gpgprismgr4cycllem10 - Mathbox for Alexander van der Vekens

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Theorem gpgprismgr4cycllem10 48417

Description: Lemma 10 for gpgprismgr4cycl0 48419. (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 6838	. . . . . 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 12806	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℕ)
5		1elfzo1ceilhalf1 47650	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 1 ∈ (1..^(⌈‘(𝑁 / 2))))
6	4, 5	jca 511	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (𝑁 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(𝑁 / 2)))))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝑁 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(𝑁 / 2)))))
8		eqid 2737	. . . . . . 7 ⊢ (1..^(⌈‘(𝑁 / 2))) = (1..^(⌈‘(𝑁 / 2)))
9		eqid 2737	. . . . . . 7 ⊢ (0..^𝑁) = (0..^𝑁)
10	8, 9	gpgiedg 48357	. . . . . 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 17	. . . . 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 2772	. . . 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 6837	. . 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 48410	. . . . 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 48408	. . . . . . . 8 ⊢ (♯‘𝐹) = 4
17	16	oveq2i 7371	. . . . . . 7 ⊢ (0..^(♯‘𝐹)) = (0..^4)
18	17	eleq2i 2829	. . . . . 6 ⊢ (𝑋 ∈ (0..^(♯‘𝐹)) ↔ 𝑋 ∈ (0..^4))
19	18	anbi2i 624	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) ↔ (𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^4)))
20		eqeq1 2741	. . . . . . . 8 ⊢ (𝑒 = (𝐹‘𝑋) → (𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ↔ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩}))
21		eqeq1 2741	. . . . . . . 8 ⊢ (𝑒 = (𝐹‘𝑋) → (𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩}))
22		eqeq1 2741	. . . . . . . 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 1438	. . . . . . 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 3161	. . . . . 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 3647	. . . . 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 292	. . . 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 7121	. . . 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 17	. . 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 2772	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = (𝐹‘𝑋))
30		fzo0to42pr 13673	. . . . . 6 ⊢ (0..^4) = ({0, 1} ∪ {2, 3})
31	30	eleq2i 2829	. . . . 5 ⊢ (𝑋 ∈ (0..^4) ↔ 𝑋 ∈ ({0, 1} ∪ {2, 3}))
32		elun 4106	. . . . 5 ⊢ (𝑋 ∈ ({0, 1} ∪ {2, 3}) ↔ (𝑋 ∈ {0, 1} ∨ 𝑋 ∈ {2, 3}))
33	18, 31, 32	3bitri 297	. . . 4 ⊢ (𝑋 ∈ (0..^(♯‘𝐹)) ↔ (𝑋 ∈ {0, 1} ∨ 𝑋 ∈ {2, 3}))
34		elpri 4605	. . . . . 6 ⊢ (𝑋 ∈ {0, 1} → (𝑋 = 0 ∨ 𝑋 = 1))
35		prex 5383	. . . . . . . . . 10 ⊢ {⟨0, 0⟩, ⟨0, 1⟩} ∈ V
36		s4fv0 14822	. . . . . . . . . 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 6836	. . . . . . . . 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 6836	. . . . . . . . . . 11 ⊢ (𝑃‘0) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘0)
41		opex 5413	. . . . . . . . . . . 12 ⊢ ⟨0, 0⟩ ∈ V
42		df-s5 14778	. . . . . . . . . . . . 13 ⊢ ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩ = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩ ++ ⟨“⟨0, 0⟩”⟩)
43		s4cli 14809	. . . . . . . . . . . . 13 ⊢ ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩ ∈ Word V
44		s4len 14826	. . . . . . . . . . . . 13 ⊢ (♯‘⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩) = 4
45		s4fv0 14822	. . . . . . . . . . . . 13 ⊢ (⟨0, 0⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘0) = ⟨0, 0⟩)
