expdioph - Mathbox for Stefan O'Rear

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Theorem expdioph 42064

Description: The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)

**Assertion**
Ref	Expression
expdioph	⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} ∈ (Dioph‘3)

**Proof of Theorem expdioph**
Step	Hyp	Ref	Expression
1		pm4.42 1050	. . . 4 ⊢ ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ (𝑎‘2) ∈ ℕ) ∨ ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ ¬ (𝑎‘2) ∈ ℕ)))
2		ancom 459	. . . . . 6 ⊢ (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ (𝑎‘2) ∈ ℕ) ↔ ((𝑎‘2) ∈ ℕ ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))))
3		elmapi 8845	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → 𝑎:(1...3)⟶ℕ₀)
4		df-2 12279	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
5		df-3 12280	. . . . . . . . . . . . . . . 16 ⊢ 3 = (2 + 1)
6		ssid 4003	. . . . . . . . . . . . . . . 16 ⊢ (1...3) ⊆ (1...3)
7	5, 6	jm2.27dlem5 42054	. . . . . . . . . . . . . . 15 ⊢ (1...2) ⊆ (1...3)
8	4, 7	jm2.27dlem5 42054	. . . . . . . . . . . . . 14 ⊢ (1...1) ⊆ (1...3)
9		1nn 12227	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ
10	9	jm2.27dlem3 42052	. . . . . . . . . . . . . 14 ⊢ 1 ∈ (1...1)
11	8, 10	sselii 3978	. . . . . . . . . . . . 13 ⊢ 1 ∈ (1...3)
12		ffvelcdm 7082	. . . . . . . . . . . . 13 ⊢ ((𝑎:(1...3)⟶ℕ₀ ∧ 1 ∈ (1...3)) → (𝑎‘1) ∈ ℕ₀)
13	3, 11, 12	sylancl 584	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (𝑎‘1) ∈ ℕ₀)
14	13	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (𝑎‘1) ∈ ℕ₀)
15		elnn0 12478	. . . . . . . . . . 11 ⊢ ((𝑎‘1) ∈ ℕ₀ ↔ ((𝑎‘1) ∈ ℕ ∨ (𝑎‘1) = 0))
16	14, 15	sylib 217	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘1) ∈ ℕ ∨ (𝑎‘1) = 0))
17		elnn1uz2 12913	. . . . . . . . . . . 12 ⊢ ((𝑎‘1) ∈ ℕ ↔ ((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)))
18	17	biimpi 215	. . . . . . . . . . 11 ⊢ ((𝑎‘1) ∈ ℕ → ((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)))
19	18	orim1i 906	. . . . . . . . . 10 ⊢ (((𝑎‘1) ∈ ℕ ∨ (𝑎‘1) = 0) → (((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∨ (𝑎‘1) = 0))
20	16, 19	syl 17	. . . . . . . . 9 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∨ (𝑎‘1) = 0))
21	20	biantrurd 531	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ ((((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∨ (𝑎‘1) = 0) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))))
22		andir 1005	. . . . . . . . . 10 ⊢ (((((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∨ (𝑎‘1) = 0) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))))
23		andir 1005	. . . . . . . . . . 11 ⊢ ((((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ (((𝑎‘1) = 1 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ∨ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))))
24	23	orbi1i 910	. . . . . . . . . 10 ⊢ (((((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ↔ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ∨ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))))
25	22, 24	bitri 274	. . . . . . . . 9 ⊢ (((((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∨ (𝑎‘1) = 0) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ∨ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))))
26		nnz 12583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘2) ∈ ℕ → (𝑎‘2) ∈ ℤ)
27		1exp 14061	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘2) ∈ ℤ → (1↑(𝑎‘2)) = 1)
28	26, 27	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘2) ∈ ℕ → (1↑(𝑎‘2)) = 1)
29	28	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (1↑(𝑎‘2)) = 1)
30	29	eqeq2d 2741	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘3) = (1↑(𝑎‘2)) ↔ (𝑎‘3) = 1))
31		oveq1 7418	. . . . . . . . . . . . . . 15 ⊢ ((𝑎‘1) = 1 → ((𝑎‘1)↑(𝑎‘2)) = (1↑(𝑎‘2)))
32	31	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘1) = 1 → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = (1↑(𝑎‘2))))
33	32	bibi1d 342	. . . . . . . . . . . . 13 ⊢ ((𝑎‘1) = 1 → (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = 1) ↔ ((𝑎‘3) = (1↑(𝑎‘2)) ↔ (𝑎‘3) = 1)))
34	30, 33	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘1) = 1 → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = 1)))
35	34	pm5.32d 575	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (((𝑎‘1) = 1 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((𝑎‘1) = 1 ∧ (𝑎‘3) = 1)))
