Theorem znval 13949

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 13931	. . 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 13909	. . . . 5 ⊢ ℤring ∈ Ring
4	3	a1i 9	. . . 4 ⊢ (𝑛 = 𝑁 → ℤring ∈ Ring)
5		vex 2755	. . . . . . 7 ⊢ 𝑧 ∈ V
6		rspex 13807	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (RSpan‘𝑧) ∈ V)
7	6	elv 2756	. . . . . . . . 9 ⊢ (RSpan‘𝑧) ∈ V
8		vex 2755	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
9	8	snex 4203	. . . . . . . . 9 ⊢ {𝑛} ∈ V
10	7, 9	fvex 5554	. . . . . . . 8 ⊢ ((RSpan‘𝑧)‘{𝑛}) ∈ V
11		eqgex 13177	. . . . . . . 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 12805	. . . . . . 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 5538	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
19		znval.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (RSpan‘ℤring)
20	18, 19	eqtr4di 2240	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
21		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → 𝑛 = 𝑁)
22	21	sneqd 3620	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → {𝑛} = {𝑁})
23	20, 22	fveq12d 5541	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
24	17, 23	oveq12d 5915	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
25	17, 24	oveq12d 5915	. . . . . . . 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 2240	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
28	16, 27	sylan9eqr 2244	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
29		eqid 2189	. . . . . . . . . . . 12 ⊢ (ℤRHom‘𝑠) = (ℤRHom‘𝑠)
30	29	zrhex 13935	. . . . . . . . . . 11 ⊢ (𝑠 ∈ V → (ℤRHom‘𝑠) ∈ V)
31	30	elv 2756	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑠) ∈ V
32	31	resex 4966	. . . . . . . . 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 5538	. . . . . . . . . . . . . 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 2198	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
38	36	oveq2d 5913	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
39	37, 38	ifbieq2d 3573	. . . . . . . . . . . . . . 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 2240	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
42	35, 41	reseq12d 4926	. . . . . . . . . . . . 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 2240	. . . . . . . . . . . 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 2244	. . . . . . . . . . 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 4806	. . . . . . . . . 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 4821	. . . . . . . . . 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 4809	. . . . . . . . 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 2240	. . . . . . . 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 3118	. . . . . . 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 3800	. . . . . 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 5915	. . . . 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 3118	. . . 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 3118	. . 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 13807	. . . . . . . . . 10 ⊢ (ℤring ∈ Ring → (RSpan‘ℤring) ∈ V)
58	3, 57	ax-mp 5	. . . . . . . . 9 ⊢ (RSpan‘ℤring) ∈ V
59	19, 58	eqeltri 2262	. . . . . . . 8 ⊢ 𝑆 ∈ V
60		snexg 4202	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → {𝑁} ∈ V)
61		fvexg 5553	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
62	59, 60, 61	sylancr 414	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘{𝑁}) ∈ V)
63		eqgex 13177	. . . . . . 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 12805	. . . . . 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 2276	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑈 ∈ V)
68		plendxnn 12717	. . . . 5 ⊢ (le‘ndx) ∈ ℕ
69	68	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (le‘ndx) ∈ ℕ)
70		eqid 2189	. . . . . . . . . . 11 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
71	70	zrhex 13935	. . . . . . . . . 10 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
72	67, 71	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) ∈ V)
73		resexg 4965	. . . . . . . . 9 ⊢ ((ℤRHom‘𝑈) ∈ V → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
74	72, 73	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
75	43, 74	eqeltrid 2276	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝐹 ∈ V)
76		xrex 9888	. . . . . . . . 9 ⊢ ℝ* ∈ V
77	76, 76	xpex 4759	. . . . . . . 8 ⊢ (ℝ* × ℝ*) ∈ V
78		lerelxr 8051	. . . . . . . 8 ⊢ ≤ ⊆ (ℝ* × ℝ*)
79	77, 78	ssexi 4156	. . . . . . 7 ⊢ ≤ ∈ V
80		coexg 5191	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
81	75, 79, 80	sylancl 413	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝐹 ∘ ≤ ) ∈ V)
82		cnvexg 5184	. . . . . . 7 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
83	75, 82	syl 14	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ◡𝐹 ∈ V)
84		coexg 5191	. . . . . 6 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
85	81, 83, 84	syl2anc 411	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ∈ V)
86	49, 85	eqeltrid 2276	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ≤ ∈ V)
87		setsex 12547	. . . 4 ⊢ ((𝑈 ∈ V ∧ (le‘ndx) ∈ ℕ ∧ ≤ ∈ V) → (𝑈 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V)
88	67, 69, 86, 87	syl3anc 1249	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑈 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V)
89	2, 55, 56, 88	fvmptd3 5630	. 2 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
90	1, 89	eqtrid 2234	1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  Vcvv 2752  ⦋csb 3072  ifcif 3549  {csn 3607  ⟨cop 3610   × cxp 4642  ^◡ccnv 4643   ↾ cres 4646   ∘ ccom 4648  ‘cfv 5235  (class class class)co 5897  0cc0 7842  ℝ^*cxr 8022   ≤ cle 8024  ℕcn 8950  ℕ₀cn0 9207  ℤcz 9284  ..^cfzo 10174  ndxcnx 12512   sSet csts 12513  lecple 12599   /_s cqus 12780   ~_QG cqg 13125  Ringcrg 13367  RSpancrsp 13801  ℤ_ringczring 13906  ℤRHomczrh 13926  ℤ/nℤczn 13928

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-addf 7964  ax-mulf 7965

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-tp 3615  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-ec 6562  df-map 6677  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-5 9012  df-6 9013  df-7 9014  df-8 9015  df-9 9016  df-n0 9208  df-z 9285  df-dec 9416  df-uz 9560  df-fz 10041  df-cj 10886  df-struct 12517  df-ndx 12518  df-slot 12519  df-base 12521  df-sets 12522  df-iress 12523  df-plusg 12605  df-mulr 12606  df-starv 12607  df-sca 12608  df-vsca 12609  df-ip 12610  df-ple 12612  df-0g 12766  df-iimas 12782  df-qus 12783  df-mgm 12835  df-sgrp 12880  df-mnd 12893  df-grp 12963  df-minusg 12964  df-subg 13126  df-eqg 13128  df-cmn 13242  df-mgp 13292  df-ur 13331  df-ring 13369  df-cring 13370  df-rhm 13519  df-subrg 13583  df-lsp 13720  df-sra 13768  df-rgmod 13769  df-rsp 13803  df-icnfld 13882  df-zring 13907  df-zrh 13929  df-zn 13931

This theorem is referenced by:  znle  13950  znval2  13951  znbaslemnn  13952