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Theorem 4fvwrd4 13585

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 12433	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
3		elnn0uz 12814	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ ↔ 0 ∈ (ℤ_≥‘0))
4	2, 3	mpbi 230	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
5		3nn0 12436	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
6		elnn0uz 12814	. . . . . . . . . . 11 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (ℤ_≥‘0))
7	5, 6	mpbi 230	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘0)
8		uzss 12792	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘0) → (ℤ_≥‘3) ⊆ (ℤ_≥‘0))
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘3) ⊆ (ℤ_≥‘0)
10	9	sseli 3939	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘0))
11		eluzfz 13456	. . . . . . . 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 7039	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘0) ∈ 𝑉)
15		clel5 3628	. . . . 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 12815	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
18		1z 12539	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
19		3z 12542	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
20		1le3 12369	. . . . . . . . . . 11 ⊢ 1 ≤ 3
21		eluz2 12775	. . . . . . . . . . 11 ⊢ (3 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 3 ∈ ℤ ∧ 1 ≤ 3))
22	18, 19, 20, 21	mpbir3an 1342	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘1)
23		uzss 12792	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘1) → (ℤ_≥‘3) ⊆ (ℤ_≥‘1))
24	22, 23	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘3) ⊆ (ℤ_≥‘1)
25	24	sseli 3939	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘1))
26		eluzfz 13456	. . . . . . . 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 7039	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘1) ∈ 𝑉)
30		clel5 3628	. . . . 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 12816	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
34		uzuzle23 12819	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘2))
35		eluzfz 13456	. . . . . . 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 7039	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘2) ∈ 𝑉)
39		clel5 3628	. . . 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 13456	. . . . . . 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 7039	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘3) ∈ 𝑉)
45		clel5 3628	. . . 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 3167	. . . . . 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 3167	. . . . . . 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 3109	. . 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 3167	. . . . 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 3165	. . . . . 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 3109	. . 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 3165	. . . . . 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 3167	. . . . . . 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 3165	. . . 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 3165	. . . . 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 3911 class class class wbr 5102 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 0cc0 11044 1c1 11045 ≤ cle 11185 2c2 12217 3c3 12218 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ...cfz 13444

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 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

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 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445

This theorem is referenced by: (None)

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