rabren3dioph - Mathbox for Stefan O'Rear

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Theorem rabren3dioph 41539

Description: Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)

**Hypotheses**
Ref	Expression
rabren3dioph.a	⊢ (((𝑎‘1) = (𝑏‘𝑋) ∧ (𝑎‘2) = (𝑏‘𝑌) ∧ (𝑎‘3) = (𝑏‘𝑍)) → (𝜑 ↔ 𝜓))
rabren3dioph.b	⊢ 𝑋 ∈ (1...𝑁)
rabren3dioph.c	⊢ 𝑌 ∈ (1...𝑁)
rabren3dioph.d	⊢ 𝑍 ∈ (1...𝑁)

**Assertion**
Ref	Expression
rabren3dioph	⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁))

Distinct variable groups: 𝜓,𝑎 𝜑,𝑏 𝑋,𝑎,𝑏 𝑌,𝑎,𝑏 𝑍,𝑎,𝑏 𝑁,𝑎,𝑏 Allowed substitution hints: 𝜑(𝑎) 𝜓(𝑏)

**Proof of Theorem rabren3dioph**
Step	Hyp	Ref	Expression
1		vex 3479	. . . . 5 ⊢ 𝑏 ∈ V
2		tpex 7731	. . . . 5 ⊢ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩} ∈ V
3	1, 2	coex 7918	. . . 4 ⊢ (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) ∈ V
4		1ne2 12417	. . . . . . . . . 10 ⊢ 1 ≠ 2
5		1re 11211	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
6		1lt3 12382	. . . . . . . . . . 11 ⊢ 1 < 3
7	5, 6	ltneii 11324	. . . . . . . . . 10 ⊢ 1 ≠ 3
8		2re 12283	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
9		2lt3 12381	. . . . . . . . . . 11 ⊢ 2 < 3
10	8, 9	ltneii 11324	. . . . . . . . . 10 ⊢ 2 ≠ 3
11		1ex 11207	. . . . . . . . . . 11 ⊢ 1 ∈ V
12		2ex 12286	. . . . . . . . . . 11 ⊢ 2 ∈ V
13		3ex 12291	. . . . . . . . . . 11 ⊢ 3 ∈ V
14		rabren3dioph.b	. . . . . . . . . . . 12 ⊢ 𝑋 ∈ (1...𝑁)
15	14	elexi 3494	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
16		rabren3dioph.c	. . . . . . . . . . . 12 ⊢ 𝑌 ∈ (1...𝑁)
17	16	elexi 3494	. . . . . . . . . . 11 ⊢ 𝑌 ∈ V
18		rabren3dioph.d	. . . . . . . . . . . 12 ⊢ 𝑍 ∈ (1...𝑁)
19	18	elexi 3494	. . . . . . . . . . 11 ⊢ 𝑍 ∈ V
20	11, 12, 13, 15, 17, 19	fntp 6607	. . . . . . . . . 10 ⊢ ((1 ≠ 2 ∧ 1 ≠ 3 ∧ 2 ≠ 3) → {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩} Fn {1, 2, 3})
21	4, 7, 10, 20	mp3an 1462	. . . . . . . . 9 ⊢ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩} Fn {1, 2, 3}
22	11	tpid1 4772	. . . . . . . . 9 ⊢ 1 ∈ {1, 2, 3}
23		fvco2 6986	. . . . . . . . 9 ⊢ (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩} Fn {1, 2, 3} ∧ 1 ∈ {1, 2, 3}) → ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘1) = (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘1)))
24	21, 22, 23	mp2an 691	. . . . . . . 8 ⊢ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘1) = (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘1))
25	11, 15	fvtp1 7193	. . . . . . . . . 10 ⊢ ((1 ≠ 2 ∧ 1 ≠ 3) → ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘1) = 𝑋)
26	4, 7, 25	mp2an 691	. . . . . . . . 9 ⊢ ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘1) = 𝑋
27	26	fveq2i 6892	. . . . . . . 8 ⊢ (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘1)) = (𝑏‘𝑋)
28	24, 27	eqtri 2761	. . . . . . 7 ⊢ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘1) = (𝑏‘𝑋)
29	12	tpid2 4774	. . . . . . . . 9 ⊢ 2 ∈ {1, 2, 3}
30		fvco2 6986	. . . . . . . . 9 ⊢ (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩} Fn {1, 2, 3} ∧ 2 ∈ {1, 2, 3}) → ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘2) = (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘2)))
