modmkpkne - Mathbox for Alexander van der Vekens

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Theorem modmkpkne 48104

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 12631	. . . 4 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
2	1	3adant1 1148	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
3		zaddcl 12629	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 + 𝐾) ∈ ℤ)
4	3	3adant2 1149	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 + 𝐾) ∈ ℤ)
5		simpl 487	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑁 ∈ ℕ)
6		difmod0 16340	. . 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 1450	. 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 12591	. . . . . . . . . 10 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℂ)
9	8	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑌 ∈ ℂ)
10		zcn 12591	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
11	10	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℂ)
12		zcn 12591	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ)
13	12	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑋 ∈ ℂ)
14	9, 11, 13, 11	subsubadd23 11616	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (𝐾 + 𝐾)))
15	10	2timesd 12482	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) = (𝐾 + 𝐾))
16	15	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 + 𝐾) = (2 · 𝐾))
17	16	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 + 𝐾) = (2 · 𝐾))
18	17	oveq2d 7426	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) − (𝐾 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
19	14, 18	eqtrd 2798	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
20	19	adantl 486	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
21	20	oveq1d 7425	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) − (2 · 𝐾)) mod 𝑁))
22	21	eqeq1d 2765	. . . 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 12631	. . . . . . 7 ⊢ ((𝑌 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
24	23	ancoms 463	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
25	24	3adant3 1150	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
26		2z 12621	. . . . . . . 8 ⊢ 2 ∈ ℤ
27	26	a1i 11	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℤ)
28		id 23	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℤ)
29	27, 28	zmulcld 12701	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℤ)
30	29	3ad2ant3 1153	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (2 · 𝐾) ∈ ℤ)
31		difmod0 16340	. . . . 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 1450	. . . 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 282	. . 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 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑌 ∈ ℂ)
35	11	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈ ℂ)
36	13	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑋 ∈ ℂ)
37	34, 35, 36, 35	addsubsub23 11617	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 + 𝐾) − (𝑋 − 𝐾)) = ((𝑌 − 𝑋) + (𝐾 + 𝐾)))
38	17	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + 𝐾) = (2 · 𝐾))
39	38	oveq2d 7426	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 − 𝑋) + (𝐾 + 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
40	37, 39	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 + 𝐾) − (𝑋 − 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
41	40	oveq1d 7425	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) + (2 · 𝐾)) mod 𝑁))
42	41	eqeq1d 2765	. . . . . . 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 16339	. . . . . . . 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 1450	. . . . . . 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 282	. . . . . 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 485	. . . . 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 12629	. . . . . . . 8 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
48	47	3adant1 1148	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
49		zsubcl 12631	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
50	49	3adant2 1149	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
51		difmod0 16340	. . . . . . 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 1450	. . . . . 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 485	. . . . 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 2767	. . . . . 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 12399	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
56	55	eqcomi 2772	. . . . . . . . . . . . 13 ⊢ 4 = (2 · 2)
57	56	oveq1i 7420	. . . . . . . . . . . 12 ⊢ (4 · 𝐾) = ((2 · 2) · 𝐾)
58		2cnd 12314	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℂ)
59	58, 58, 10	mulassd 11227	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = (2 · (2 · 𝐾)))
60	29	zcnd 12696	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℂ)
61	60	2timesd 12482	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → (2 · (2 · 𝐾)) = ((2 · 𝐾) + (2 · 𝐾)))
62	59, 61	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
63	57, 62	eqtrid 2810	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (4 · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
64	63	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (4 · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
65	64	adantl 486	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (4 · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
66	65	oveq1d 7425	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((4 · 𝐾) mod 𝑁) = (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁))
67	66	eqeq1d 2765	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((4 · 𝐾) mod 𝑁) = 0 ↔ (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁) = 0))
68		summodnegmod 16339	. . . . . . . 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 1450	. . . . . . 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 283	. . . . . 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 519	. . . . 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 312	. . . 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 417	. . 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 243	. 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 263	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 (class class class)co 7410 ℂcc 11093 0cc0 11095 + caddc 11098 · cmul 11100 − cmin 11436 -cneg 11437 ℕcn 12228 2c2 12290 4c4 12292 ℤcz 12586 mod cmo 13898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-fl 13821 df-mod 13899 df-dvds 16306

This theorem is referenced by: modm1p1ne 48113 gpgedg2iv 48832

W3C validator