46		0nn0 12420	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ₀
47		4pos 12256	. . . . . . . . . . . . 13 ⊢ 0 < 4
48	42, 43, 44, 45, 46, 47	cats1fv 14786	. . . . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (𝑃‘0) = ⟨0, 0⟩
51	39	fveq1i 6836	. . . . . . . . . . 11 ⊢ (𝑃‘1) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘1)
52		opex 5413	. . . . . . . . . . . 12 ⊢ ⟨0, 1⟩ ∈ V
53		s4fv1 14823	. . . . . . . . . . . . 13 ⊢ (⟨0, 1⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘1) = ⟨0, 1⟩)
54		1nn0 12421	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
55		1lt4 12320	. . . . . . . . . . . . 13 ⊢ 1 < 4
56	42, 43, 44, 53, 54, 55	cats1fv 14786	. . . . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (𝑃‘1) = ⟨0, 1⟩
59	50, 58	preq12i 4696	. . . . . . . . 9 ⊢ {(𝑃‘0), (𝑃‘1)} = {⟨0, 0⟩, ⟨0, 1⟩}
60	37, 38, 59	3eqtr4i 2770	. . . . . . . 8 ⊢ (𝐹‘0) = {(𝑃‘0), (𝑃‘1)}
61		fveq2 6835	. . . . . . . 8 ⊢ (𝑋 = 0 → (𝐹‘𝑋) = (𝐹‘0))
62		fveq2 6835	. . . . . . . . 9 ⊢ (𝑋 = 0 → (𝑃‘𝑋) = (𝑃‘0))
63		fv0p1e1 12267	. . . . . . . . 9 ⊢ (𝑋 = 0 → (𝑃‘(𝑋 + 1)) = (𝑃‘1))
64	62, 63	preq12d 4699	. . . . . . . 8 ⊢ (𝑋 = 0 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘0), (𝑃‘1)})
65	60, 61, 64	3eqtr4a 2798	. . . . . . 7 ⊢ (𝑋 = 0 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
66		prex 5383	. . . . . . . . . 10 ⊢ {⟨0, 1⟩, ⟨1, 1⟩} ∈ V
67		s4fv1 14823	. . . . . . . . . 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 6836	. . . . . . . . 9 ⊢ (𝐹‘1) = (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘1)
70	39	fveq1i 6836	. . . . . . . . . . 11 ⊢ (𝑃‘2) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘2)
71		opex 5413	. . . . . . . . . . . 12 ⊢ ⟨1, 1⟩ ∈ V
72		s4fv2 14824	. . . . . . . . . . . . 13 ⊢ (⟨1, 1⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘2) = ⟨1, 1⟩)
73		2nn0 12422	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
74		2lt4 12319	. . . . . . . . . . . . 13 ⊢ 2 < 4
75	42, 43, 44, 72, 73, 74	cats1fv 14786	. . . . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (𝑃‘2) = ⟨1, 1⟩
78	58, 77	preq12i 4696	. . . . . . . . 9 ⊢ {(𝑃‘1), (𝑃‘2)} = {⟨0, 1⟩, ⟨1, 1⟩}
79	68, 69, 78	3eqtr4i 2770	. . . . . . . 8 ⊢ (𝐹‘1) = {(𝑃‘1), (𝑃‘2)}
80		fveq2 6835	. . . . . . . 8 ⊢ (𝑋 = 1 → (𝐹‘𝑋) = (𝐹‘1))
81		fveq2 6835	. . . . . . . . 9 ⊢ (𝑋 = 1 → (𝑃‘𝑋) = (𝑃‘1))
82		oveq1 7367	. . . . . . . . . . 11 ⊢ (𝑋 = 1 → (𝑋 + 1) = (1 + 1))
83		1p1e2 12269	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
84	82, 83	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑋 = 1 → (𝑋 + 1) = 2)
85	84	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑋 = 1 → (𝑃‘(𝑋 + 1)) = (𝑃‘2))
86	81, 85	preq12d 4699	. . . . . . . 8 ⊢ (𝑋 = 1 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘1), (𝑃‘2)})
87	79, 80, 86	3eqtr4a 2798	. . . . . . 7 ⊢ (𝑋 = 1 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
88	65, 87	jaoi 858	. . . . . 6 ⊢ ((𝑋 = 0 ∨ 𝑋 = 1) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
89	34, 88	syl 17	. . . . 5 ⊢ (𝑋 ∈ {0, 1} → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
90		elpri 4605	. . . . . 6 ⊢ (𝑋 ∈ {2, 3} → (𝑋 = 2 ∨ 𝑋 = 3))
91		prex 5383	. . . . . . . . . 10 ⊢ {⟨1, 1⟩, ⟨1, 0⟩} ∈ V
92		s4fv2 14824	. . . . . . . . . 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 6836	. . . . . . . . 9 ⊢ (𝐹‘2) = (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘2)
95	39	fveq1i 6836	. . . . . . . . . . 11 ⊢ (𝑃‘3) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘3)
96		opex 5413	. . . . . . . . . . . 12 ⊢ ⟨1, 0⟩ ∈ V
97		s4fv3 14825	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ∈ V → (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩”⟩‘3) = ⟨1, 0⟩)