36		iba 526	. . . . . . . . . . . . 13 ⊢ ((𝑎‘2) ∈ ℕ → ((𝑎‘1) ∈ (ℤ_≥‘2) ↔ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ)))
37	36	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘1) ∈ (ℤ_≥‘2) ↔ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ)))
38	37	anbi1d 628	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))))
39	35, 38	orbi12d 915	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((((𝑎‘1) = 1 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ∨ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ↔ (((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))))))
40		0exp 14067	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘2) ∈ ℕ → (0↑(𝑎‘2)) = 0)
41	40	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (0↑(𝑎‘2)) = 0)
42	41	eqeq2d 2741	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘3) = (0↑(𝑎‘2)) ↔ (𝑎‘3) = 0))
43		oveq1 7418	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘1) = 0 → ((𝑎‘1)↑(𝑎‘2)) = (0↑(𝑎‘2)))
44	43	eqeq2d 2741	. . . . . . . . . . . . 13 ⊢ ((𝑎‘1) = 0 → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = (0↑(𝑎‘2))))
45	44	bibi1d 342	. . . . . . . . . . . 12 ⊢ ((𝑎‘1) = 0 → (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = 0) ↔ ((𝑎‘3) = (0↑(𝑎‘2)) ↔ (𝑎‘3) = 0)))
46	42, 45	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘1) = 0 → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = 0)))
47	46	pm5.32d 575	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (((𝑎‘1) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)))
48	39, 47	orbi12d 915	. . . . . . . . 9 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (((((𝑎‘1) = 1 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ∨ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ↔ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))))
49	25, 48	bitrid 282	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → (((((𝑎‘1) = 1 ∨ (𝑎‘1) ∈ (ℤ_≥‘2)) ∨ (𝑎‘1) = 0) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))))
50	21, 49	bitrd 278	. . . . . . 7 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘2) ∈ ℕ) → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))))
51	50	pm5.32da 577	. . . . . 6 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (((𝑎‘2) ∈ ℕ ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)))))
52	2, 51	bitrid 282	. . . . 5 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ (𝑎‘2) ∈ ℕ) ↔ ((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)))))
53		ancom 459	. . . . . 6 ⊢ (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ ¬ (𝑎‘2) ∈ ℕ) ↔ (¬ (𝑎‘2) ∈ ℕ ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))))
54		2nn 12289	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
55	54	jm2.27dlem3 42052	. . . . . . . . . . 11 ⊢ 2 ∈ (1...2)
56	7, 55	sselii 3978	. . . . . . . . . 10 ⊢ 2 ∈ (1...3)
57		ffvelcdm 7082	. . . . . . . . . 10 ⊢ ((𝑎:(1...3)⟶ℕ₀ ∧ 2 ∈ (1...3)) → (𝑎‘2) ∈ ℕ₀)
58	3, 56, 57	sylancl 584	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (𝑎‘2) ∈ ℕ₀)
59		elnn0 12478	. . . . . . . . . . 11 ⊢ ((𝑎‘2) ∈ ℕ₀ ↔ ((𝑎‘2) ∈ ℕ ∨ (𝑎‘2) = 0))
60		pm2.53 847	. . . . . . . . . . 11 ⊢ (((𝑎‘2) ∈ ℕ ∨ (𝑎‘2) = 0) → (¬ (𝑎‘2) ∈ ℕ → (𝑎‘2) = 0))
61	59, 60	sylbi 216	. . . . . . . . . 10 ⊢ ((𝑎‘2) ∈ ℕ₀ → (¬ (𝑎‘2) ∈ ℕ → (𝑎‘2) = 0))
62		0nnn 12252	. . . . . . . . . . 11 ⊢ ¬ 0 ∈ ℕ
63		eleq1 2819	. . . . . . . . . . 11 ⊢ ((𝑎‘2) = 0 → ((𝑎‘2) ∈ ℕ ↔ 0 ∈ ℕ))
64	62, 63	mtbiri 326	. . . . . . . . . 10 ⊢ ((𝑎‘2) = 0 → ¬ (𝑎‘2) ∈ ℕ)
65	61, 64	impbid1 224	. . . . . . . . 9 ⊢ ((𝑎‘2) ∈ ℕ₀ → (¬ (𝑎‘2) ∈ ℕ ↔ (𝑎‘2) = 0))
66	58, 65	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (¬ (𝑎‘2) ∈ ℕ ↔ (𝑎‘2) = 0))
67	66	anbi1d 628	. . . . . . 7 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → ((¬ (𝑎‘2) ∈ ℕ ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((𝑎‘2) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))))
68	13	nn0cnd 12538	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (𝑎‘1) ∈ ℂ)
69	68	exp0d 14109	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → ((𝑎‘1)↑0) = 1)
70	69	eqeq2d 2741	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → ((𝑎‘3) = ((𝑎‘1)↑0) ↔ (𝑎‘3) = 1))
71		oveq2 7419	. . . . . . . . . . 11 ⊢ ((𝑎‘2) = 0 → ((𝑎‘1)↑(𝑎‘2)) = ((𝑎‘1)↑0))
72	71	eqeq2d 2741	. . . . . . . . . 10 ⊢ ((𝑎‘2) = 0 → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = ((𝑎‘1)↑0)))
