modmkpkne - Mathbox for Alexander van der Vekens

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Theorem modmkpkne 47352

Description: If an integer minus a constant equals another integer plus the constant modulo 𝑁, then the first integer plus the constant equals the second integer minus the constant modulo 𝑁 iff the fourfold of the constant is a multiple of 𝑁. (Contributed by AV, 15-Nov-2025.)

**Assertion**
Ref	Expression
modmkpkne	⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) mod 𝑁) = ((𝑋 + 𝐾) mod 𝑁) → (((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0)))

**Proof of Theorem modmkpkne**
Step	Hyp	Ref	Expression
1		zsubcl 12581	. . . 4 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
2	1	3adant1 1130	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
3		zaddcl 12579	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 + 𝐾) ∈ ℤ)
4	3	3adant2 1131	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 + 𝐾) ∈ ℤ)
5		simpl 482	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑁 ∈ ℕ)
6		difmod0 16263	. . 3 ⊢ (((𝑌 − 𝐾) ∈ ℤ ∧ (𝑋 + 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝐾) mod 𝑁) = ((𝑋 + 𝐾) mod 𝑁)))
7	2, 4, 5, 6	syl2an23an 1425	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝐾) mod 𝑁) = ((𝑋 + 𝐾) mod 𝑁)))
8		zcn 12540	. . . . . . . . . 10 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℂ)
9	8	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑌 ∈ ℂ)
10		zcn 12540	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
11	10	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℂ)
12		zcn 12540	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ)
13	12	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑋 ∈ ℂ)
14	9, 11, 13, 11	subsubadd23 11591	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (𝐾 + 𝐾)))
15	10	2timesd 12431	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) = (𝐾 + 𝐾))
16	15	eqcomd 2736	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 + 𝐾) = (2 · 𝐾))
17	16	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 + 𝐾) = (2 · 𝐾))
18	17	oveq2d 7405	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) − (𝐾 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
19	14, 18	eqtrd 2765	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
20	19	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
21	20	oveq1d 7404	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) − (2 · 𝐾)) mod 𝑁))
22	21	eqeq1d 2732	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = 0 ↔ (((𝑌 − 𝑋) − (2 · 𝐾)) mod 𝑁) = 0))
23		zsubcl 12581	. . . . . . 7 ⊢ ((𝑌 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
24	23	ancoms 458	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
25	24	3adant3 1132	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
26		2z 12571	. . . . . . . 8 ⊢ 2 ∈ ℤ
27	26	a1i 11	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℤ)
28		id 22	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℤ)
29	27, 28	zmulcld 12650	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℤ)
30	29	3ad2ant3 1135	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (2 · 𝐾) ∈ ℤ)
31		difmod0 16263	. . . . 5 ⊢ (((𝑌 − 𝑋) ∈ ℤ ∧ (2 · 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((((𝑌 − 𝑋) − (2 · 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁)))
32	25, 30, 5, 31	syl2an23an 1425	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 − 𝑋) − (2 · 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁)))
33	22, 32	bitrd 279	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁)))
34	9	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑌 ∈ ℂ)
35	11	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈ ℂ)
36	13	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑋 ∈ ℂ)
37	34, 35, 36, 35	addsubsub23 11592	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 + 𝐾) − (𝑋 − 𝐾)) = ((𝑌 − 𝑋) + (𝐾 + 𝐾)))
38	17	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + 𝐾) = (2 · 𝐾))
39	38	oveq2d 7405	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 − 𝑋) + (𝐾 + 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
40	37, 39	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 + 𝐾) − (𝑋 − 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
41	40	oveq1d 7404	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) + (2 · 𝐾)) mod 𝑁))
42	41	eqeq1d 2732	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = 0 ↔ (((𝑌 − 𝑋) + (2 · 𝐾)) mod 𝑁) = 0))
43		summodnegmod 16262	. . . . . . . 8 ⊢ (((𝑌 − 𝑋) ∈ ℤ ∧ (2 · 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((((𝑌 − 𝑋) + (2 · 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝑋) mod 𝑁) = (-(2 · 𝐾) mod 𝑁)))
