gpgprismgriedgdmss - Mathbox for Alexander van der Vekens

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Theorem gpgprismgriedgdmss 48151

Description: A subset of the index of edges of the generalized Petersen graph GPG(N,1). (Contributed by AV, 2-Nov-2025.)

**Assertion**
Ref	Expression
gpgprismgriedgdmss	⊢ (𝑁 ∈ (ℤ_≥‘3) → ({{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ∪ {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}}) ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))

Proof of Theorem gpgprismgriedgdmss Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluz3nn 12787	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
2		lbfzo0 13599	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
3	1, 2	sylibr 234	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 0 ∈ (0..^𝑁))
4		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, 𝑥⟩ = ⟨0, 0⟩)
5		oveq1 7353	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
6		0p1e1 12242	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
7	5, 6	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
8	7	oveq1d 7361	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 + 1) mod 𝑁) = (1 mod 𝑁))
9	8	opeq2d 4829	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, (1 mod 𝑁)⟩)
10	4, 9	preq12d 4691	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
11	10	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 0 → ({⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩}))
12	11	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 0) → ({⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩}))
13		uzuzle23 12782	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
14		eluz2b1 12817	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))
15		zre 12472	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
16	15	anim1i 615	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
17	14, 16	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
18		1mod 13807	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1)
19	13, 17, 18	3syl 18	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (1 mod 𝑁) = 1)
20	19	eqcomd 2737	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 = (1 mod 𝑁))
21	20	opeq2d 4829	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨0, 1⟩ = ⟨0, (1 mod 𝑁)⟩)
22	21	preq2d 4690	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
23	3, 12, 22	rspcedvd 3574	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩})
24	23	3mix1d 1337	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
25		3r19.43 3101	. . . . 5 ⊢ (∃𝑥 ∈ (0..^𝑁)({⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}) ↔ (∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
26	24, 25	sylibr 234	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁)({⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
27		eqid 2731	. . . . 5 ⊢ (0..^𝑁) = (0..^𝑁)
28		eqid 2731	. . . . 5 ⊢ (𝑁 gPetersenGr 1) = (𝑁 gPetersenGr 1)
29	27, 28	gpgprismgriedgdmel 48150	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ({⟨0, 0⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)) ↔ ∃𝑥 ∈ (0..^𝑁)({⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 0⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
30	26, 29	mpbird 257	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
31		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, 𝑥⟩ = ⟨1, 0⟩)
32	4, 31	preq12d 4691	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 0⟩, ⟨1, 0⟩})
33	32	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 0 → ({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}))
34	33	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 0) → ({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}))
35		eqid 2731	. . . . . . . 8 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
36	35	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩})
37	3, 34, 36	rspcedvd 3574	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
38	37	3mix2d 1338	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
39		3r19.43 3101	. . . . 5 ⊢ (∃𝑥 ∈ (0..^𝑁)({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}) ↔ (∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
40	38, 39	sylibr 234	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁)({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
41	27, 28	gpgprismgriedgdmel 48150	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ({⟨0, 0⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)) ↔ ∃𝑥 ∈ (0..^𝑁)({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
42	40, 41	mpbird 257	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
43	30, 42	prssd 4771	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
44		1nn0 12397	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
45	44	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ ℕ₀)
46		eluz2gt1 12818	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 < 𝑁)
47	13, 46	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 < 𝑁)
48		elfzo0 13600	. . . . . . . 8 ⊢ (1 ∈ (0..^𝑁) ↔ (1 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝑁))
49	45, 1, 47, 48	syl3anbrc 1344	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ (0..^𝑁))
50		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨0, 𝑥⟩ = ⟨0, 1⟩)
51		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨1, 𝑥⟩ = ⟨1, 1⟩)
52	50, 51	preq12d 4691	. . . . . . . . 9 ⊢ (𝑥 = 1 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 1⟩, ⟨1, 1⟩})
53	52	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 1 → ({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩}))
54	53	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 1) → ({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩}))
55		prcom 4682	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩}
56	55	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩})
57	49, 54, 56	rspcedvd 3574	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
58	57	3mix2d 1338	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
59		3r19.43 3101	. . . . 5 ⊢ (∃𝑥 ∈ (0..^𝑁)({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}) ↔ (∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
60	58, 59	sylibr 234	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁)({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
61	27, 28	gpgprismgriedgdmel 48150	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ({⟨1, 1⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)) ↔ ∃𝑥 ∈ (0..^𝑁)({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
62	60, 61	mpbird 257	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
63	8	opeq2d 4829	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, ((𝑥 + 1) mod 𝑁)⟩ = ⟨1, (1 mod 𝑁)⟩)
64	31, 63	preq12d 4691	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
65	64	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = 0 → ({⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} ↔ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩}))
66	65	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 0) → ({⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} ↔ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩}))
67		prcom 4682	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, 1⟩}
68	20	opeq2d 4829	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨1, 1⟩ = ⟨1, (1 mod 𝑁)⟩)
69	68	preq2d 4690	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
70	67, 69	eqtrid 2778	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
71	3, 66, 70	rspcedvd 3574	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})
72	71	3mix3d 1339	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
73		3r19.43 3101	. . . . 5 ⊢ (∃𝑥 ∈ (0..^𝑁)({⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}) ↔ (∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
74	72, 73	sylibr 234	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁)({⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
75	27, 28	gpgprismgriedgdmel 48150	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ({⟨1, 1⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)) ↔ ∃𝑥 ∈ (0..^𝑁)({⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
76	74, 75	mpbird 257	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
77	62, 76	prssd 4771	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
78	43, 77	unssd 4139	1 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ({{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ∪ {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}}) ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 ∪ cun 3895 ⊆ wss 3897 {cpr 4575 ⟨cop 4579 class class class wbr 5089 dom cdm 5614 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 < clt 11146 ℕcn 12125 2c2 12180 3c3 12181 ℕ₀cn0 12381 ℤcz 12468 ℤ_≥cuz 12732 ..^cfzo 13554 mod cmo 13773 iEdgciedg 28975 gPetersenGr cgpg 48139

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-oadd 8389 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-ico 13251 df-fz 13408 df-fzo 13555 df-fl 13696 df-ceil 13697 df-mod 13774 df-hash 14238 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-edgf 28967 df-iedg 28977 df-gpg 48140

This theorem is referenced by: gpgprismgr4cycllem8 48201

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