73	72	bibi1d 342	. . . . . . . . 9 ⊢ ((𝑎‘2) = 0 → (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = 1) ↔ ((𝑎‘3) = ((𝑎‘1)↑0) ↔ (𝑎‘3) = 1)))
74	70, 73	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → ((𝑎‘2) = 0 → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (𝑎‘3) = 1)))
75	74	pm5.32d 575	. . . . . . 7 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (((𝑎‘2) = 0 ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1)))
76	67, 75	bitrd 278	. . . . . 6 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → ((¬ (𝑎‘2) ∈ ℕ ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))) ↔ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1)))
77	53, 76	bitrid 282	. . . . 5 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ ¬ (𝑎‘2) ∈ ℕ) ↔ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1)))
78	52, 77	orbi12d 915	. . . 4 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → ((((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ (𝑎‘2) ∈ ℕ) ∨ ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ∧ ¬ (𝑎‘2) ∈ ℕ)) ↔ (((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))) ∨ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1))))
79	1, 78	bitrid 282	. . 3 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → ((𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)) ↔ (((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))) ∨ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1))))
80	79	rabbiia 3434	. 2 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} = {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))) ∨ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1))}
81		3nn0 12494	. . . . 5 ⊢ 3 ∈ ℕ₀
82		ovex 7444	. . . . . 6 ⊢ (1...3) ∈ V
83		mzpproj 41777	. . . . . 6 ⊢ (((1...3) ∈ V ∧ 2 ∈ (1...3)) → (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘2)) ∈ (mzPoly‘(1...3)))
84	82, 56, 83	mp2an 688	. . . . 5 ⊢ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘2)) ∈ (mzPoly‘(1...3))
85		elnnrabdioph 41847	. . . . 5 ⊢ ((3 ∈ ℕ₀ ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘2)) ∈ (mzPoly‘(1...3))) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘2) ∈ ℕ} ∈ (Dioph‘3))
86	81, 84, 85	mp2an 688	. . . 4 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘2) ∈ ℕ} ∈ (Dioph‘3)
87		mzpproj 41777	. . . . . . . . 9 ⊢ (((1...3) ∈ V ∧ 1 ∈ (1...3)) → (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘1)) ∈ (mzPoly‘(1...3)))
88	82, 11, 87	mp2an 688	. . . . . . . 8 ⊢ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘1)) ∈ (mzPoly‘(1...3))
89		1z 12596	. . . . . . . . 9 ⊢ 1 ∈ ℤ
90		mzpconstmpt 41780	. . . . . . . . 9 ⊢ (((1...3) ∈ V ∧ 1 ∈ ℤ) → (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ 1) ∈ (mzPoly‘(1...3)))
91	82, 89, 90	mp2an 688	. . . . . . . 8 ⊢ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ 1) ∈ (mzPoly‘(1...3))
92		eqrabdioph 41817	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘1)) ∈ (mzPoly‘(1...3)) ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ 1) ∈ (mzPoly‘(1...3))) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘1) = 1} ∈ (Dioph‘3))
93	81, 88, 91, 92	mp3an 1459	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘1) = 1} ∈ (Dioph‘3)
94		3nn 12295	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
95	94	jm2.27dlem3 42052	. . . . . . . . 9 ⊢ 3 ∈ (1...3)
96		mzpproj 41777	. . . . . . . . 9 ⊢ (((1...3) ∈ V ∧ 3 ∈ (1...3)) → (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘3)) ∈ (mzPoly‘(1...3)))
97	82, 95, 96	mp2an 688	. . . . . . . 8 ⊢ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘3)) ∈ (mzPoly‘(1...3))
98		eqrabdioph 41817	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘3)) ∈ (mzPoly‘(1...3)) ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ 1) ∈ (mzPoly‘(1...3))) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = 1} ∈ (Dioph‘3))
99	81, 97, 91, 98	mp3an 1459	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = 1} ∈ (Dioph‘3)
100		anrabdioph 41820	. . . . . . 7 ⊢ (({𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘1) = 1} ∈ (Dioph‘3) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = 1} ∈ (Dioph‘3)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) = 1 ∧ (𝑎‘3) = 1)} ∈ (Dioph‘3))
101	93, 99, 100	mp2an 688	. . . . . 6 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) = 1 ∧ (𝑎‘3) = 1)} ∈ (Dioph‘3)
