modmkpkne - Mathbox for Alexander van der Vekens

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Theorem modmkpkne 48137

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 12647	. . . 4 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
2	1	3adant1 1148	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝐾) ∈ ℤ)
3		zaddcl 12645	. . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 + 𝐾) ∈ ℤ)
4	3	3adant2 1149	. . 3 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 + 𝐾) ∈ ℤ)
5		simpl 488	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑁 ∈ ℕ)
6		difmod0 16363	. . 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 12607	. . . . . . . . . 10 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℂ)
9	8	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑌 ∈ ℂ)
10		zcn 12607	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
11	10	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℂ)
12		zcn 12607	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℤ → 𝑋 ∈ ℂ)
13	12	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → 𝑋 ∈ ℂ)
14	9, 11, 13, 11	subsubadd23 11632	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (𝐾 + 𝐾)))
15	10	2timesd 12498	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) = (𝐾 + 𝐾))
16	15	eqcomd 2771	. . . . . . . . . 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 7432	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝑋) − (𝐾 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
19	14, 18	eqtrd 2800	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
20	19	adantl 487	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 − 𝐾) − (𝑋 + 𝐾)) = ((𝑌 − 𝑋) − (2 · 𝐾)))
21	20	oveq1d 7431	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 − 𝐾) − (𝑋 + 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) − (2 · 𝐾)) mod 𝑁))
22	21	eqeq1d 2767	. . . 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 12647	. . . . . . 7 ⊢ ((𝑌 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
24	23	ancoms 464	. . . . . 6 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
25	24	3adant3 1150	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 − 𝑋) ∈ ℤ)
26		2z 12637	. . . . . . . 8 ⊢ 2 ∈ ℤ
27	26	a1i 11	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℤ)
28		id 23	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℤ)
29	27, 28	zmulcld 12718	. . . . . 6 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℤ)
30	29	3ad2ant3 1153	. . . . 5 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (2 · 𝐾) ∈ ℤ)
31		difmod0 16363	. . . . 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 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑌 ∈ ℂ)
35	11	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝐾 ∈ ℂ)
36	13	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → 𝑋 ∈ ℂ)
37	34, 35, 36, 35	addsubsub23 11633	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 + 𝐾) − (𝑋 − 𝐾)) = ((𝑌 − 𝑋) + (𝐾 + 𝐾)))
38	17	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 + 𝐾) = (2 · 𝐾))
39	38	oveq2d 7432	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 − 𝑋) + (𝐾 + 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
40	37, 39	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((𝑌 + 𝐾) − (𝑋 − 𝐾)) = ((𝑌 − 𝑋) + (2 · 𝐾)))
41	40	oveq1d 7431	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((𝑌 + 𝐾) − (𝑋 − 𝐾)) mod 𝑁) = (((𝑌 − 𝑋) + (2 · 𝐾)) mod 𝑁))
42	41	eqeq1d 2767	. . . . . . 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 16362	. . . . . . . 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 486	. . . . 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 12645	. . . . . . . 8 ⊢ ((𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
48	47	3adant1 1148	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑌 + 𝐾) ∈ ℤ)
49		zsubcl 12647	. . . . . . . 8 ⊢ ((𝑋 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
50	49	3adant2 1149	. . . . . . 7 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑋 − 𝐾) ∈ ℤ)
51		difmod0 16363	. . . . . . 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 486	. . . . 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 2769	. . . . . 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 12415	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
56	55	eqcomi 2774	. . . . . . . . . . . . 13 ⊢ 4 = (2 · 2)
57	56	oveq1i 7426	. . . . . . . . . . . 12 ⊢ (4 · 𝐾) = ((2 · 2) · 𝐾)
58		2cnd 12330	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → 2 ∈ ℂ)
59	58, 58, 10	mulassd 11243	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = (2 · (2 · 𝐾)))
60	29	zcnd 12713	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → (2 · 𝐾) ∈ ℂ)
61	60	2timesd 12498	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℤ → (2 · (2 · 𝐾)) = ((2 · 𝐾) + (2 · 𝐾)))
62	59, 61	eqtrd 2800	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℤ → ((2 · 2) · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
63	57, 62	eqtrid 2812	. . . . . . . . . . 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 487	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (4 · 𝐾) = ((2 · 𝐾) + (2 · 𝐾)))
66	65	oveq1d 7431	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ((4 · 𝐾) mod 𝑁) = (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁))
67	66	eqeq1d 2767	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (((4 · 𝐾) mod 𝑁) = 0 ↔ (((2 · 𝐾) + (2 · 𝐾)) mod 𝑁) = 0))
68		summodnegmod 16362	. . . . . . . 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 520	. . . . 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 418	. . 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 (class class class)co 7416 ℂcc 11109 0cc0 11111 + caddc 11114 · cmul 11116 − cmin 11452 -cneg 11453 ℕcn 12244 2c2 12306 4c4 12308 ℤcz 12602 mod cmo 13916

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9405 df-inf 9406 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-n0 12516 df-z 12603 df-uz 12875 df-rp 13029 df-fl 13839 df-mod 13917 df-dvds 16329

This theorem is used by: modm1p1ne 48146 gpgedg2iv 48865

W3C validator