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Theorem 4fvwrd4 13551

Description: The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)

**Assertion**
Ref	Expression
4fvwrd4	⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))

Distinct variable groups: 𝑃,𝑎,𝑏,𝑐,𝑑 𝑉,𝑎,𝑏,𝑐,𝑑 Allowed substitution hints: 𝐿(𝑎,𝑏,𝑐,𝑑)

**Proof of Theorem 4fvwrd4**
Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 𝑃:(0...𝐿)⟶𝑉)
2		0nn0 12399	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
3		elnn0uz 12780	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ ↔ 0 ∈ (ℤ_≥‘0))
4	2, 3	mpbi 230	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
5		3nn0 12402	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
6		elnn0uz 12780	. . . . . . . . . . 11 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (ℤ_≥‘0))
7	5, 6	mpbi 230	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘0)
8		uzss 12758	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘0) → (ℤ_≥‘3) ⊆ (ℤ_≥‘0))
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘3) ⊆ (ℤ_≥‘0)
10	9	sseli 3931	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘0))
11		eluzfz 13422	. . . . . . . 8 ⊢ ((0 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘0)) → 0 ∈ (0...𝐿))
12	4, 10, 11	sylancr 587	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 0 ∈ (0...𝐿))
13	12	adantr 480	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 0 ∈ (0...𝐿))
14	1, 13	ffvelcdmd 7019	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘0) ∈ 𝑉)
15		clel5 3620	. . . . 5 ⊢ ((𝑃‘0) ∈ 𝑉 ↔ ∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎)
16	14, 15	sylib 218	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎)
17		1eluzge0 12781	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
18		1z 12505	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
19		3z 12508	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
20		1le3 12335	. . . . . . . . . . 11 ⊢ 1 ≤ 3
21		eluz2 12741	. . . . . . . . . . 11 ⊢ (3 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 3 ∈ ℤ ∧ 1 ≤ 3))
22	18, 19, 20, 21	mpbir3an 1342	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘1)
23		uzss 12758	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘1) → (ℤ_≥‘3) ⊆ (ℤ_≥‘1))
24	22, 23	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘3) ⊆ (ℤ_≥‘1)
25	24	sseli 3931	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘1))
26		eluzfz 13422	. . . . . . . 8 ⊢ ((1 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘1)) → 1 ∈ (0...𝐿))
27	17, 25, 26	sylancr 587	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 1 ∈ (0...𝐿))
28	27	adantr 480	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 1 ∈ (0...𝐿))
29	1, 28	ffvelcdmd 7019	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘1) ∈ 𝑉)
30		clel5 3620	. . . . 5 ⊢ ((𝑃‘1) ∈ 𝑉 ↔ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏)
31	29, 30	sylib 218	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏)
32	16, 31	jca 511	. . 3 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏))
33		2eluzge0 12782	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
34		uzuzle23 12785	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘2))
35		eluzfz 13422	. . . . . . 7 ⊢ ((2 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘2)) → 2 ∈ (0...𝐿))
36	33, 34, 35	sylancr 587	. . . . . 6 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 2 ∈ (0...𝐿))
37	36	adantr 480	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 2 ∈ (0...𝐿))
38	1, 37	ffvelcdmd 7019	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘2) ∈ 𝑉)
39		clel5 3620	. . . 4 ⊢ ((𝑃‘2) ∈ 𝑉 ↔ ∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐)
40	38, 39	sylib 218	. . 3 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐)
41		eluzfz 13422	. . . . . . 7 ⊢ ((3 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘3)) → 3 ∈ (0...𝐿))
42	7, 41	mpan 690	. . . . . 6 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 3 ∈ (0...𝐿))
43	42	adantr 480	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 3 ∈ (0...𝐿))
44	1, 43	ffvelcdmd 7019	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘3) ∈ 𝑉)
45		clel5 3620	. . . 4 ⊢ ((𝑃‘3) ∈ 𝑉 ↔ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)
46	44, 45	sylib 218	. . 3 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)
47	32, 40, 46	jca32 515	. 2 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
48		r19.42v 3161	. . . . . 6 ⊢ (∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑑 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
49		r19.42v 3161	. . . . . . 7 ⊢ (∃𝑑 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) ↔ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑))
50	49	anbi2i 623	. . . . . 6 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑑 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
51	48, 50	bitri 275	. . . . 5 ⊢ (∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
52	51	rexbii 3076	. . . 4 ⊢ (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ ∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
53	52	2rexbii 3105	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
54		r19.42v 3161	. . . . 5 ⊢ (∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑐 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
55		r19.41v 3159	. . . . . 6 ⊢ (∃𝑐 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑) ↔ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑))
56	55	anbi2i 623	. . . . 5 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑐 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
57	54, 56	bitri 275	. . . 4 ⊢ (∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
58	57	2rexbii 3105	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
59		r19.41v 3159	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (∃𝑏 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
60		r19.42v 3161	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ↔ ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏))
61	60	anbi1i 624	. . . . . 6 ⊢ ((∃𝑏 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
62	59, 61	bitri 275	. . . . 5 ⊢ (∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
63	62	rexbii 3076	. . . 4 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ∃𝑎 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
64		r19.41v 3159	. . . 4 ⊢ (∃𝑎 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (∃𝑎 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
65		r19.41v 3159	. . . . 5 ⊢ (∃𝑎 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ↔ (∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏))
66	65	anbi1i 624	. . . 4 ⊢ ((∃𝑎 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
67	63, 64, 66	3bitri 297	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
68	53, 58, 67	3bitri 297	. 2 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
69	47, 68	sylibr 234	1 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 ⊆ wss 3903 class class class wbr 5092 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 0cc0 11009 1c1 11010 ≤ cle 11150 2c2 12183 3c3 12184 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ...cfz 13410

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411

This theorem is referenced by: (None)

W3C validator