gpgprismgr4cycllem10 - Mathbox for Alexander van der Vekens

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Theorem gpgprismgr4cycllem10 48897

Description: Lemma 10 for gpgprismgr4cycl0 48899. (Contributed by AV, 5-Nov-2025.)

**Hypotheses**
Ref	Expression
gpgprismgr4cycl.p	⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩
gpgprismgr4cycl.f	⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩
gpgprismgr4cycl.g	⊢ 𝐺 = (𝑁 gPetersenGr 1)

**Assertion**
Ref	Expression
gpgprismgr4cycllem10	⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})

Proof of Theorem gpgprismgr4cycllem10 Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gpgprismgr4cycl.g	. . . . . . 7 ⊢ 𝐺 = (𝑁 gPetersenGr 1)
2	1	fveq2i 6884	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘(𝑁 gPetersenGr 1))
3	2	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (iEdg‘𝐺) = (iEdg‘(𝑁 gPetersenGr 1)))
4		eluz3nn 12917	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
5		1elfzo1ceilhalf1 48106	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ (1..^(⌈‘(𝑁 / 2))))
6	4, 5	jca 520	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (𝑁 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(𝑁 / 2)))))
7	6	adantr 485	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝑁 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(𝑁 / 2)))))
8		eqid 2763	. . . . . . 7 ⊢ (1..^(⌈‘(𝑁 / 2))) = (1..^(⌈‘(𝑁 / 2)))
9		eqid 2763	. . . . . . 7 ⊢ (0..^𝑁) = (0..^𝑁)
10	8, 9	gpgiedg 48837	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(𝑁 / 2)))) → (iEdg‘(𝑁 gPetersenGr 1)) = ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})}))
11	7, 10	syl 18	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (iEdg‘(𝑁 gPetersenGr 1)) = ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})}))
12	3, 11	eqtrd 2798	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (iEdg‘𝐺) = ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})}))
13	12	fveq1d 6883	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = (( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})})‘(𝐹‘𝑋)))
14		gpgprismgr4cycl.f	. . . . . 6 ⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩
15	14	gpgprismgr4cycllem3 48890	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^4)) → ((𝐹‘𝑋) ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∧ ∃𝑥 ∈ (0..^𝑁)((𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ (𝐹‘𝑋) = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
16	14	gpgprismgr4cycllem1 48888	. . . . . . . 8 ⊢ (♯‘𝐹) = 4
17	16	oveq2i 7421	. . . . . . 7 ⊢ (0..^(♯‘𝐹)) = (0..^4)
18	17	eleq2i 2855	. . . . . 6 ⊢ (𝑋 ∈ (0..^(♯‘𝐹)) ↔ 𝑋 ∈ (0..^4))
19	18	anbi2i 634	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) ↔ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^4)))
20		eqeq1 2767	. . . . . . . 8 ⊢ (𝑒 = (𝐹‘𝑋) → (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩}))
21		eqeq1 2767	. . . . . . . 8 ⊢ (𝑒 = (𝐹‘𝑋) → (𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩}))
