modm1p1ne - Mathbox for Alexander van der Vekens

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Theorem modm1p1ne 47361

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 12805	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁))
2		4nn0 12467	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ₀
3	2	a1i 11	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ ℕ₀)
4		simp2 1137	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℤ)
5		0red 11183	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 ∈ ℝ)
6		5re 12274	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℝ
7	6	a1i 11	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 5 ∈ ℝ)
8		zre 12539	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
9	8	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℝ)
10		5pos 12296	. . . . . . . . . . . . . 14 ⊢ 0 < 5
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 < 5)
12		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 5 ≤ 𝑁)
13	5, 7, 9, 11, 12	ltletrd 11340	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 0 < 𝑁)
14		elnnz 12545	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
15	4, 13, 14	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 𝑁 ∈ ℕ)
16		4re 12271	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ ℝ)
18		4lt5 12364	. . . . . . . . . . . . 13 ⊢ 4 < 5
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 < 5)
20	17, 7, 9, 19, 12	ltletrd 11340	. . . . . . . . . . 11 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 < 𝑁)
21		elfzo0 13667	. . . . . . . . . . 11 ⊢ (4 ∈ (0..^𝑁) ↔ (4 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 4 < 𝑁))
22	3, 15, 20, 21	syl3anbrc 1344	. . . . . . . . . 10 ⊢ ((5 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 5 ≤ 𝑁) → 4 ∈ (0..^𝑁))
23	1, 22	sylbi 217	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 4 ∈ (0..^𝑁))
24		zmodidfzoimp 13869	. . . . . . . . 9 ⊢ (4 ∈ (0..^𝑁) → (4 mod 𝑁) = 4)
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (4 mod 𝑁) = 4)
26		4ne0 12295	. . . . . . . . 9 ⊢ 4 ≠ 0
27	26	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 4 ≠ 0)
28	25, 27	eqnetrd 2993	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (4 mod 𝑁) ≠ 0)
29		df-ne 2927	. . . . . . . 8 ⊢ (((4 · 1) mod 𝑁) ≠ 0 ↔ ¬ ((4 · 1) mod 𝑁) = 0)
30		4cn 12272	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
31	30	mulridi 11184	. . . . . . . . . 10 ⊢ (4 · 1) = 4
32	31	oveq1i 7399	. . . . . . . . 9 ⊢ ((4 · 1) mod 𝑁) = (4 mod 𝑁)
33	32	neeq1i 2990	. . . . . . . 8 ⊢ (((4 · 1) mod 𝑁) ≠ 0 ↔ (4 mod 𝑁) ≠ 0)
34	29, 33	bitr3i 277	. . . . . . 7 ⊢ (¬ ((4 · 1) mod 𝑁) = 0 ↔ (4 mod 𝑁) ≠ 0)
35	28, 34	sylibr 234	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘5) → ¬ ((4 · 1) mod 𝑁) = 0)
36	35	3ad2ant1 1133	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → ¬ ((4 · 1) mod 𝑁) = 0)
37	36	adantr 480	. . . 4 ⊢ (((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ ((𝑌 − 1) mod 𝑁) = ((𝑋 + 1) mod 𝑁)) → ¬ ((4 · 1) mod 𝑁) = 0)
38		uzuzle35 12852	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ (ℤ_≥‘3))
39		eluz3nn 12854	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
40	38, 39	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘5) → 𝑁 ∈ ℕ)
41	40	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑁 ∈ ℕ)
42		elfzoelz 13626	. . . . . . . 8 ⊢ (𝑋 ∈ (0..^𝑁) → 𝑋 ∈ ℤ)
43		modm1nep1.i	. . . . . . . 8 ⊢ 𝐼 = (0..^𝑁)
44	42, 43	eleq2s 2847	. . . . . . 7 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ ℤ)
45	44	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑋 ∈ ℤ)
46		elfzoelz 13626	. . . . . . . 8 ⊢ (𝑌 ∈ (0..^𝑁) → 𝑌 ∈ ℤ)
47	46, 43	eleq2s 2847	. . . . . . 7 ⊢ (𝑌 ∈ 𝐼 → 𝑌 ∈ ℤ)
48	47	3ad2ant3 1135	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 𝑌 ∈ ℤ)
49		1zzd 12570	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘5) ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) → 1 ∈ ℤ)
50		modmkpkne 47352	. . . . . 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 1374	. . . . 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 406	. . . 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 325	. . 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 2933	. 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 412	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 ℝcr 11073 0cc0 11074 1c1 11075 + caddc 11077 · cmul 11079 < clt 11214 ≤ cle 11215 − cmin 11411 ℕcn 12187 3c3 12243 4c4 12244 5c5 12245 ℕ₀cn0 12448 ℤcz 12535 ℤ_≥cuz 12799 ..^cfzo 13621 mod cmo 13837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-sup 9399 df-inf 9400 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-n0 12449 df-z 12536 df-uz 12800 df-rp 12958 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-dvds 16229

This theorem is referenced by: gpgedg2ov 48047

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