modm1p1ne - Mathbox for Alexander van der Vekens

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Theorem modm1p1ne 48264

Description: If an integer minus one equals another integer plus one modulo an integer greater than 4, then the first integer plus one is not equal to the second integer minus one modulo the same modulus. (Contributed by AV, 15-Nov-2025.)

**Hypothesis**
Ref	Expression
modm1nep1.i	⊢ 𝐼 = (0..^𝑁)

**Assertion**
Ref	Expression
modm1p1ne	⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → ((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁)))

**Proof of Theorem modm1p1ne**
Step	Hyp	Ref	Expression
1		eluz2 12893	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁))
2		4nn0 12547	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
3	2	a1i 11	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ ℕ₀)
4		simp2 1155	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℤ)
5		0red 11235	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 ∈ ℝ)
6		5re 12352	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℝ
7	6	a1i 11	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 5 ∈ ℝ)
8		zre 12619	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
9	8	3ad2ant2 1152	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℝ)
10		5pos 12377	. . . . . . . . . . . . . 14 ⊢ 0 < 5
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 < 5)
12		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 5 ≤ 𝑁)
13	5, 7, 9, 11, 12	ltletrd 11394	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 < 𝑁)
14		elnnz 12625	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
15	4, 13, 14	sylanbrc 595	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℕ)
16		4re 12349	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ ℝ)
18		4lt5 12444	. . . . . . . . . . . . 13 ⊢ 4 < 5
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 < 5)
20	17, 7, 9, 19, 12	ltletrd 11394	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 < 𝑁)
21		elfzo0 13756	. . . . . . . . . . 11 ⊢ (4 ∈ (0..^𝑁) ↔ (4 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 4 < 𝑁))
22	3, 15, 20, 21	syl3anbrc 1362	. . . . . . . . . 10 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ (0..^𝑁))
23	1, 22	sylbi 220	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 4 ∈ (0..^𝑁))
24		zmodidfzoimp 13962	. . . . . . . . 9 ⊢ (4 ∈ (0..^𝑁) → (4 mod 𝑁) = 4)
25	23, 24	syl 18	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (4 mod 𝑁) = 4)
26		4ne0 12376	. . . . . . . . 9 ⊢ 4 ≠ 0
27	26	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 4 ≠ 0)
28	25, 27	eqnetrd 3022	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (4 mod 𝑁) ≠ 0)
29		df-ne 2956	. . . . . . . 8 ⊢ (((4 · 1) mod 𝑁) ≠ 0 ↔ ¬ ((4 · 1) mod 𝑁) = 0)
30		4cn 12350	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
31	30	mulridi 11237	. . . . . . . . . 10 ⊢ (4 · 1) = 4
32	31	oveq1i 7423	. . . . . . . . 9 ⊢ ((4 · 1) mod 𝑁) = (4 mod 𝑁)
33	32	neeq1i 3019	. . . . . . . 8 ⊢ (((4 · 1) mod 𝑁) ≠ 0 ↔ (4 mod 𝑁) ≠ 0)
34	29, 33	bitr3i 280	. . . . . . 7 ⊢ (¬ ((4 · 1) mod 𝑁) = 0 ↔ (4 mod 𝑁) ≠ 0)
35	28, 34	sylibr 237	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘5) → ¬ ((4 · 1) mod 𝑁) = 0)
36	35	3ad2ant1 1151	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → ¬ ((4 · 1) mod 𝑁) = 0)
37	36	adantr 486	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → ¬ ((4 · 1) mod 𝑁) = 0)
38		uzuzle35 12936	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ (ℤ_≥‘3))
39		eluz3nn 12938	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
40	38, 39	syl 18	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ ℕ)
41	40	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑁 ∈ ℕ)
42		elfzoelz 13714	. . . . . . . 8 ⊢ (𝑋 ∈ (0..^𝑁) → 𝑋 ∈ ℤ)
43		modm1nep1.i	. . . . . . . 8 ⊢ 𝐼 = (0..^𝑁)
44	42, 43	eleq2s 2878	. . . . . . 7 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ ℤ)
45	44	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑋 ∈ ℤ)
46		elfzoelz 13714	. . . . . . . 8 ⊢ (𝑌 ∈ (0..^𝑁) → 𝑌 ∈ ℤ)
47	46, 43	eleq2s 2878	. . . . . . 7 ⊢ (𝑌 ∈ 𝐼 → 𝑌 ∈ ℤ)
48	47	3ad2ant3 1153	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ ℤ)
49		1zzd 12649	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 1 ∈ ℤ)
50		modmkpkne 48255	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 1 ∈ ℤ)) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) ↔ ((4 · 1) mod 𝑁) = 0)))
51	41, 45, 48, 49, 50	syl13anc 1399	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) ↔ ((4 · 1) mod 𝑁) = 0)))
52	51	imp 412	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → (((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁) ↔ ((4 · 1) mod 𝑁) = 0))
53	37, 52	mtbird 328	. . 3 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → ¬ ((𝑌 + 1) mod 𝑁) = ((𝑋 − 1) mod 𝑁))
54	53	neqned 2962	. 2 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → ((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁))
55	54	ex 418	1 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → (((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁) → ((𝑌 + 1) mod 𝑁) ≠ ((𝑋 − 1) mod 𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ‘cfv 6533 (class class class)co 7413 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 < clt 11267 ≤ cle 11268 − cmin 11465 ℕcn 12257 3c3 12320 4c4 12321 5c5 12322 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12887 ..^cfzo 13709 mod cmo 13930

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-n0 12529 df-z 12616 df-uz 12888 df-rp 13043 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-dvds 16343

This theorem is used by: gpgedg2ov 48982

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