Theorem erclwwlktr 28386

Description: ∼ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlk.r	⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlktr	⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)

Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤,𝑦   𝑧,𝑛,𝑢,𝑤,𝑥

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑧,𝑤,𝑢,𝑛)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem erclwwlktr

Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3436	. 2 ⊢ 𝑥 ∈ V
2		vex 3436	. 2 ⊢ 𝑦 ∈ V
3		vex 3436	. 2 ⊢ 𝑧 ∈ V
4		erclwwlk.r	. . . . . 6 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
5	4	erclwwlkeqlen 28383	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
6	5	3adant3 1131	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
7	4	erclwwlkeqlen 28383	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
8	7	3adant1 1129	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
9	4	erclwwlkeq 28382	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
10	9	3adant1 1129	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
11	4	erclwwlkeq 28382	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
12	11	3adant3 1131	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
13		simpr1 1193	. . . . . . . . . . . . . . 15 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥 ∈ (ClWWalks‘𝐺))
14		simplr2 1215	. . . . . . . . . . . . . . 15 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧 ∈ (ClWWalks‘𝐺))
15		oveq2 7283	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
16	15	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
17	16	cbvrexvw 3384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑚))
18		oveq2 7283	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
19	18	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
20	19	cbvrexvw 3384	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑘))
21		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
22	21	clwwlkbp 28349	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑧 ∈ Word (Vtx‘𝐺) ∧ 𝑧 ≠ ∅))
23	22	simp2d 1142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (ClWWalks‘𝐺) → 𝑧 ∈ Word (Vtx‘𝐺))
24	23	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑧 ∈ Word (Vtx‘𝐺))
25		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
26	24, 25	cshwcsh2id 14541	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
27	26	exp5l 447	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑚 ∈ (0...(♯‘𝑦)) → (𝑥 = (𝑦 cyclShift 𝑚) → (𝑘 ∈ (0...(♯‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
28	27	imp41 426	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) ∧ 𝑚 ∈ (0...(♯‘𝑦))) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
29	28	rexlimdva 3213	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) ∧ 𝑚 ∈ (0...(♯‘𝑦))) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (∃𝑘 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
30	29	rexlimdva2 3216	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑚 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑘 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
31	20, 30	syl7bi 254	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑚 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
32	17, 31	syl5bi 241	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
33	32	exp31 420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → (𝑧 ∈ (ClWWalks‘𝐺) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
34	33	com15 101	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → (𝑧 ∈ (ClWWalks‘𝐺) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
35	34	impcom 408	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
36	35	3adant1 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
37	36	impcom 408	. . . . . . . . . . . . . . . . . 18 ⊢ ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
38	37	com13 88	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) → (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
39	38	3impia 1116	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
40	39	impcom 408	. . . . . . . . . . . . . . 15 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
41	13, 14, 40	3jca 1127	. . . . . . . . . . . . . 14 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
42	4	erclwwlkeq 28382	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
43	42	3adant2 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
44	41, 43	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∼ 𝑧))
45	44	exp31 420	. . . . . . . . . . . 12 ⊢ (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∼ 𝑧))))
46	45	com24 95	. . . . . . . . . . 11 ⊢ (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧))))
47	46	ex 413	. . . . . . . . . 10 ⊢ ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
48	47	com4t 93	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
49	12, 48	sylbid 239	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
50	49	com25 99	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧)))))
51	10, 50	sylbid 239	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧)))))
52	8, 51	mpdd 43	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧))))
53	52	com24 95	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧))))
54	6, 53	mpdd 43	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧)))
55	54	impd 411	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))
56	1, 2, 3, 55	mp3an 1460	1 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065  Vcvv 3432  ∅c0 4256   class class class wbr 5074  {copab 5136  ‘cfv 6433  (class class class)co 7275  0cc0 10871  ...cfz 13239  ♯chash 14044  Word cword 14217   cyclShift ccsh 14501  Vtxcvtx 27366  ClWWalkscclwwlk 28345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-hash 14045  df-word 14218  df-concat 14274  df-substr 14354  df-pfx 14384  df-csh 14502  df-clwwlk 28346

This theorem is referenced by:  erclwwlk  28387