Theorem znval 14910

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 14890	. . 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 14867	. . . . 5 ⊢ ℤring ∈ Ring
4	3	a1i 9	. . . 4 ⊢ (𝑛 = 𝑁 → ℤring ∈ Ring)
5		vex 2818	. . . . . . 7 ⊢ 𝑧 ∈ V
6		rspex 14748	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (RSpan‘𝑧) ∈ V)
7	6	elv 2819	. . . . . . . . 9 ⊢ (RSpan‘𝑧) ∈ V
8		vex 2818	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
9	8	snex 4303	. . . . . . . . 9 ⊢ {𝑛} ∈ V
10	7, 9	fvex 5695	. . . . . . . 8 ⊢ ((RSpan‘𝑧)‘{𝑛}) ∈ V
11		eqgex 13974	. . . . . . . 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 13589	. . . . . . 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 5679	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
19		znval.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (RSpan‘ℤring)
20	18, 19	eqtr4di 2285	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
21		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → 𝑛 = 𝑁)
22	21	sneqd 3707	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → {𝑛} = {𝑁})
23	20, 22	fveq12d 5682	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
24	17, 23	oveq12d 6076	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
25	17, 24	oveq12d 6076	. . . . . . . 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 2285	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
28	16, 27	sylan9eqr 2289	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
29		eqid 2234	. . . . . . . . . . . 12 ⊢ (ℤRHom‘𝑠) = (ℤRHom‘𝑠)
30	29	zrhex 14895	. . . . . . . . . . 11 ⊢ (𝑠 ∈ V → (ℤRHom‘𝑠) ∈ V)
31	30	elv 2819	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑠) ∈ V
32	31	resex 5084	. . . . . . . . 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 5679	. . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
38	36	oveq2d 6074	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
39	37, 38	ifbieq2d 3651	. . . . . . . . . . . . . . 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 2285	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
42	35, 41	reseq12d 5044	. . . . . . . . . . . . 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 2285	. . . . . . . . . . . 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 2289	. . . . . . . . . . 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 4921	. . . . . . . . . 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 4936	. . . . . . . . . 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 4924	. . . . . . . . 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 2285	. . . . . . . 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 3188	. . . . . . 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 3895	. . . . . 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 6076	. . . . 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 3188	. . . 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 3188	. . 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 14748	. . . . . . . . . 10 ⊢ (ℤring ∈ Ring → (RSpan‘ℤring) ∈ V)
58	3, 57	ax-mp 5	. . . . . . . . 9 ⊢ (RSpan‘ℤring) ∈ V
59	19, 58	eqeltri 2307	. . . . . . . 8 ⊢ 𝑆 ∈ V
60		snexg 4302	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → {𝑁} ∈ V)
61		fvexg 5694	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
62	59, 60, 61	sylancr 414	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘{𝑁}) ∈ V)
63		eqgex 13974	. . . . . . 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 13589	. . . . . 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 2321	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑈 ∈ V)
68		plendxnn 13500	. . . . 5 ⊢ (le‘ndx) ∈ ℕ
69	68	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (le‘ndx) ∈ ℕ)
70		eqid 2234	. . . . . . . . . . 11 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
71	70	zrhex 14895	. . . . . . . . . 10 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
72	67, 71	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) ∈ V)
73		resexg 5083	. . . . . . . . 9 ⊢ ((ℤRHom‘𝑈) ∈ V → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
74	72, 73	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
75	43, 74	eqeltrid 2321	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝐹 ∈ V)
76		xrex 10208	. . . . . . . . 9 ⊢ ℝ* ∈ V
77	76, 76	xpex 4871	. . . . . . . 8 ⊢ (ℝ* × ℝ*) ∈ V
78		lerelxr 8352	. . . . . . . 8 ⊢ ≤ ⊆ (ℝ* × ℝ*)
79	77, 78	ssexi 4253	. . . . . . 7 ⊢ ≤ ∈ V
80		coexg 5312	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
81	75, 79, 80	sylancl 413	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝐹 ∘ ≤ ) ∈ V)
82		cnvexg 5305	. . . . . . 7 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
83	75, 82	syl 14	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ◡𝐹 ∈ V)
84		coexg 5312	. . . . . 6 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
85	81, 83, 84	syl2anc 411	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
86	49, 85	eqeltrid 2321	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ≤ ∈ V)
87		setsex 13328	. . . 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 5776	. 2 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
90	1, 89	eqtrid 2279	1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ⦋csb 3141  ifcif 3624  {csn 3694  ⟨cop 3697   × cxp 4752  ^◡ccnv 4753   ↾ cres 4756   ∘ ccom 4758  ‘cfv 5357  (class class class)co 6058  0cc0 8143  ℝ^*cxr 8323   ≤ cle 8325  ℕcn 9254  ℕ₀cn0 9513  ℤcz 9594  ..^cfzo 10498  ndxcnx 13293   sSet csts 13294  lecple 13381   /_s cqus 13566   ~_QG cqg 13922  Ringcrg 14239  RSpancrsp 14742  ℤ_ringczring 14864  ℤRHomczrh 14885  ℤ/nℤczn 14887

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-addf 8265  ax-mulf 8266

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-ec 6782  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-rp 10005  df-fz 10362  df-cj 11552  df-abs 11709  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-starv 13389  df-sca 13390  df-vsca 13391  df-ip 13392  df-tset 13393  df-ple 13394  df-ds 13396  df-unif 13397  df-0g 13555  df-topgen 13557  df-iimas 13567  df-qus 13568  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-subg 13923  df-eqg 13925  df-cmn 14039  df-mgp 14160  df-ur 14203  df-ring 14241  df-cring 14242  df-rhm 14397  df-subrg 14465  df-lsp 14661  df-sra 14709  df-rgmod 14710  df-rsp 14744  df-bl 14820  df-mopn 14821  df-fg 14823  df-metu 14824  df-cnfld 14831  df-zring 14865  df-zrh 14888  df-zn 14890

This theorem is referenced by:  znle  14911  znval2  14912  znbaslemnn  14913