22		eqeq1 2767	. . . . . . . 8 ⊢ (𝑒 = (𝐹‘𝑋) → (𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩} ↔ (𝐹‘𝑋) = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}))
23	20, 21, 22	3orbi123d 1463	. . . . . . 7 ⊢ (𝑒 = (𝐹‘𝑋) → ((𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}) ↔ ((𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ (𝐹‘𝑋) = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
24	23	rexbidv 3189	. . . . . 6 ⊢ (𝑒 = (𝐹‘𝑋) → (∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩}) ↔ ∃𝑥 ∈ (0..^𝑁)((𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ (𝐹‘𝑋) = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
25	24	elrab 3650	. . . . 5 ⊢ ((𝐹‘𝑋) ∈ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})} ↔ ((𝐹‘𝑋) ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∧ ∃𝑥 ∈ (0..^𝑁)((𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ (𝐹‘𝑋) = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
26	15, 19, 25	3imtr4i 295	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑋) ∈ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})})
27		fvresi 7171	. . . 4 ⊢ ((𝐹‘𝑋) ∈ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})} → (( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})})‘(𝐹‘𝑋)) = (𝐹‘𝑋))
28	26, 27	syl 18	. . 3 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∣ ∃𝑥 ∈ (0..^𝑁)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})})‘(𝐹‘𝑋)) = (𝐹‘𝑋))
29	13, 28	eqtrd 2798	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = (𝐹‘𝑋))
30		fzo0to42pr 13787	. . . . . 6 ⊢ (0..^4) = ({0, 1} ∪ {2, 3})
31	30	eleq2i 2855	. . . . 5 ⊢ (𝑋 ∈ (0..^4) ↔ 𝑋 ∈ ({0, 1} ∪ {2, 3}))
32		elun 4107	. . . . 5 ⊢ (𝑋 ∈ ({0, 1} ∪ {2, 3}) ↔ (𝑋 ∈ {0, 1} ∨ 𝑋 ∈ {2, 3}))
33	18, 31, 32	3bitri 300	. . . 4 ⊢ (𝑋 ∈ (0..^(♯‘𝐹)) ↔ (𝑋 ∈ {0, 1} ∨ 𝑋 ∈ {2, 3}))
34		elpri 4613	. . . . . 6 ⊢ (𝑋 ∈ {0, 1} → (𝑋 = 0 ∨ 𝑋 = 1))
35		prex 5409	. . . . . . . . . 10 ⊢ {⟨0, 0⟩, ⟨0, 1⟩} ∈ V
36		s4fv0 14937	. . . . . . . . . 10 ⊢ ({⟨0, 0⟩, ⟨0, 1⟩} ∈ V → (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘0) = {⟨0, 0⟩, ⟨0, 1⟩})
37	35, 36	ax-mp 5	. . . . . . . . 9 ⊢ (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘0) = {⟨0, 0⟩, ⟨0, 1⟩}
38	14	fveq1i 6882	. . . . . . . . 9 ⊢ (𝐹‘0) = (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘0)
39		gpgprismgr4cycl.p	. . . . . . . . . . . 12 ⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩
40	39	fveq1i 6882	. . . . . . . . . . 11 ⊢ (𝑃‘0) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘0)
41		opex 5445	. . . . . . . . . . . 12 ⊢ ⟨0, 0⟩ ∈ V
42		df-s5 14893	. . . . . . . . . . . . 13 ⊢ ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩ = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩ ++ ⟨“⟨0, 0⟩”⟩)
43		s4cli 14924	. . . . . . . . . . . . 13 ⊢ ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩ ∈ Word V
44		s4len 14941	. . . . . . . . . . . . 13 ⊢ (♯‘⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩) = 4
45		s4fv0 14937	. . . . . . . . . . . . 13 ⊢ (⟨0, 0⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘0) = ⟨0, 0⟩)