44	25, 30, 5, 43	syl2an23an 1425	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 − 𝑋) + (2 · 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝑋) mod 𝑁) = (-(2 · 𝐾) mod 𝑁)))
45	42, 44	bitrd 279	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝑋) mod 𝑁) = (-(2 · 𝐾) mod 𝑁)))
46	45	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) ∧ ((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁)) → ((((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 − 𝑋) mod 𝑁) = (-(2 · 𝐾) mod 𝑁)))
47		zaddcl 12579	. . . . . . . 8 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
48	47	3adant1 1130	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
49		zsubcl 12581	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
50	49	3adant2 1131	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
51		difmod0 16263	. . . . . . 7 ⊢ (((𝑌 + 𝐾) ∈ ℤ ∧ (𝑋 − 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁)))
52	48, 50, 5, 51	syl2an23an 1425	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁)))
53	52	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) ∧ ((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁)) → ((((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = 0 ↔ ((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁)))
54		eqeq1 2734	. . . . . 6 ⊢ (((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁) → (((𝑌 − 𝑋) mod 𝑁) = (-(2 · 𝐾) mod 𝑁) ↔ ((2 · 𝐾) mod 𝑁) = (-(2 · 𝐾) mod 𝑁)))
55		2t2e4 12351	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
56	55	eqcomi 2739	. . . . . . . . . . . . 13 ⊢ 4 = (2 · 2)
57	56	oveq1i 7399	. . . . . . . . . . . 12 ⊢ (4 · 𝐾) = ((2 · 2) · 𝐾)
58		2cnd 12265	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℂ)
59	58, 58, 10	mulassd 11203	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = (2 · (2 · 𝐾)))
60	29	zcnd 12645	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℂ)
61	60	2timesd 12431	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → (2 · (2 · 𝐾)) = ((2 · 𝐾) + (2 · 𝐾)))
62	59, 61	eqtrd 2765	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
63	57, 62	eqtrid 2777	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (4 · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
64	63	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (4 · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
65	64	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (4 · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
66	65	oveq1d 7404	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((4 · 𝐾) mod 𝑁) = (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁))
67	66	eqeq1d 2732	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((4 · 𝐾) mod 𝑁) = 0 ↔ (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁) = 0))
68		summodnegmod 16262	. . . . . . . 8 ⊢ (((2 · 𝐾) ∈ ℤ ∧ (2 · 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((((2 · 𝐾) + (2 · 𝐾)) mod 𝑁) = 0 ↔ ((2 · 𝐾) mod 𝑁) = (-(2 · 𝐾) mod 𝑁)))
69	30, 30, 5, 68	syl2an23an 1425	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((2 · 𝐾) + (2 · 𝐾)) mod 𝑁) = 0 ↔ ((2 · 𝐾) mod 𝑁) = (-(2 · 𝐾) mod 𝑁)))
70	67, 69	bitr2d 280	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((2 · 𝐾) mod 𝑁) = (-(2 · 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0))
71	54, 70	sylan9bbr 510	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) ∧ ((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁)) → (((𝑌 − 𝑋) mod 𝑁) = (-(2 · 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0))
72	46, 53, 71	3bitr3d 309	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) ∧ ((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁)) → (((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0))
73	72	ex 412	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝑋) mod 𝑁) = ((2 · 𝐾) mod 𝑁) → (((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0)))
74	33, 73	sylbid 240	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = 0 → (((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0)))
75	7, 74	sylbird 260	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) mod 𝑁) = ((𝑋 + 𝐾) mod 𝑁) → (((𝑌 + 𝐾) mod 𝑁) = ((𝑋 − 𝐾) mod 𝑁) ↔ ((4 · 𝐾) mod 𝑁) = 0)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 (class class class)co 7389 ℂcc 11072 0cc0 11074 + caddc 11077 · cmul 11079 − cmin 11411 -cneg 11412 ℕcn 12187 2c2 12242 4c4 12244 ℤcz 12535 mod cmo 13837

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 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-sup 9399 df-inf 9400 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-n0 12449 df-z 12536 df-uz 12800 df-rp 12958 df-fl 13760 df-mod 13838 df-dvds 16229

This theorem is referenced by: modm1p1ne 47361 gpgedg2iv 48048

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