98		3nn0 12423	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
99		3lt4 12318	. . . . . . . . . . . . 13 ⊢ 3 < 4
100	42, 43, 44, 97, 98, 99	cats1fv 14786	. . . . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (𝑃‘3) = ⟨1, 0⟩
103	77, 102	preq12i 4696	. . . . . . . . 9 ⊢ {(𝑃‘2), (𝑃‘3)} = {⟨1, 1⟩, ⟨1, 0⟩}
104	93, 94, 103	3eqtr4i 2770	. . . . . . . 8 ⊢ (𝐹‘2) = {(𝑃‘2), (𝑃‘3)}
105		fveq2 6835	. . . . . . . 8 ⊢ (𝑋 = 2 → (𝐹‘𝑋) = (𝐹‘2))
106		fveq2 6835	. . . . . . . . 9 ⊢ (𝑋 = 2 → (𝑃‘𝑋) = (𝑃‘2))
107		oveq1 7367	. . . . . . . . . . 11 ⊢ (𝑋 = 2 → (𝑋 + 1) = (2 + 1))
108		2p1e3 12286	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
109	107, 108	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑋 = 2 → (𝑋 + 1) = 3)
110	109	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑋 = 2 → (𝑃‘(𝑋 + 1)) = (𝑃‘3))
111	106, 110	preq12d 4699	. . . . . . . 8 ⊢ (𝑋 = 2 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘2), (𝑃‘3)})
112	104, 105, 111	3eqtr4a 2798	. . . . . . 7 ⊢ (𝑋 = 2 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
113		prex 5383	. . . . . . . . . 10 ⊢ {⟨1, 0⟩, ⟨0, 0⟩} ∈ V
114		s4fv3 14825	. . . . . . . . . 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 6836	. . . . . . . . 9 ⊢ (𝐹‘3) = (⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩‘3)
117	39	fveq1i 6836	. . . . . . . . . . 11 ⊢ (𝑃‘4) = (⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩‘4)
118	42, 43, 44	cats1fvn 14785	. . . . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (𝑃‘4) = ⟨0, 0⟩
121	102, 120	preq12i 4696	. . . . . . . . 9 ⊢ {(𝑃‘3), (𝑃‘4)} = {⟨1, 0⟩, ⟨0, 0⟩}
122	115, 116, 121	3eqtr4i 2770	. . . . . . . 8 ⊢ (𝐹‘3) = {(𝑃‘3), (𝑃‘4)}
123		fveq2 6835	. . . . . . . 8 ⊢ (𝑋 = 3 → (𝐹‘𝑋) = (𝐹‘3))
124		fveq2 6835	. . . . . . . . 9 ⊢ (𝑋 = 3 → (𝑃‘𝑋) = (𝑃‘3))
125		oveq1 7367	. . . . . . . . . . 11 ⊢ (𝑋 = 3 → (𝑋 + 1) = (3 + 1))
126		3p1e4 12289	. . . . . . . . . . 11 ⊢ (3 + 1) = 4
127	125, 126	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑋 = 3 → (𝑋 + 1) = 4)
128	127	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑋 = 3 → (𝑃‘(𝑋 + 1)) = (𝑃‘4))
129	124, 128	preq12d 4699	. . . . . . . 8 ⊢ (𝑋 = 3 → {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))} = {(𝑃‘3), (𝑃‘4)})
130	122, 123, 129	3eqtr4a 2798	. . . . . . 7 ⊢ (𝑋 = 3 → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
131	112, 130	jaoi 858	. . . . . 6 ⊢ ((𝑋 = 2 ∨ 𝑋 = 3) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
132	90, 131	syl 17	. . . . 5 ⊢ (𝑋 ∈ {2, 3} → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
133	89, 132	jaoi 858	. . . 4 ⊢ ((𝑋 ∈ {0, 1} ∨ 𝑋 ∈ {2, 3}) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
134	33, 133	sylbi 217	. . 3 ⊢ (𝑋 ∈ (0..^(♯‘𝐹)) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
135	134	adantl 481	. 2 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑋) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
136	29, 135	eqtrd 2772	1 ⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 {crab 3400 Vcvv 3441 ∪ cun 3900 𝒫 cpw 4555 {cpr 4583 ⟨cop 4587 I cid 5519 × cxp 5623 ↾ cres 5627 ‘cfv 6493 (class class class)co 7360 0cc0 11030 1c1 11031 + caddc 11033 / cdiv 11798 ℕcn 12149 2c2 12204 3c3 12205 4c4 12206 ℤ_≥cuz 12755 ..^cfzo 13574 ⌈cceil 13715 mod cmo 13793 ♯chash 14257 ⟨“cs4 14770 ⟨“cs5 14771 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-word 14441 df-concat 14498 df-s1 14524 df-s2 14775 df-s3 14776 df-s4 14777 df-s5 14778 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: gpgprismgr4cycllem11 48418

W3C validator