46		0nn0 12523	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ₀
47		4pos 12355	. . . . . . . . . . . . 13 ⊢ 0 < 4
48	42, 43, 44, 45, 46, 47	cats1fv 14901	. . . . . . . . . . . 12 ⊢ (⟨0, 0⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘0) = ⟨0, 0⟩)
49	41, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘0) = ⟨0, 0⟩
50	40, 49	eqtri 2786	. . . . . . . . . 10 ⊢ (𝑃‘0) = ⟨0, 0⟩
51	39	fveq1i 6882	. . . . . . . . . . 11 ⊢ (𝑃‘1) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘1)
52		opex 5445	. . . . . . . . . . . 12 ⊢ ⟨0, 1⟩ ∈ V
53		s4fv1 14938	. . . . . . . . . . . . 13 ⊢ (⟨0, 1⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘1) = ⟨0, 1⟩)
54		1nn0 12524	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
55		1lt4 12423	. . . . . . . . . . . . 13 ⊢ 1 < 4
56	42, 43, 44, 53, 54, 55	cats1fv 14901	. . . . . . . . . . . 12 ⊢ (⟨0, 1⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘1) = ⟨0, 1⟩)
57	52, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘1) = ⟨0, 1⟩
58	51, 57	eqtri 2786	. . . . . . . . . 10 ⊢ (𝑃‘1) = ⟨0, 1⟩
59	50, 58	preq12i 4704	. . . . . . . . 9 ⊢ {(𝑃‘0), (𝑃‘1)} = {⟨0, 0⟩, ⟨0, 1⟩}
60	37, 38, 59	3eqtr4i 2796	. . . . . . . 8 ⊢ (𝐹‘0) = {(𝑃‘0), (𝑃‘1)}
61		fveq2 6881	. . . . . . . 8 ⊢ (𝑋 = 0 → (𝐹‘𝑋) = (𝐹‘0))
62		fveq2 6881	. . . . . . . . 9 ⊢ (𝑋 = 0 → (𝑃‘𝑋) = (𝑃‘0))
63		fv0p1e1 12366	. . . . . . . . 9 ⊢ (𝑋 = 0 → (𝑃‘(𝑋 + 1)) = (𝑃‘1))
64	62, 63	preq12d 4707	. . . . . . . 8 ⊢ (𝑋 = 0 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘0), (𝑃‘1)})
65	60, 61, 64	3eqtr4a 2824	. . . . . . 7 ⊢ (𝑋 = 0 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
66		prex 5409	. . . . . . . . . 10 ⊢ {⟨0, 1⟩, ⟨1, 1⟩} ∈ V
67		s4fv1 14938	. . . . . . . . . 10 ⊢ ({⟨0, 1⟩, ⟨1, 1⟩} ∈ V → (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘1) = {⟨0, 1⟩, ⟨1, 1⟩})
68	66, 67	ax-mp 5	. . . . . . . . 9 ⊢ (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘1) = {⟨0, 1⟩, ⟨1, 1⟩}
69	14	fveq1i 6882	. . . . . . . . 9 ⊢ (𝐹‘1) = (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘1)
70	39	fveq1i 6882	. . . . . . . . . . 11 ⊢ (𝑃‘2) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘2)
71		opex 5445	. . . . . . . . . . . 12 ⊢ ⟨1, 1⟩ ∈ V
72		s4fv2 14939	. . . . . . . . . . . . 13 ⊢ (⟨1, 1⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘2) = ⟨1, 1⟩)
73		2nn0 12525	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
74		2lt4 12422	. . . . . . . . . . . . 13 ⊢ 2 < 4
75	42, 43, 44, 72, 73, 74	cats1fv 14901	. . . . . . . . . . . 12 ⊢ (⟨1, 1⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘2) = ⟨1, 1⟩)
76	71, 75	ax-mp 5	. . . . . . . . . . 11 ⊢ (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘2) = ⟨1, 1⟩
77	70, 76	eqtri 2786	. . . . . . . . . 10 ⊢ (𝑃‘2) = ⟨1, 1⟩
78	58, 77	preq12i 4704	. . . . . . . . 9 ⊢ {(𝑃‘1), (𝑃‘2)} = {⟨0, 1⟩, ⟨1, 1⟩}
79	68, 69, 78	3eqtr4i 2796	. . . . . . . 8 ⊢ (𝐹‘1) = {(𝑃‘1), (𝑃‘2)}
80		fveq2 6881	. . . . . . . 8 ⊢ (𝑋 = 1 → (𝐹‘𝑋) = (𝐹‘1))
