Theorem znval 14971

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 14951	. . 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 14928	. . . . 5 ⊢ ℤring ∈ Ring
4	3	a1i 9	. . . 4 ⊢ (𝑛 = 𝑁 → ℤring ∈ Ring)
5		vex 2824	. . . . . . 7 ⊢ 𝑧 ∈ V
6		rspex 14811	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (RSpan‘𝑧) ∈ V)
7	6	elv 2825	. . . . . . . . 9 ⊢ (RSpan‘𝑧) ∈ V
8		vex 2824	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
9	8	snex 4322	. . . . . . . . 9 ⊢ {𝑛} ∈ V
10	7, 9	fvex 5715	. . . . . . . 8 ⊢ ((RSpan‘𝑧)‘{𝑛}) ∈ V
11		eqgex 14024	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ ((RSpan‘𝑧)‘{𝑛}) ∈ V) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V)
12	5, 10, 11	mp2an 430	. . . . . . 7 ⊢ (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V
13		qusex 13646	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) ∈ V) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ V)
14	5, 12, 13	mp2an 430	. . . . . 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 5699	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
19		znval.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (RSpan‘ℤring)
20	18, 19	eqtr4di 2289	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
21		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → 𝑛 = 𝑁)
22	21	sneqd 3722	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → {𝑛} = {𝑁})
23	20, 22	fveq12d 5702	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
24	17, 23	oveq12d 6103	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
25	17, 24	oveq12d 6103	. . . . . . . 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 2289	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
28	16, 27	sylan9eqr 2293	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
29		eqid 2238	. . . . . . . . . . . 12 ⊢ (ℤRHom‘𝑠) = (ℤRHom‘𝑠)
30	29	zrhex 14956	. . . . . . . . . . 11 ⊢ (𝑠 ∈ V → (ℤRHom‘𝑠) ∈ V)
31	30	elv 2825	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑠) ∈ V
32	31	resex 5104	. . . . . . . . 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 5699	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (ℤRHom‘𝑠) = (ℤRHom‘𝑈))
36		simpll 531	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑛 = 𝑁)
37	36	eqeq1d 2247	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
38	36	oveq2d 6101	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
39	37, 38	ifbieq2d 3665	. . . . . . . . . . . . . . 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 2289	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
42	35, 41	reseq12d 5064	. . . . . . . . . . . . 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 2289	. . . . . . . . . . . 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 2293	. . . . . . . . . . 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 4941	. . . . . . . . . 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 4956	. . . . . . . . . 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 4944	. . . . . . . . 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 2289	. . . . . . . 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 3194	. . . . . . 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 3911	. . . . . 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 6103	. . . . 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 3194	. . . 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 3194	. . 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 14811	. . . . . . . . . 10 ⊢ (ℤring ∈ Ring → (RSpan‘ℤring) ∈ V)
58	3, 57	ax-mp 5	. . . . . . . . 9 ⊢ (RSpan‘ℤring) ∈ V
59	19, 58	eqeltri 2311	. . . . . . . 8 ⊢ 𝑆 ∈ V
60		snexg 4321	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → {𝑁} ∈ V)
61		fvexg 5714	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
62	59, 60, 61	sylancr 418	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘{𝑁}) ∈ V)
63		eqgex 14024	. . . . . . 7 ⊢ ((ℤring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ V) → (ℤring ~QG (𝑆‘{𝑁})) ∈ V)
64	3, 62, 63	sylancr 418	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤring ~QG (𝑆‘{𝑁})) ∈ V)
65		qusex 13646	. . . . . 6 ⊢ ((ℤring ∈ Ring ∧ (ℤring ~QG (𝑆‘{𝑁})) ∈ V) → (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ∈ V)
66	3, 64, 65	sylancr 418	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ∈ V)
67	26, 66	eqeltrid 2325	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑈 ∈ V)
68		plendxnn 13557	. . . . 5 ⊢ (le‘ndx) ∈ ℕ
69	68	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (le‘ndx) ∈ ℕ)
70		eqid 2238	. . . . . . . . . . 11 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
71	70	zrhex 14956	. . . . . . . . . 10 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
72	67, 71	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) ∈ V)
73		resexg 5103	. . . . . . . . 9 ⊢ ((ℤRHom‘𝑈) ∈ V → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
74	72, 73	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
75	43, 74	eqeltrid 2325	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝐹 ∈ V)
76		xrex 10258	. . . . . . . . 9 ⊢ ℝ* ∈ V
77	76, 76	xpex 4891	. . . . . . . 8 ⊢ (ℝ* × ℝ*) ∈ V
78		lerelxr 8388	. . . . . . . 8 ⊢ ≤ ⊆ (ℝ* × ℝ*)
79	77, 78	ssexi 4271	. . . . . . 7 ⊢ ≤ ∈ V
80		coexg 5332	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
81	75, 79, 80	sylancl 417	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝐹 ∘ ≤ ) ∈ V)
82		cnvexg 5325	. . . . . . 7 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
83	75, 82	syl 14	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ◡𝐹 ∈ V)
84		coexg 5332	. . . . . 6 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
85	81, 83, 84	syl2anc 415	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
86	49, 85	eqeltrid 2325	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ≤ ∈ V)
87		setsex 13384	. . . 4 ⊢ ((𝑈 ∈ V ∧ (le‘ndx) ∈ ℕ ∧ ≤ ∈ V) → (𝑈 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V)
88	67, 69, 86, 87	syl3anc 1278	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑈 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V)
89	2, 55, 56, 88	fvmptd3 5799	. 2 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
90	1, 89	eqtrid 2283	1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ⦋csb 3147  ifcif 3638  {csn 3709  ⟨cop 3712   × cxp 4772  ^◡ccnv 4773   ↾ cres 4776   ∘ ccom 4778  ‘cfv 5377  (class class class)co 6085  0cc0 8179  ℝ^*cxr 8359   ≤ cle 8361  ℕcn 9304  ℕ₀cn0 9563  ℤcz 9644  ..^cfzo 10549  ndxcnx 13349   sSet csts 13350  lecple 13438   /_s cqus 13623   ~_QG cqg 13972  Ringcrg 14300  RSpancrsp 14805  ℤ_ringczring 14925  ℤRHomczrh 14946  ℤ/nℤczn 14948

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-ec 6809  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-rp 10055  df-fz 10412  df-cj 11607  df-abs 11765  df-struct 13354  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-starv 13446  df-sca 13447  df-vsca 13448  df-ip 13449  df-tset 13450  df-ple 13451  df-ds 13453  df-unif 13454  df-0g 13612  df-topgen 13614  df-iimas 13624  df-qus 13625  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-subg 13973  df-eqg 13975  df-cmn 14089  df-mgp 14218  df-ur 14263  df-ring 14302  df-cring 14303  df-rhm 14459  df-subrg 14527  df-lsp 14724  df-sra 14772  df-rgmod 14773  df-rsp 14807  df-bl 14883  df-mopn 14884  df-fg 14886  df-metu 14887  df-cnfld 14894  df-zring 14926  df-zrh 14949  df-zn 14951

This theorem is used by:  znle  14972  znval2  14973  znbaslemnn  14974