gpgprismgriedgdmss - Mathbox for Alexander van der Vekens

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Theorem gpgprismgriedgdmss 48046

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 12790	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
2		lbfzo0 13602	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
3	1, 2	sylibr 234	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 0 ∈ (0..^𝑁))
4		opeq2 4825	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, 𝑥⟩ = ⟨0, 0⟩)
5		oveq1 7356	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
6		0p1e1 12245	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
7	5, 6	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
8	7	oveq1d 7364	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 + 1) mod 𝑁) = (1 mod 𝑁))
9	8	opeq2d 4831	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, (1 mod 𝑁)⟩)
10	4, 9	preq12d 4693	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
11	10	eqeq2d 2740	. . . . . . . 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 12785	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
14		eluz2b1 12820	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))
15		zre 12475	. . . . . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 = (1 mod 𝑁))
21	20	opeq2d 4831	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨0, 1⟩ = ⟨0, (1 mod 𝑁)⟩)
22	21	preq2d 4692	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
23	3, 12, 22	rspcedvd 3579	. . . . . 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 3098	. . . . 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 2729	. . . . 5 ⊢ (0..^𝑁) = (0..^𝑁)
28		eqid 2729	. . . . 5 ⊢ (𝑁 gPetersenGr 1) = (𝑁 gPetersenGr 1)
29	27, 28	gpgprismgriedgdmel 48045	. . . 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 4825	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, 𝑥⟩ = ⟨1, 0⟩)
32	4, 31	preq12d 4693	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 0⟩, ⟨1, 0⟩})
33	32	eqeq2d 2740	. . . . . . . 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 2729	. . . . . . . 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 3579	. . . . . 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 3098	. . . . 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 48045	. . . 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 4773	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
44		1nn0 12400	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
45	44	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ ℕ₀)
46		eluz2gt1 12821	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 < 𝑁)
47	13, 46	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 < 𝑁)
48		elfzo0 13603	. . . . . . . 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 4825	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨0, 𝑥⟩ = ⟨0, 1⟩)
51		opeq2 4825	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨1, 𝑥⟩ = ⟨1, 1⟩)
52	50, 51	preq12d 4693	. . . . . . . . 9 ⊢ (𝑥 = 1 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 1⟩, ⟨1, 1⟩})
53	52	eqeq2d 2740	. . . . . . . 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 4684	. . . . . . . 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 3579	. . . . . 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 3098	. . . . 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 48045	. . . 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 4831	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, ((𝑥 + 1) mod 𝑁)⟩ = ⟨1, (1 mod 𝑁)⟩)
64	31, 63	preq12d 4693	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
65	64	eqeq2d 2740	. . . . . . . 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 4684	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, 1⟩}
68	20	opeq2d 4831	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨1, 1⟩ = ⟨1, (1 mod 𝑁)⟩)
69	68	preq2d 4692	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
70	67, 69	eqtrid 2776	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
71	3, 66, 70	rspcedvd 3579	. . . . . 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 3098	. . . . 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 48045	. . . 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 4773	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
78	43, 77	unssd 4143	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 1540 ∈ wcel 2109 ∃wrex 3053 ∪ cun 3901 ⊆ wss 3903 {cpr 4579 ⟨cop 4583 class class class wbr 5092 dom cdm 5619 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 < clt 11149 ℕcn 12128 2c2 12183 3c3 12184 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ..^cfzo 13557 mod cmo 13773 iEdgciedg 28942 gPetersenGr cgpg 48034

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-oadd 8392 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-ico 13254 df-fz 13411 df-fzo 13558 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 28934 df-iedg 28944 df-gpg 48035

This theorem is referenced by: gpgprismgr4cycllem8 48096

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