gpgprismgriedgdmss - Mathbox for Alexander van der Vekens

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Theorem gpgprismgriedgdmss 48365

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 12806	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
2		lbfzo0 13619	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
3	1, 2	sylibr 234	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 0 ∈ (0..^𝑁))
4		opeq2 4831	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, 𝑥⟩ = ⟨0, 0⟩)
5		oveq1 7367	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
6		0p1e1 12266	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
7	5, 6	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
8	7	oveq1d 7375	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 + 1) mod 𝑁) = (1 mod 𝑁))
9	8	opeq2d 4837	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, (1 mod 𝑁)⟩)
10	4, 9	preq12d 4699	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
11	10	eqeq2d 2748	. . . . . . . 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 12801	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
14		eluz2b1 12836	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))
15		zre 12496	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
16	15	anim1i 616	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
17	14, 16	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
18		1mod 13827	. . . . . . . . . . 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 2743	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 = (1 mod 𝑁))
21	20	opeq2d 4837	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨0, 1⟩ = ⟨0, (1 mod 𝑁)⟩)
22	21	preq2d 4698	. . . . . . 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 1338	. . . . 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 3106	. . . . 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 2737	. . . . 5 ⊢ (0..^𝑁) = (0..^𝑁)
28		eqid 2737	. . . . 5 ⊢ (𝑁 gPetersenGr 1) = (𝑁 gPetersenGr 1)
29	27, 28	gpgprismgriedgdmel 48364	. . . 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 4831	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, 𝑥⟩ = ⟨1, 0⟩)
32	4, 31	preq12d 4699	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 0⟩, ⟨1, 0⟩})
33	32	eqeq2d 2748	. . . . . . . 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 2737	. . . . . . . 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 1339	. . . . 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 3106	. . . . 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 48364	. . . 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 4779	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
44		1nn0 12421	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
45	44	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ ℕ₀)
46		eluz2gt1 12837	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 < 𝑁)
47	13, 46	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 < 𝑁)
48		elfzo0 13620	. . . . . . . 8 ⊢ (1 ∈ (0..^𝑁) ↔ (1 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝑁))
49	45, 1, 47, 48	syl3anbrc 1345	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ (0..^𝑁))
50		opeq2 4831	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨0, 𝑥⟩ = ⟨0, 1⟩)
51		opeq2 4831	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨1, 𝑥⟩ = ⟨1, 1⟩)
52	50, 51	preq12d 4699	. . . . . . . . 9 ⊢ (𝑥 = 1 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 1⟩, ⟨1, 1⟩})
53	52	eqeq2d 2748	. . . . . . . 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 4690	. . . . . . . 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 1339	. . . . 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 3106	. . . . 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 48364	. . . 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 4837	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, ((𝑥 + 1) mod 𝑁)⟩ = ⟨1, (1 mod 𝑁)⟩)
64	31, 63	preq12d 4699	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
65	64	eqeq2d 2748	. . . . . . . 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 4690	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, 1⟩}
68	20	opeq2d 4837	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨1, 1⟩ = ⟨1, (1 mod 𝑁)⟩)
69	68	preq2d 4698	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
70	67, 69	eqtrid 2784	. . . . . . 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 1340	. . . . 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 3106	. . . . 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 48364	. . . 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 4779	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
78	43, 77	unssd 4145	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 1086 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 ∪ cun 3900 ⊆ wss 3902 {cpr 4583 ⟨cop 4587 class class class wbr 5099 dom cdm 5625 ‘cfv 6493 (class class class)co 7360 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 < clt 11170 ℕcn 12149 2c2 12204 3c3 12205 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ..^cfzo 13574 mod cmo 13793 iEdgciedg 29074 gPetersenGr cgpg 48353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 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 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-xnn0 12479 df-z 12493 df-dec 12612 df-uz 12756 df-rp 12910 df-ico 13271 df-fz 13428 df-fzo 13575 df-fl 13716 df-ceil 13717 df-mod 13794 df-hash 14258 df-struct 17078 df-slot 17113 df-ndx 17125 df-base 17141 df-edgf 29066 df-iedg 29076 df-gpg 48354

This theorem is referenced by: gpgprismgr4cycllem8 48415

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