Theorem seqlem1 4662

Description: We prove by induction that the first member of the ordered pair value of the internal sequence of seq equals its index.

Hypothesis

Ref	Expression
seqlem1.1	⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, 1) ↾ ω)

Assertion

Ref	Expression
seqlem1	⊢ (A ∈ ℕ → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘A)) = A)

Distinct variable group(s):   x,y   z,w   w,B   x,C

Proof of Theorem seqlem1

Step	Hyp	Ref	Expression
1		fveq2 2832	. . . 4 ⊢ (v = 1 → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v) = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘1))
2	1	fveq2d 2836	. . 3 ⊢ (v = 1 → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘1)))
3		id 9	. . 3 ⊢ (v = 1 → v = 1)
4	2, 3	cleq12d 1115	. 2 ⊢ (v = 1 → ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = v ↔ (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘1)) = 1))
5		fveq2 2832	. . . 4 ⊢ (v = u → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v) = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u))
6	5	fveq2d 2836	. . 3 ⊢ (v = u → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)))
7		id 9	. . 3 ⊢ (v = u → v = u)
8	6, 7	cleq12d 1115	. 2 ⊢ (v = u → ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = v ↔ (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) = u))
9		fveq2 2832	. . . 4 ⊢ (v = (u + 1) → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v) = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1)))
10	9	fveq2d 2836	. . 3 ⊢ (v = (u + 1) → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1))))
11		id 9	. . 3 ⊢ (v = (u + 1) → v = (u + 1))
12	10, 11	cleq12d 1115	. 2 ⊢ (v = (u + 1) → ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = v ↔ (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1))) = (u + 1)))
13		fveq2 2832	. . . 4 ⊢ (v = A → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v) = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘A))
14	13	fveq2d 2836	. . 3 ⊢ (v = A → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘A)))
15		id 9	. . 3 ⊢ (v = A → v = A)
16	14, 15	cleq12d 1115	. 2 ⊢ (v = A → ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘v)) = v ↔ (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘A)) = A))
17		opex 1893	. . . . 5 ⊢ ⟨1, C⟩ ∈ V
18		1z 4584	. . . . . 6 ⊢ 1 ∈ ℤ
19		seqlem1.1	. . . . . 6 ⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, 1) ↾ ω)
20	18, 19	uzrdgini 4658	. . . . 5 ⊢ (⟨1, C⟩ ∈ V → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘1) = ⟨1, C⟩)
21	17, 20	ax-mp 6	. . . 4 ⊢ ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘1) = ⟨1, C⟩
22	21	fveq2i 2835	. . 3 ⊢ (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘1)) = (1st ‘⟨1, C⟩)
23	18	elisseti 1355	. . . 4 ⊢ 1 ∈ V
24	23	op1st 3091	. . 3 ⊢ (1st ‘⟨1, C⟩) = 1
25	22, 24	eqtr 1119	. 2 ⊢ (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘1)) = 1
26		nnz 4582	. . . . . . . . 9 ⊢ ℕ = {v ∈ ℤ∣1 ≤ v}
27	26	eleq2i 1153	. . . . . . . 8 ⊢ (u ∈ ℕ ↔ u ∈ {v ∈ ℤ∣1 ≤ v})
28	18, 19	uzrdgsuc 4659	. . . . . . . 8 ⊢ (u ∈ {v ∈ ℤ∣1 ≤ v} → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1)) = ({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩} ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)))
29	27, 28	sylbi 174	. . . . . . 7 ⊢ (u ∈ ℕ → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1)) = ({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩} ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)))
30		ax-17 925	. . . . . . . . . 10 ⊢ (w = ⟨((1st ‘z) + 1), B⟩ → ∀v w = ⟨((1st ‘z) + 1), B⟩)
31		ax-17 925	. . . . . . . . . . . 12 ⊢ (w ∈ ((1st ‘v) + 1) → ∀z w ∈ ((1st ‘v) + 1))
32		hbs1 986	. . . . . . . . . . . . 13 ⊢ ([v / z]t ∈ B → ∀z[v / z]t ∈ B)
33	32	hbab 1096	. . . . . . . . . . . 12 ⊢ (w ∈ {t∣[v / z]t ∈ B} → ∀z w ∈ {t∣[v / z]t ∈ B})
34	31, 33	hbop 1879	. . . . . . . . . . 11 ⊢ (w ∈ ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩ → ∀z w ∈ ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩)
35	34	hbeleq 1173	. . . . . . . . . 10 ⊢ (w = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩ → ∀z w = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩)
36		opeq12 1878	. . . . . . . . . . . 12 ⊢ ((((1st ‘z) + 1) = ((1st ‘v) + 1) ∧ B = {t∣[v / z]t ∈ B}) → ⟨((1st ‘z) + 1), B⟩ = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩)
37		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (z = v → (1st ‘z) = (1st ‘v))
38	37	opreq1d 3012	. . . . . . . . . . . 12 ⊢ (z = v → ((1st ‘z) + 1) = ((1st ‘v) + 1))
39		sbab 1188	. . . . . . . . . . . 12 ⊢ (z = v → B = {t∣[v / z]t ∈ B})
40	36, 38, 39	sylanc 361	. . . . . . . . . . 11 ⊢ (z = v → ⟨((1st ‘z) + 1), B⟩ = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩)