31	21, 29, 30	mp2an 691	. . . . . . . 8 ⊢ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘2) = (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘2))
32	12, 17	fvtp2 7194	. . . . . . . . . 10 ⊢ ((1 ≠ 2 ∧ 2 ≠ 3) → ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘2) = 𝑌)
33	4, 10, 32	mp2an 691	. . . . . . . . 9 ⊢ ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘2) = 𝑌
34	33	fveq2i 6892	. . . . . . . 8 ⊢ (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘2)) = (𝑏‘𝑌)
35	31, 34	eqtri 2761	. . . . . . 7 ⊢ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘2) = (𝑏‘𝑌)
36	13	tpid3 4777	. . . . . . . . 9 ⊢ 3 ∈ {1, 2, 3}
37		fvco2 6986	. . . . . . . . 9 ⊢ (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩} Fn {1, 2, 3} ∧ 3 ∈ {1, 2, 3}) → ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘3) = (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘3)))
38	21, 36, 37	mp2an 691	. . . . . . . 8 ⊢ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘3) = (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘3))
39	13, 19	fvtp3 7195	. . . . . . . . . 10 ⊢ ((1 ≠ 3 ∧ 2 ≠ 3) → ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘3) = 𝑍)
40	7, 10, 39	mp2an 691	. . . . . . . . 9 ⊢ ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘3) = 𝑍
41	40	fveq2i 6892	. . . . . . . 8 ⊢ (𝑏‘({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}‘3)) = (𝑏‘𝑍)
42	38, 41	eqtri 2761	. . . . . . 7 ⊢ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘3) = (𝑏‘𝑍)
43	28, 35, 42	3pm3.2i 1340	. . . . . 6 ⊢ (((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘1) = (𝑏‘𝑋) ∧ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘2) = (𝑏‘𝑌) ∧ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘3) = (𝑏‘𝑍))
44		fveq1 6888	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → (𝑎‘1) = ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘1))
45	44	eqeq1d 2735	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → ((𝑎‘1) = (𝑏‘𝑋) ↔ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘1) = (𝑏‘𝑋)))
46		fveq1 6888	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → (𝑎‘2) = ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘2))
47	46	eqeq1d 2735	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → ((𝑎‘2) = (𝑏‘𝑌) ↔ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘2) = (𝑏‘𝑌)))
48		fveq1 6888	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → (𝑎‘3) = ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘3))
49	48	eqeq1d 2735	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → ((𝑎‘3) = (𝑏‘𝑍) ↔ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘3) = (𝑏‘𝑍)))
50	45, 47, 49	3anbi123d 1437	. . . . . 6 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → (((𝑎‘1) = (𝑏‘𝑋) ∧ (𝑎‘2) = (𝑏‘𝑌) ∧ (𝑎‘3) = (𝑏‘𝑍)) ↔ (((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘1) = (𝑏‘𝑋) ∧ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘2) = (𝑏‘𝑌) ∧ ((𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩})‘3) = (𝑏‘𝑍))))
51	43, 50	mpbiri 258	. . . . 5 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → ((𝑎‘1) = (𝑏‘𝑋) ∧ (𝑎‘2) = (𝑏‘𝑌) ∧ (𝑎‘3) = (𝑏‘𝑍)))