102		expdiophlem2 42063	. . . . . 6 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))} ∈ (Dioph‘3)
103		orrabdioph 41821	. . . . . 6 ⊢ (({𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) = 1 ∧ (𝑎‘3) = 1)} ∈ (Dioph‘3) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))} ∈ (Dioph‘3)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))))} ∈ (Dioph‘3))
104	101, 102, 103	mp2an 688	. . . . 5 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))))} ∈ (Dioph‘3)
105		eq0rabdioph 41816	. . . . . . 7 ⊢ ((3 ∈ ℕ₀ ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘1)) ∈ (mzPoly‘(1...3))) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘1) = 0} ∈ (Dioph‘3))
106	81, 88, 105	mp2an 688	. . . . . 6 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘1) = 0} ∈ (Dioph‘3)
107		eq0rabdioph 41816	. . . . . . 7 ⊢ ((3 ∈ ℕ₀ ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘3)) ∈ (mzPoly‘(1...3))) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = 0} ∈ (Dioph‘3))
108	81, 97, 107	mp2an 688	. . . . . 6 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = 0} ∈ (Dioph‘3)
109		anrabdioph 41820	. . . . . 6 ⊢ (({𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘1) = 0} ∈ (Dioph‘3) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = 0} ∈ (Dioph‘3)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)} ∈ (Dioph‘3))
110	106, 108, 109	mp2an 688	. . . . 5 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)} ∈ (Dioph‘3)
111		orrabdioph 41821	. . . . 5 ⊢ (({𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))))} ∈ (Dioph‘3) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)} ∈ (Dioph‘3)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))} ∈ (Dioph‘3))
112	104, 110, 111	mp2an 688	. . . 4 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))} ∈ (Dioph‘3)
113		anrabdioph 41820	. . . 4 ⊢ (({𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘2) ∈ ℕ} ∈ (Dioph‘3) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))} ∈ (Dioph‘3)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)))} ∈ (Dioph‘3))
114	86, 112, 113	mp2an 688	. . 3 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)))} ∈ (Dioph‘3)
115		eq0rabdioph 41816	. . . . 5 ⊢ ((3 ∈ ℕ₀ ∧ (𝑎 ∈ (ℤ ↑_m (1...3)) ↦ (𝑎‘2)) ∈ (mzPoly‘(1...3))) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘2) = 0} ∈ (Dioph‘3))
116	81, 84, 115	mp2an 688	. . . 4 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘2) = 0} ∈ (Dioph‘3)
117		anrabdioph 41820	. . . 4 ⊢ (({𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘2) = 0} ∈ (Dioph‘3) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = 1} ∈ (Dioph‘3)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1)} ∈ (Dioph‘3))
118	116, 99, 117	mp2an 688	. . 3 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1)} ∈ (Dioph‘3)
119		orrabdioph 41821	. . 3 ⊢ (({𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0)))} ∈ (Dioph‘3) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1)} ∈ (Dioph‘3)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))) ∨ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1))} ∈ (Dioph‘3))
120	114, 118, 119	mp2an 688	. 2 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (((𝑎‘2) ∈ ℕ ∧ ((((𝑎‘1) = 1 ∧ (𝑎‘3) = 1) ∨ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))) ∨ ((𝑎‘1) = 0 ∧ (𝑎‘3) = 0))) ∨ ((𝑎‘2) = 0 ∧ (𝑎‘3) = 1))} ∈ (Dioph‘3)
121	80, 120	eqeltri 2827	1 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} ∈ (Dioph‘3)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 {crab 3430 Vcvv 3472 ↦ cmpt 5230 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ↑_m cmap 8822 0cc0 11112 1c1 11113 ℕcn 12216 2c2 12271 3c3 12272 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13488 ↑cexp 14031 mzPolycmzp 41762 Diophcdioph 41795

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 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

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-dvds 16202 df-gcd 16440 df-prm 16613 df-numer 16675 df-denom 16676 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-mzpcl 41763 df-mzp 41764 df-dioph 41796 df-squarenn 41881 df-pell1qr 41882 df-pell14qr 41883 df-pell1234qr 41884 df-pellfund 41885 df-rmx 41942 df-rmy 41943

This theorem is referenced by: (None)

W3C validator