znleval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > znleval		Structured version Visualization version GIF version

Theorem znleval 21116

Description: The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)

**Hypotheses**
Ref	Expression
znle2.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
znle2.f	⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊)
znle2.w	⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
znle2.l	⊢ ≤ = (le‘𝑌)
znleval.x	⊢ 𝑋 = (Base‘𝑌)

**Assertion**
Ref	Expression
znleval	⊢ (𝑁 ∈ ℕ₀ → (𝐴 ≤ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵))))

Proof of Theorem znleval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		znle2.y	. . . . . . 7 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
2		znle2.f	. . . . . . 7 ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊)
3		znle2.w	. . . . . . 7 ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
4		znle2.l	. . . . . . 7 ⊢ ≤ = (le‘𝑌)
5	1, 2, 3, 4	znle2 21115	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))
6		relco 6107	. . . . . . . 8 ⊢ Rel ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)
7		relssdmrn 6267	. . . . . . . 8 ⊢ (Rel ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ (dom ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) × ran ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)))
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ (dom ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) × ran ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))
9		dmcoss 5970	. . . . . . . . 9 ⊢ dom ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ dom ^◡𝐹
10		df-rn 5687	. . . . . . . . . 10 ⊢ ran 𝐹 = dom ^◡𝐹
11		znleval.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝑌)
12	1, 11, 2, 3	znf1o 21113	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝑊–1-1-onto→𝑋)
13		f1ofo 6840	. . . . . . . . . . 11 ⊢ (𝐹:𝑊–1-1-onto→𝑋 → 𝐹:𝑊–onto→𝑋)
14		forn 6808	. . . . . . . . . . 11 ⊢ (𝐹:𝑊–onto→𝑋 → ran 𝐹 = 𝑋)
15	12, 13, 14	3syl 18	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ran 𝐹 = 𝑋)
16	10, 15	eqtr3id 2786	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → dom ^◡𝐹 = 𝑋)
17	9, 16	sseqtrid 4034	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → dom ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ 𝑋)
18		rncoss 5971	. . . . . . . . 9 ⊢ ran ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ ran (𝐹 ∘ ≤ )
19		rncoss 5971	. . . . . . . . . 10 ⊢ ran (𝐹 ∘ ≤ ) ⊆ ran 𝐹
20	19, 15	sseqtrid 4034	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → ran (𝐹 ∘ ≤ ) ⊆ 𝑋)
21	18, 20	sstrid 3993	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → ran ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ 𝑋)
22		xpss12 5691	. . . . . . . 8 ⊢ ((dom ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ 𝑋 ∧ ran ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ 𝑋) → (dom ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) × ran ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)) ⊆ (𝑋 × 𝑋))