52		rabren3dioph.a	. . . . 5 ⊢ (((𝑎‘1) = (𝑏‘𝑋) ∧ (𝑎‘2) = (𝑏‘𝑌) ∧ (𝑎‘3) = (𝑏‘𝑍)) → (𝜑 ↔ 𝜓))
53	51, 52	syl 17	. . . 4 ⊢ (𝑎 = (𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) → (𝜑 ↔ 𝜓))
54	3, 53	sbcie 3820	. . 3 ⊢ ([(𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) / 𝑎]𝜑 ↔ 𝜓)
55	54	rabbii 3439	. 2 ⊢ {𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ [(𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) / 𝑎]𝜑} = {𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝜓}
56	11, 12, 13, 15, 17, 19, 4, 7, 10	ftp 7152	. . . . 5 ⊢ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:{1, 2, 3}⟶{𝑋, 𝑌, 𝑍}
57		1z 12589	. . . . . . . 8 ⊢ 1 ∈ ℤ
58		fztp 13554	. . . . . . . 8 ⊢ (1 ∈ ℤ → (1...(1 + 2)) = {1, (1 + 1), (1 + 2)})
59	57, 58	ax-mp 5	. . . . . . 7 ⊢ (1...(1 + 2)) = {1, (1 + 1), (1 + 2)}
60		1p2e3 12352	. . . . . . . 8 ⊢ (1 + 2) = 3
61	60	oveq2i 7417	. . . . . . 7 ⊢ (1...(1 + 2)) = (1...3)
62		eqidd 2734	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 = 1)
63		1p1e2 12334	. . . . . . . . . 10 ⊢ (1 + 1) = 2
64	63	a1i 11	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (1 + 1) = 2)
65	60	a1i 11	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (1 + 2) = 3)
66	62, 64, 65	tpeq123d 4752	. . . . . . . 8 ⊢ (1 ∈ ℤ → {1, (1 + 1), (1 + 2)} = {1, 2, 3})
67	57, 66	ax-mp 5	. . . . . . 7 ⊢ {1, (1 + 1), (1 + 2)} = {1, 2, 3}
68	59, 61, 67	3eqtr3i 2769	. . . . . 6 ⊢ (1...3) = {1, 2, 3}
69	68	feq2i 6707	. . . . 5 ⊢ ({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:(1...3)⟶{𝑋, 𝑌, 𝑍} ↔ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:{1, 2, 3}⟶{𝑋, 𝑌, 𝑍})
70	56, 69	mpbir 230	. . . 4 ⊢ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:(1...3)⟶{𝑋, 𝑌, 𝑍}
71	14, 16, 18	3pm3.2i 1340	. . . . 5 ⊢ (𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑍 ∈ (1...𝑁))
72	15, 17, 19	tpss 4838	. . . . 5 ⊢ ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑍 ∈ (1...𝑁)) ↔ {𝑋, 𝑌, 𝑍} ⊆ (1...𝑁))
73	71, 72	mpbi 229	. . . 4 ⊢ {𝑋, 𝑌, 𝑍} ⊆ (1...𝑁)
74		fss 6732	. . . 4 ⊢ (({⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:(1...3)⟶{𝑋, 𝑌, 𝑍} ∧ {𝑋, 𝑌, 𝑍} ⊆ (1...𝑁)) → {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:(1...3)⟶(1...𝑁))
75	70, 73, 74	mp2an 691	. . 3 ⊢ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:(1...3)⟶(1...𝑁)
76		rabrenfdioph 41538	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}:(1...3)⟶(1...𝑁) ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ [(𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) / 𝑎]𝜑} ∈ (Dioph‘𝑁))
77	75, 76	mp3an2 1450	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ [(𝑏 ∘ {⟨1, 𝑋⟩, ⟨2, 𝑌⟩, ⟨3, 𝑍⟩}) / 𝑎]𝜑} ∈ (Dioph‘𝑁))
78	55, 77	eqeltrrid 2839	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {crab 3433 [wsbc 3777 ⊆ wss 3948 {ctp 4632 ⟨cop 4634 ∘ ccom 5680 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ↑_m cmap 8817 1c1 11108 + caddc 11110 2c2 12264 3c3 12265 ℕ₀cn0 12469 ℤcz 12555 ...cfz 13481 Diophcdioph 41479

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-hash 14288 df-mzpcl 41447 df-mzp 41448 df-dioph 41480

This theorem is referenced by: rmxdioph 41741 expdiophlem2 41747

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