gpgprismgriedgdmss - Mathbox for Alexander van der Vekens

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Theorem gpgprismgriedgdmss 48557

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 12834	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
2		lbfzo0 13649	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
3	1, 2	sylibr 236	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 0 ∈ (0..^𝑁))
4		opeq2 4808	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, 𝑥⟩ = ⟨0, 0⟩)
5		oveq1 7367	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
6		0p1e1 12293	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
7	5, 6	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
8	7	oveq1d 7375	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 + 1) mod 𝑁) = (1 mod 𝑁))
9	8	opeq2d 4814	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, (1 mod 𝑁)⟩)
10	4, 9	preq12d 4676	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
11	10	eqeq2d 2752	. . . . . . . 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 483	. . . . . . 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 12829	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
14		eluz2b1 12864	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))
15		zre 12523	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
16	15	anim1i 622	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
17	14, 16	sylbi 219	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
18		1mod 13857	. . . . . . . . . . 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 2747	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 = (1 mod 𝑁))
21	20	opeq2d 4814	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨0, 1⟩ = ⟨0, (1 mod 𝑁)⟩)
22	21	preq2d 4675	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
23	3, 12, 22	rspcedvd 3564	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩})
24	23	3mix1d 1344	. . . . 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 3110	. . . . 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 236	. . . 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 2741	. . . . 5 ⊢ (0..^𝑁) = (0..^𝑁)
28		eqid 2741	. . . . 5 ⊢ (𝑁 gPetersenGr 1) = (𝑁 gPetersenGr 1)
29	27, 28	gpgprismgriedgdmel 48556	. . . 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 259	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
31		opeq2 4808	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, 𝑥⟩ = ⟨1, 0⟩)
32	4, 31	preq12d 4676	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 0⟩, ⟨1, 0⟩})
33	32	eqeq2d 2752	. . . . . . . 8 ⊢ (𝑥 = 0 → ({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}))
34	33	adantl 483	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 0) → ({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}))
35		eqid 2741	. . . . . . . 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 3564	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
38	37	3mix2d 1345	. . . . 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 3110	. . . . 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 236	. . . 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 48556	. . . 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 259	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
43	30, 42	prssd 4756	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
44		1nn0 12448	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
45	44	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ ℕ₀)
46		eluz2gt1 12865	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 < 𝑁)
47	13, 46	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 < 𝑁)
48		elfzo0 13650	. . . . . . . 8 ⊢ (1 ∈ (0..^𝑁) ↔ (1 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝑁))
49	45, 1, 47, 48	syl3anbrc 1351	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ (0..^𝑁))
50		opeq2 4808	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨0, 𝑥⟩ = ⟨0, 1⟩)
51		opeq2 4808	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨1, 𝑥⟩ = ⟨1, 1⟩)
52	50, 51	preq12d 4676	. . . . . . . . 9 ⊢ (𝑥 = 1 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 1⟩, ⟨1, 1⟩})
53	52	eqeq2d 2752	. . . . . . . 8 ⊢ (𝑥 = 1 → ({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩}))
54	53	adantl 483	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 1) → ({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩}))
55		prcom 4667	. . . . . . . 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 3564	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
58	57	3mix2d 1345	. . . . 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 3110	. . . . 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 236	. . . 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 48556	. . . 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 259	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
63	8	opeq2d 4814	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, ((𝑥 + 1) mod 𝑁)⟩ = ⟨1, (1 mod 𝑁)⟩)
64	31, 63	preq12d 4676	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
65	64	eqeq2d 2752	. . . . . . . 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 483	. . . . . . 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 4667	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, 1⟩}
68	20	opeq2d 4814	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨1, 1⟩ = ⟨1, (1 mod 𝑁)⟩)
69	68	preq2d 4675	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
70	67, 69	eqtrid 2788	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
71	3, 66, 70	rspcedvd 3564	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})
72	71	3mix3d 1346	. . . . 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 3110	. . . . 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 236	. . . 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 48556	. . . 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 259	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
77	62, 76	prssd 4756	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
78	43, 77	unssd 4124	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 208 ∧ wa 397 ∨ w3o 1092 = wceq 1548 ∈ wcel 2121 ∃wrex 3065 ∪ cun 3883 ⊆ wss 3885 {cpr 4560 ⟨cop 4564 class class class wbr 5075 dom cdm 5621 ‘cfv 6489 (class class class)co 7360 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 < clt 11174 ℕcn 12169 2c2 12231 3c3 12232 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 ..^cfzo 13603 mod cmo 13823 iEdgciedg 29088 gPetersenGr cgpg 48545

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-ico 13299 df-fz 13457 df-fzo 13604 df-fl 13746 df-ceil 13747 df-mod 13824 df-hash 14288 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-edgf 29080 df-iedg 29090 df-gpg 48546

This theorem is referenced by: gpgprismgr4cycllem8 48607

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