23	17, 21, 22	syl2anc 584	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (dom ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) × ran ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)) ⊆ (𝑋 × 𝑋))
24	8, 23	sstrid 3993	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ⊆ (𝑋 × 𝑋))
25	5, 24	eqsstrd 4020	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ≤ ⊆ (𝑋 × 𝑋))
26	25	ssbrd 5191	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ≤ 𝐵 → 𝐴(𝑋 × 𝑋)𝐵))
27		brxp 5725	. . . 4 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
28	26, 27	imbitrdi 250	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ≤ 𝐵 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
29	28	pm4.71rd 563	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ≤ 𝐵 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ≤ 𝐵)))
30	5	adantr 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))
31	30	breqd 5159	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ≤ 𝐵 ↔ 𝐴((𝐹 ∘ ≤ ) ∘ ^◡𝐹)𝐵))
32		brcog 5866	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴((𝐹 ∘ ≤ ) ∘ ^◡𝐹)𝐵 ↔ ∃𝑥(𝐴^◡𝐹𝑥 ∧ 𝑥(𝐹 ∘ ≤ )𝐵)))
33	32	adantl 482	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴((𝐹 ∘ ≤ ) ∘ ^◡𝐹)𝐵 ↔ ∃𝑥(𝐴^◡𝐹𝑥 ∧ 𝑥(𝐹 ∘ ≤ )𝐵)))
34		eqcom 2739	. . . . . . . . 9 ⊢ (𝑥 = (^◡𝐹‘𝐴) ↔ (^◡𝐹‘𝐴) = 𝑥)
35	12	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐹:𝑊–1-1-onto→𝑋)
36		f1ocnv 6845	. . . . . . . . . . 11 ⊢ (𝐹:𝑊–1-1-onto→𝑋 → ^◡𝐹:𝑋–1-1-onto→𝑊)
37		f1ofn 6834	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝑋–1-1-onto→𝑊 → ^◡𝐹 Fn 𝑋)
38	35, 36, 37	3syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ^◡𝐹 Fn 𝑋)
39		simprl 769	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
40		fnbrfvb 6944	. . . . . . . . . 10 ⊢ ((^◡𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((^◡𝐹‘𝐴) = 𝑥 ↔ 𝐴^◡𝐹𝑥))
41	38, 39, 40	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((^◡𝐹‘𝐴) = 𝑥 ↔ 𝐴^◡𝐹𝑥))
42	34, 41	bitr2id 283	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴^◡𝐹𝑥 ↔ 𝑥 = (^◡𝐹‘𝐴)))
43	42	anbi1d 630	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴^◡𝐹𝑥 ∧ 𝑥(𝐹 ∘ ≤ )𝐵) ↔ (𝑥 = (^◡𝐹‘𝐴) ∧ 𝑥(𝐹 ∘ ≤ )𝐵)))
44	43	exbidv 1924	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∃𝑥(𝐴^◡𝐹𝑥 ∧ 𝑥(𝐹 ∘ ≤ )𝐵) ↔ ∃𝑥(𝑥 = (^◡𝐹‘𝐴) ∧ 𝑥(𝐹 ∘ ≤ )𝐵)))
45	33, 44	bitrd 278	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴((𝐹 ∘ ≤ ) ∘ ^◡𝐹)𝐵 ↔ ∃𝑥(𝑥 = (^◡𝐹‘𝐴) ∧ 𝑥(𝐹 ∘ ≤ )𝐵)))
46		fvex 6904	. . . . . . 7 ⊢ (^◡𝐹‘𝐴) ∈ V
47		breq1 5151	. . . . . . 7 ⊢ (𝑥 = (^◡𝐹‘𝐴) → (𝑥(𝐹 ∘ ≤ )𝐵 ↔ (^◡𝐹‘𝐴)(𝐹 ∘ ≤ )𝐵))
48	46, 47	ceqsexv 3525	. . . . . 6 ⊢ (∃𝑥(𝑥 = (^◡𝐹‘𝐴) ∧ 𝑥(𝐹 ∘ ≤ )𝐵) ↔ (^◡𝐹‘𝐴)(𝐹 ∘ ≤ )𝐵)
49		simprr 771	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
50		brcog 5866	. . . . . . . 8 ⊢ (((^◡𝐹‘𝐴) ∈ V ∧ 𝐵 ∈ 𝑋) → ((^◡𝐹‘𝐴)(𝐹 ∘ ≤ )𝐵 ↔ ∃𝑥((^◡𝐹‘𝐴) ≤ 𝑥 ∧ 𝑥𝐹𝐵)))
51	46, 49, 50	sylancr 587	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((^◡𝐹‘𝐴)(𝐹 ∘ ≤ )𝐵 ↔ ∃𝑥((^◡𝐹‘𝐴) ≤ 𝑥 ∧ 𝑥𝐹𝐵)))
52		fvex 6904	. . . . . . . . 9 ⊢ (^◡𝐹‘𝐵) ∈ V
53		breq2 5152	. . . . . . . . 9 ⊢ (𝑥 = (^◡𝐹‘𝐵) → ((^◡𝐹‘𝐴) ≤ 𝑥 ↔ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵)))
54	52, 53	ceqsexv 3525	. . . . . . . 8 ⊢ (∃𝑥(𝑥 = (^◡𝐹‘𝐵) ∧ (^◡𝐹‘𝐴) ≤ 𝑥) ↔ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵))
