	sylancl 413	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝐹 ∘ ≤ ) ∈ V)
82		cnvexg 5281	. . . . . . 7 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
83	75, 82	syl 14	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ◡𝐹 ∈ V)
84		coexg 5288	. . . . . 6 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
85	81, 83, 84	syl2anc 411	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
86	49, 85	eqeltrid 2318	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ≤ ∈ V)
87		setsex 13177	. . . 4 ⊢ ((𝑈 ∈ V ∧ (le‘ndx) ∈ ℕ ∧ ≤ ∈ V) → (𝑈 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V)
88	67, 69, 86, 87	syl3anc 1274	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑈 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V)
89	2, 55, 56, 88	fvmptd3 5749	. 2 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
90	1, 89	eqtrid 2276	1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  Vcvv 2803  ⦋csb 3128  ifcif 3607  {csn 3673  ⟨cop 3676   × cxp 4729  ^◡ccnv 4730   ↾ cres 4733   ∘ ccom 4735  ‘cfv 5333  (class class class)co 6028  0cc0 8075  ℝ^*cxr 8255   ≤ cle 8257  ℕcn 9185  ℕ₀cn0 9444  ℤcz 9523  ..^cfzo 10422  ndxcnx 13142   sSet csts 13143  lecple 13230   /_s cqus 13446   ~_QG cqg 13819  Ringcrg 14073  RSpancrsp 14547  ℤ_ringczring 14669  ℤRHomczrh 14690  ℤ/nℤczn 14692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-addf 8197  ax-mulf 8198

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-ec 6747  df-map 6862  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-z 9524  df-dec 9656  df-uz 9800  df-rp 9933  df-fz 10289  df-cj 11465  df-abs 11622  df-struct 13147  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-starv 13238  df-sca 13239  df-vsca 13240  df-ip 13241  df-tset 13242  df-ple 13243  df-ds 13245  df-unif 13246  df-0g 13404  df-topgen 13406  df-iimas 13448  df-qus 13449  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-subg 13820  df-eqg 13822  df-cmn 13936  df-mgp 13998  df-ur 14037  df-ring 14075  df-cring 14076  df-rhm 14230  df-subrg 14297  df-lsp 14466  df-sra 14514  df-rgmod 14515  df-rsp 14549  df-bl 14625  df-mopn 14626  df-fg 14628  df-metu 14629  df-cnfld 14636  df-zring 14670  df-zrh 14693  df-zn 14695

This theorem is referenced by:  znle  14716  znval2  14717  znbaslemnn  14718