modm1p1ne - Mathbox for Alexander van der Vekens

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Theorem modm1p1ne 47997

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 12864	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁))
2		4nn0 12519	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
3	2	a1i 11	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ ℕ₀)
4		simp2 1153	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℤ)
5		0red 11207	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 ∈ ℝ)
6		5re 12324	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℝ
7	6	a1i 11	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 5 ∈ ℝ)
8		zre 12591	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
9	8	3ad2ant2 1150	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℝ)
10		5pos 12349	. . . . . . . . . . . . . 14 ⊢ 0 < 5
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 < 5)
12		simp3 1154	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 5 ≤ 𝑁)
13	5, 7, 9, 11, 12	ltletrd 11366	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 < 𝑁)
14		elnnz 12597	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
15	4, 13, 14	sylanbrc 594	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℕ)
16		4re 12321	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ ℝ)
18		4lt5 12416	. . . . . . . . . . . . 13 ⊢ 4 < 5
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 < 5)
20	17, 7, 9, 19, 12	ltletrd 11366	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 < 𝑁)
21		elfzo0 13725	. . . . . . . . . . 11 ⊢ (4 ∈ (0..^𝑁) ↔ (4 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 4 < 𝑁))
22	3, 15, 20, 21	syl3anbrc 1360	. . . . . . . . . 10 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ (0..^𝑁))
23	1, 22	sylbi 220	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 4 ∈ (0..^𝑁))
24		zmodidfzoimp 13930	. . . . . . . . 9 ⊢ (4 ∈ (0..^𝑁) → (4 mod 𝑁) = 4)
25	23, 24	syl 18	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (4 mod 𝑁) = 4)
26		4ne0 12348	. . . . . . . . 9 ⊢ 4 ≠ 0
27	26	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 4 ≠ 0)
28	25, 27	eqnetrd 3031	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (4 mod 𝑁) ≠ 0)
29		df-ne 2965	. . . . . . . 8 ⊢ (((4 · 1) mod 𝑁) ≠ 0 ↔ ¬ ((4 · 1) mod 𝑁) = 0)
30		4cn 12322	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
31	30	mulridi 11209	. . . . . . . . . 10 ⊢ (4 · 1) = 4
32	31	oveq1i 7418	. . . . . . . . 9 ⊢ ((4 · 1) mod 𝑁) = (4 mod 𝑁)
33	32	neeq1i 3028	. . . . . . . 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 1149	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → ¬ ((4 · 1) mod 𝑁) = 0)
37	36	adantr 485	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → ¬ ((4 · 1) mod 𝑁) = 0)
38		uzuzle35 12907	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ (ℤ_≥‘3))
39		eluz3nn 12909	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
40	38, 39	syl 18	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ ℕ)
41	40	3ad2ant1 1149	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑁 ∈ ℕ)
42		elfzoelz 13683	. . . . . . . 8 ⊢ (𝑋 ∈ (0..^𝑁) → 𝑋 ∈ ℤ)
43		modm1nep1.i	. . . . . . . 8 ⊢ 𝐼 = (0..^𝑁)
44	42, 43	eleq2s 2887	. . . . . . 7 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ ℤ)
45	44	3ad2ant2 1150	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑋 ∈ ℤ)
46		elfzoelz 13683	. . . . . . . 8 ⊢ (𝑌 ∈ (0..^𝑁) → 𝑌 ∈ ℤ)
47	46, 43	eleq2s 2887	. . . . . . 7 ⊢ (𝑌 ∈ 𝐼 → 𝑌 ∈ ℤ)
48	47	3ad2ant3 1151	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ ℤ)
49		1zzd 12621	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 1 ∈ ℤ)
50		modmkpkne 47988	. . . . . 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 1397	. . . . 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 411	. . . 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 2971	. 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 417	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 class class class wbr 5110 ‘cfv 6534 (class class class)co 7408 ℝcr 11095 0cc0 11096 1c1 11097 + caddc 11099 · cmul 11101 < clt 11239 ≤ cle 11240 − cmin 11437 ℕcn 12229 3c3 12292 4c4 12293 5c5 12294 ℕ₀cn0 12500 ℤcz 12587 ℤ_≥cuz 12858 ..^cfzo 13678 mod cmo 13898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 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 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-n0 12501 df-z 12588 df-uz 12859 df-rp 13013 df-fz 13532 df-fzo 13679 df-fl 13821 df-mod 13899 df-dvds 16307

This theorem is referenced by: gpgedg2ov 48715

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