55		eqcom 2739	. . . . . . . . . . . . 13 ⊢ (𝑥 = (^◡𝐹‘𝐵) ↔ (^◡𝐹‘𝐵) = 𝑥)
56		fnbrfvb 6944	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐹 Fn 𝑋 ∧ 𝐵 ∈ 𝑋) → ((^◡𝐹‘𝐵) = 𝑥 ↔ 𝐵^◡𝐹𝑥))
57	38, 49, 56	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((^◡𝐹‘𝐵) = 𝑥 ↔ 𝐵^◡𝐹𝑥))
58	55, 57	bitrid 282	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑥 = (^◡𝐹‘𝐵) ↔ 𝐵^◡𝐹𝑥))
59		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
60		brcnvg 5879	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ V) → (𝐵^◡𝐹𝑥 ↔ 𝑥𝐹𝐵))
61	49, 59, 60	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵^◡𝐹𝑥 ↔ 𝑥𝐹𝐵))
62	58, 61	bitrd 278	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑥 = (^◡𝐹‘𝐵) ↔ 𝑥𝐹𝐵))
63	62	anbi1d 630	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑥 = (^◡𝐹‘𝐵) ∧ (^◡𝐹‘𝐴) ≤ 𝑥) ↔ (𝑥𝐹𝐵 ∧ (^◡𝐹‘𝐴) ≤ 𝑥)))
64	63	biancomd 464	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑥 = (^◡𝐹‘𝐵) ∧ (^◡𝐹‘𝐴) ≤ 𝑥) ↔ ((^◡𝐹‘𝐴) ≤ 𝑥 ∧ 𝑥𝐹𝐵)))
65	64	exbidv 1924	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∃𝑥(𝑥 = (^◡𝐹‘𝐵) ∧ (^◡𝐹‘𝐴) ≤ 𝑥) ↔ ∃𝑥((^◡𝐹‘𝐴) ≤ 𝑥 ∧ 𝑥𝐹𝐵)))
66	54, 65	bitr3id 284	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵) ↔ ∃𝑥((^◡𝐹‘𝐴) ≤ 𝑥 ∧ 𝑥𝐹𝐵)))
67	51, 66	bitr4d 281	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((^◡𝐹‘𝐴)(𝐹 ∘ ≤ )𝐵 ↔ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵)))
68	48, 67	bitrid 282	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∃𝑥(𝑥 = (^◡𝐹‘𝐴) ∧ 𝑥(𝐹 ∘ ≤ )𝐵) ↔ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵)))
69	31, 45, 68	3bitrd 304	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ≤ 𝐵 ↔ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵)))
70	69	pm5.32da 579	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ≤ 𝐵) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵))))
71		df-3an 1089	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵)) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵)))
72	70, 71	bitr4di 288	. 2 ⊢ (𝑁 ∈ ℕ₀ → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ≤ 𝐵) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵))))
73	29, 72	bitrd 278	1 ⊢ (𝑁 ∈ ℕ₀ → (𝐴 ≤ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (^◡𝐹‘𝐴) ≤ (^◡𝐹‘𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 ifcif 4528 class class class wbr 5148 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Rel wrel 5681 Fn wfn 6538 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 0cc0 11112 ≤ cle 11251 ℕ₀cn0 12474 ℤcz 12560 ..^cfzo 13629 Basecbs 17146 lecple 17206 ℤRHomczrh 21055 ℤ/nℤczn 21058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-dvds 16200 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-0g 17389 df-imas 17456 df-qus 17457 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-nsg 19006 df-eqg 19007 df-ghm 19092 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-rnghom 20255 df-subrg 20321 df-lmod 20477 df-lss 20548 df-lsp 20588 df-sra 20791 df-rgmod 20792 df-lidl 20793 df-rsp 20794 df-2idl 20863 df-cnfld 20951 df-zring 21024 df-zrh 21059 df-zn 21062

This theorem is referenced by: znleval2 21117

W3C validator