81		fveq2 6881	. . . . . . . . 9 ⊢ (𝑋 = 1 → (𝑃‘𝑋) = (𝑃‘1))
82		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑋 = 1 → (𝑋 + 1) = (1 + 1))
83		1p1e2 12368	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
84	82, 83	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑋 = 1 → (𝑋 + 1) = 2)
85	84	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑋 = 1 → (𝑃‘(𝑋 + 1)) = (𝑃‘2))
86	81, 85	preq12d 4707	. . . . . . . 8 ⊢ (𝑋 = 1 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘1), (𝑃‘2)})
87	79, 80, 86	3eqtr4a 2824	. . . . . . 7 ⊢ (𝑋 = 1 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
88	65, 87	jaoi 870	. . . . . 6 ⊢ ((𝑋 = 0 ∨ 𝑋 = 1) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
89	34, 88	syl 18	. . . . 5 ⊢ (𝑋 ∈ {0, 1} → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
90		elpri 4613	. . . . . 6 ⊢ (𝑋 ∈ {2, 3} → (𝑋 = 2 ∨ 𝑋 = 3))
91		prex 5409	. . . . . . . . . 10 ⊢ {⟨1, 1⟩, ⟨1, 0⟩} ∈ V
92		s4fv2 14939	. . . . . . . . . 10 ⊢ ({⟨1, 1⟩, ⟨1, 0⟩} ∈ V → (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘2) = {⟨1, 1⟩, ⟨1, 0⟩})
93	91, 92	ax-mp 5	. . . . . . . . 9 ⊢ (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘2) = {⟨1, 1⟩, ⟨1, 0⟩}
94	14	fveq1i 6882	. . . . . . . . 9 ⊢ (𝐹‘2) = (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘2)
95	39	fveq1i 6882	. . . . . . . . . . 11 ⊢ (𝑃‘3) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘3)
96		opex 5445	. . . . . . . . . . . 12 ⊢ ⟨1, 0⟩ ∈ V
97		s4fv3 14940	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘3) = ⟨1, 0⟩)
98		3nn0 12526	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
99		3lt4 12421	. . . . . . . . . . . . 13 ⊢ 3 < 4
100	42, 43, 44, 97, 98, 99	cats1fv 14901	. . . . . . . . . . . 12 ⊢ (⟨1, 0⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘3) = ⟨1, 0⟩)
101	96, 100	ax-mp 5	. . . . . . . . . . 11 ⊢ (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘3) = ⟨1, 0⟩
102	95, 101	eqtri 2786	. . . . . . . . . 10 ⊢ (𝑃‘3) = ⟨1, 0⟩
103	77, 102	preq12i 4704	. . . . . . . . 9 ⊢ {(𝑃‘2), (𝑃‘3)} = {⟨1, 1⟩, ⟨1, 0⟩}
104	93, 94, 103	3eqtr4i 2796	. . . . . . . 8 ⊢ (𝐹‘2) = {(𝑃‘2), (𝑃‘3)}
105		fveq2 6881	. . . . . . . 8 ⊢ (𝑋 = 2 → (𝐹‘𝑋) = (𝐹‘2))
106		fveq2 6881	. . . . . . . . 9 ⊢ (𝑋 = 2 → (𝑃‘𝑋) = (𝑃‘2))
107		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑋 = 2 → (𝑋 + 1) = (2 + 1))
108		2p1e3 12386	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
109	107, 108	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑋 = 2 → (𝑋 + 1) = 3)
110	109	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑋 = 2 → (𝑃‘(𝑋 + 1)) = (𝑃‘3))
111	106, 110	preq12d 4707	. . . . . . . 8 ⊢ (𝑋 = 2 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘2), (𝑃‘3)})
112	104, 105, 111	3eqtr4a 2824	. . . . . . 7 ⊢ (𝑋 = 2 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
113		prex 5409	. . . . . . . . . 10 ⊢ {⟨1, 0⟩, ⟨0, 0⟩} ∈ V
114		s4fv3 14940	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨0, 0⟩} ∈ V → (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘3) = {⟨1, 0⟩, ⟨0, 0⟩})
