modmkpkne - Mathbox for Alexander van der Vekens

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Theorem modmkpkne 47607

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 12533	. . . 4 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
2	1	3adant1 1130	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
3		zaddcl 12531	. . . 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 16214	. . 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 12493	. . . . . . . . . 10 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℂ)
9	8	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑌 ∈ ℂ)
10		zcn 12493	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
11	10	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℂ)
12		zcn 12493	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ)
13	12	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑋 ∈ ℂ)
14	9, 11, 13, 11	subsubadd23 11544	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (𝐾 + 𝐾)))
15	10	2timesd 12384	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) = (𝐾 + 𝐾))
16	15	eqcomd 2742	. . . . . . . . . 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 7374	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) − (𝐾 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
19	14, 18	eqtrd 2771	. . . . . . 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 7373	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) − (2 · 𝐾)) mod 𝑁))
22	21	eqeq1d 2738	. . . 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 12533	. . . . . . 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 12523	. . . . . . . 8 ⊢ 2 ∈ ℤ
27	26	a1i 11	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℤ)
28		id 22	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℤ)
29	27, 28	zmulcld 12602	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℤ)
30	29	3ad2ant3 1135	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (2 · 𝐾) ∈ ℤ)
31		difmod0 16214	. . . . 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 11545	. . . . . . . . . 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 7374	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 − 𝑋) + (𝐾 + 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
40	37, 39	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 + 𝐾) − (𝑋 − 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
41	40	oveq1d 7373	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) + (2 · 𝐾)) mod 𝑁))
42	41	eqeq1d 2738	. . . . . . 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 16213	. . . . . . . 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 12531	. . . . . . . 8 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
48	47	3adant1 1130	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
49		zsubcl 12533	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
50	49	3adant2 1131	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
51		difmod0 16214	. . . . . . 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 2740	. . . . . 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 12304	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
56	55	eqcomi 2745	. . . . . . . . . . . . 13 ⊢ 4 = (2 · 2)
57	56	oveq1i 7368	. . . . . . . . . . . 12 ⊢ (4 · 𝐾) = ((2 · 2) · 𝐾)
58		2cnd 12223	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℂ)
59	58, 58, 10	mulassd 11155	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = (2 · (2 · 𝐾)))
60	29	zcnd 12597	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℂ)
61	60	2timesd 12384	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → (2 · (2 · 𝐾)) = ((2 · 𝐾) + (2 · 𝐾)))
62	59, 61	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
63	57, 62	eqtrid 2783	. . . . . . . . . . 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 7373	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((4 · 𝐾) mod 𝑁) = (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁))
67	66	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((4 · 𝐾) mod 𝑁) = 0 ↔ (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁) = 0))
68		summodnegmod 16213	. . . . . . . 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 1541 ∈ wcel 2113 (class class class)co 7358 ℂcc 11024 0cc0 11026 + caddc 11029 · cmul 11031 − cmin 11364 -cneg 11365 ℕcn 12145 2c2 12200 4c4 12202 ℤcz 12488 mod cmo 13789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-fl 13712 df-mod 13790 df-dvds 16180

This theorem is referenced by: modm1p1ne 47616 gpgedg2iv 48313

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