gpgprismgriedgdmss - Mathbox for Alexander van der Vekens

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Theorem gpgprismgriedgdmss 48895

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 12934	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
2		lbfzo0 13750	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
3	1, 2	sylibr 237	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 0 ∈ (0..^𝑁))
4		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, 𝑥⟩ = ⟨0, 0⟩)
5		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
6		0p1e1 12381	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
7	5, 6	eqtrdi 2816	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
8	7	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑥 + 1) mod 𝑁) = (1 mod 𝑁))
9	8	opeq2d 4847	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨0, ((𝑥 + 1) mod 𝑁)⟩ = ⟨0, (1 mod 𝑁)⟩)
10	4, 9	preq12d 4709	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
11	10	eqeq2d 2776	. . . . . . . 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 487	. . . . . . 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 12929	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
14		eluz2b1 12964	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))
15		zre 12615	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
16	15	anim1i 627	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
17	14, 16	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 ∈ ℝ ∧ 1 < 𝑁))
18		1mod 13959	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1)
19	13, 17, 18	3syl 19	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (1 mod 𝑁) = 1)
20	19	eqcomd 2771	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 = (1 mod 𝑁))
21	20	opeq2d 4847	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨0, 1⟩ = ⟨0, (1 mod 𝑁)⟩)
22	21	preq2d 4708	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 0⟩, ⟨0, (1 mod 𝑁)⟩})
23	3, 12, 22	rspcedvd 3585	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩})
24	23	3mix1d 1355	. . . . 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 3136	. . . . 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 237	. . . 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 2765	. . . . 5 ⊢ (0..^𝑁) = (0..^𝑁)
28		eqid 2765	. . . . 5 ⊢ (𝑁 gPetersenGr 1) = (𝑁 gPetersenGr 1)
29	27, 28	gpgprismgriedgdmel 48894	. . . 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 260	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
31		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, 𝑥⟩ = ⟨1, 0⟩)
32	4, 31	preq12d 4709	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 0⟩, ⟨1, 0⟩})
33	32	eqeq2d 2776	. . . . . . . 8 ⊢ (𝑥 = 0 → ({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}))
34	33	adantl 487	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 0) → ({⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}))
35		eqid 2765	. . . . . . . 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 3585	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
38	37	3mix2d 1356	. . . . 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 3136	. . . . 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 237	. . . 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 48894	. . . 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 260	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨0, 0⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
43	30, 42	prssd 4790	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
44		1nn0 12540	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
45	44	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ ℕ₀)
46		eluz2gt1 12965	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 1 < 𝑁)
47	13, 46	syl 18	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 < 𝑁)
48		elfzo0 13751	. . . . . . . 8 ⊢ (1 ∈ (0..^𝑁) ↔ (1 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝑁))
49	45, 1, 47, 48	syl3anbrc 1362	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ (0..^𝑁))
50		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨0, 𝑥⟩ = ⟨0, 1⟩)
51		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ⟨1, 𝑥⟩ = ⟨1, 1⟩)
52	50, 51	preq12d 4709	. . . . . . . . 9 ⊢ (𝑥 = 1 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 1⟩, ⟨1, 1⟩})
53	52	eqeq2d 2776	. . . . . . . 8 ⊢ (𝑥 = 1 → ({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩}))
54	53	adantl 487	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑥 = 1) → ({⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 1⟩, ⟨1, 1⟩}))
55		prcom 4700	. . . . . . . 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 3585	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨0, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
58	57	3mix2d 1356	. . . . 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 3136	. . . . 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 237	. . . 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 48894	. . . 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 260	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨0, 1⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
63	8	opeq2d 4847	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ⟨1, ((𝑥 + 1) mod 𝑁)⟩ = ⟨1, (1 mod 𝑁)⟩)
64	31, 63	preq12d 4709	. . . . . . . . 9 ⊢ (𝑥 = 0 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
65	64	eqeq2d 2776	. . . . . . . 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 487	. . . . . . 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 4700	. . . . . . . 8 ⊢ {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, 1⟩}
68	20	opeq2d 4847	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ⟨1, 1⟩ = ⟨1, (1 mod 𝑁)⟩)
69	68	preq2d 4708	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
70	67, 69	eqtrid 2812	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 0⟩, ⟨1, (1 mod 𝑁)⟩})
71	3, 66, 70	rspcedvd 3585	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ∃𝑥 ∈ (0..^𝑁){⟨1, 1⟩, ⟨1, 0⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})
72	71	3mix3d 1357	. . . . 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 3136	. . . . 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 237	. . . 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 48894	. . . 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 260	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘3) → {⟨1, 1⟩, ⟨1, 0⟩} ∈ dom (iEdg‘(𝑁 gPetersenGr 1)))
77	62, 76	prssd 4790	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 = wceq 1570 ∈ wcel 2146 ∃wrex 3091 ∪ cun 3904 ⊆ wss 3906 {cpr 4593 ⟨cop 4597 class class class wbr 5111 dom cdm 5663 ‘cfv 6541 (class class class)co 7420 ℝcr 11119 0cc0 11120 1c1 11121 + caddc 11123 < clt 11263 ℕcn 12253 2c2 12315 3c3 12316 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12883 ..^cfzo 13704 mod cmo 13925 iEdgciedg 29407 gPetersenGr cgpg 48883

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7743 ax-cnex 11176 ax-resscn 11177 ax-1cn 11178 ax-icn 11179 ax-addcl 11180 ax-addrcl 11181 ax-mulcl 11182 ax-mulrcl 11183 ax-mulcom 11184 ax-addass 11185 ax-mulass 11186 ax-distr 11187 ax-i2m1 11188 ax-1ne0 11189 ax-1rid 11190 ax-rnegex 11191 ax-rrecex 11192 ax-cnre 11193 ax-pre-lttri 11194 ax-pre-lttrn 11195 ax-pre-ltadd 11196 ax-pre-mulgt0 11197 ax-pre-sup 11198

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7870 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8460 df-oadd 8464 df-er 8701 df-en 8951 df-dom 8952 df-sdom 8953 df-fin 8954 df-sup 9410 df-inf 9411 df-dju 9904 df-card 9942 df-pnf 11265 df-mnf 11266 df-xr 11267 df-ltxr 11268 df-le 11269 df-sub 11463 df-neg 11464 df-div 11892 df-nn 12254 df-2 12323 df-3 12324 df-4 12325 df-5 12326 df-6 12327 df-7 12328 df-8 12329 df-9 12330 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12733 df-uz 12884 df-rp 13038 df-ico 13399 df-fz 13557 df-fzo 13705 df-fl 13848 df-ceil 13849 df-mod 13926 df-hash 14390 df-struct 17234 df-slot 17269 df-ndx 17281 df-base 17297 df-edgf 29399 df-iedg 29409 df-gpg 48884

This theorem is used by: gpgprismgr4cycllem8 48945

W3C validator