41	40	cleq2d 1112	. . . . . . . . . 10 ⊢ (z = v → (w = ⟨((1st ‘z) + 1), B⟩ ↔ w = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩))
42	30, 35, 41	cbvopab1 2106	. . . . . . . . 9 ⊢ {⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩} = {⟨v, w⟩∣w = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩}
43	42	fveq1i 2833	. . . . . . . 8 ⊢ ({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩} ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) = ({⟨v, w⟩∣w = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩} ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u))
44		ax-17 925	. . . . . . . . 9 ⊢ (f ∈ ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) → ∀v f ∈ ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u))
45		ax-17 925	. . . . . . . . 9 ⊢ (f ∈ ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩ → ∀v f ∈ ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩)
46		fvex 2838	. . . . . . . . 9 ⊢ ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) ∈ V
47		opex 1893	. . . . . . . . 9 ⊢ ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩ ∈ V
48		opeq12 1878	. . . . . . . . . 10 ⊢ ((((1st ‘v) + 1) = ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1) ∧ {t∣[v / z]t ∈ B} = {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}) → ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩ = ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩)
49		fveq2 2832	. . . . . . . . . . 11 ⊢ (v = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) → (1st ‘v) = (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)))
50	49	opreq1d 3012	. . . . . . . . . 10 ⊢ (v = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) → ((1st ‘v) + 1) = ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1))
51		dfsbcq 1442	. . . . . . . . . . 11 ⊢ (v = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) → ([v / z]t ∈ B ↔ [((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B))
52	51	biabdv 1183	. . . . . . . . . 10 ⊢ (v = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) → {t∣[v / z]t ∈ B} = {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B})
53	48, 50, 52	sylanc 361	. . . . . . . . 9 ⊢ (v = ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) → ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩ = ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩)
54	44, 45, 46, 47, 53	fvopabf 2876	. . . . . . . 8 ⊢ ({⟨v, w⟩∣w = ⟨((1st ‘v) + 1), {t∣[v / z]t ∈ B}⟩} ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) = ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩
55	43, 54	eqtr 1119	. . . . . . 7 ⊢ ({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩} ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) = ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩
56	29, 55	syl6eq 1140	. . . . . 6 ⊢ (u ∈ ℕ → ((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1)) = ⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩)
57	56	fveq2d 2836	. . . . 5 ⊢ (u ∈ ℕ → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1))) = (1st ‘⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩))
58		oprex 3018	. . . . . 6 ⊢ ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1) ∈ V
59	58	op1st 3091	. . . . 5 ⊢ (1st ‘⟨((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1), {t∣[((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u) / z]t ∈ B}⟩) = ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1)
60	57, 59	syl6eq 1140	. . . 4 ⊢ (u ∈ ℕ → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1))) = ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1))
61		opreq1 3006	. . . 4 ⊢ ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) = u → ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) + 1) = (u + 1))
62	60, 61	sylan9eq 1144	. . 3 ⊢ ((u ∈ ℕ ∧ (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) = u) → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1))) = (u + 1))
63	62	exp 291	. 2 ⊢ (u ∈ ℕ → ((1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘u)) = u → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘(u + 1))) = (u + 1)))
64	4, 8, 12, 16, 25, 63	nnind 4434	1 ⊢ (A ∈ ℕ → (1st ‘((rec({⟨z, w⟩∣w = ⟨((1st ‘z) + 1), B⟩}, ⟨1, C⟩) ∘ ◡G) ‘A)) = A)

Syntax hints:   → wi 2   = weq 797  [wsb 852  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348  [wsbc 1440  ⟨cop 1810   class class class wbr 2054  {copab 2055  ωcom 2372  ^◡ccnv 2409   ↾ cres 2412   ∘ ccom 2414   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1^st c1st 3085  1c1 4029   + caddc 4031   ≤ cle 4092  ℕcn 4093  ℤcz 4095

This theorem is referenced by:  seqsuclem 4669

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-le 4277  df-n 4423  df-n0 4535  df-z 4564

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