115	113, 114	ax-mp 5	. . . . . . . . 9 ⊢ (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘3) = {⟨1, 0⟩, ⟨0, 0⟩}
116	14	fveq1i 6882	. . . . . . . . 9 ⊢ (𝐹‘3) = (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘3)
117	39	fveq1i 6882	. . . . . . . . . . 11 ⊢ (𝑃‘4) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘4)
118	42, 43, 44	cats1fvn 14900	. . . . . . . . . . . 12 ⊢ (⟨0, 0⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘4) = ⟨0, 0⟩)
119	41, 118	ax-mp 5	. . . . . . . . . . 11 ⊢ (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘4) = ⟨0, 0⟩
120	117, 119	eqtri 2786	. . . . . . . . . 10 ⊢ (𝑃‘4) = ⟨0, 0⟩
121	102, 120	preq12i 4704	. . . . . . . . 9 ⊢ {(𝑃‘3), (𝑃‘4)} = {⟨1, 0⟩, ⟨0, 0⟩}
122	115, 116, 121	3eqtr4i 2796	. . . . . . . 8 ⊢ (𝐹‘3) = {(𝑃‘3), (𝑃‘4)}
123		fveq2 6881	. . . . . . . 8 ⊢ (𝑋 = 3 → (𝐹‘𝑋) = (𝐹‘3))
124		fveq2 6881	. . . . . . . . 9 ⊢ (𝑋 = 3 → (𝑃‘𝑋) = (𝑃‘3))
125		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑋 = 3 → (𝑋 + 1) = (3 + 1))
126		3p1e4 12389	. . . . . . . . . . 11 ⊢ (3 + 1) = 4
127	125, 126	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑋 = 3 → (𝑋 + 1) = 4)
128	127	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑋 = 3 → (𝑃‘(𝑋 + 1)) = (𝑃‘4))
129	124, 128	preq12d 4707	. . . . . . . 8 ⊢ (𝑋 = 3 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘3), (𝑃‘4)})
130	122, 123, 129	3eqtr4a 2824	. . . . . . 7 ⊢ (𝑋 = 3 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
131	112, 130	jaoi 870	. . . . . 6 ⊢ ((𝑋 = 2 ∨ 𝑋 = 3) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
132	90, 131	syl 18	. . . . 5 ⊢ (𝑋 ∈ {2, 3} → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
133	89, 132	jaoi 870	. . . 4 ⊢ ((𝑋 ∈ {0, 1} ∨ 𝑋 ∈ {2, 3}) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
134	33, 133	sylbi 220	. . 3 ⊢ (𝑋 ∈ (0..^(♯‘𝐹)) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
135	134	adantl 486	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
136	29, 135	eqtrd 2798	1 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∨ wo 860 ∨ w3o 1102 = wceq 1570 ∈ wcel 2143 ∃wrex 3089 {crab 3416 Vcvv 3455 ∪ cun 3903 𝒫 cpw 4562 {cpr 4591 ⟨cop 4595 I cid 5555 × cxp 5659 ↾ cres 5663 ‘cfv 6536 (class class class)co 7410 0cc0 11104 1c1 11105 + caddc 11107 / cdiv 11875 ℕcn 12237 2c2 12299 3c3 12300 4c4 12301 ℤ_≥cuz 12866 ..^cfzo 13687 ⌈cceil 13829 mod cmo 13907 ♯chash 14371 ⟨“cs4 14885 ⟨“cs5 14886 iEdgciedg 29356 gPetersenGr cgpg 48833

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-dju 9892 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-xnn0 12582 df-z 12596 df-dec 12716 df-uz 12867 df-rp 13021 df-ico 13382 df-fz 13540 df-fzo 13688 df-fl 13830 df-ceil 13831 df-mod 13908 df-hash 14372 df-word 14556 df-concat 14613 df-s1 14639 df-s2 14890 df-s3 14891 df-s4 14892 df-s5 14893 df-struct 17211 df-slot 17246 df-ndx 17258 df-base 17274 df-edgf 29348 df-iedg 29358 df-gpg 48834

This theorem is used by: gpgprismgr4cycllem11 48898

W3C validator