Theorem erclwwlktr 29958

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 3454	. 2 ⊢ 𝑥 ∈ V
2		vex 3454	. 2 ⊢ 𝑦 ∈ V
3		vex 3454	. 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 29955	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
6	5	3adant3 1132	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
7	4	erclwwlkeqlen 29955	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
8	7	3adant1 1130	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
9	4	erclwwlkeq 29954	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
10	9	3adant1 1130	. . . . . . 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 29954	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
12	11	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
13		simpr1 1195	. . . . . . . . . . . . . . 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 1217	. . . . . . . . . . . . . . 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 7398	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
16	15	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
17	16	cbvrexvw 3217	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑚))
18		oveq2 7398	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
19	18	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
20	19	cbvrexvw 3217	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑘))
21		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
22	21	clwwlkbp 29921	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑧 ∈ Word (Vtx‘𝐺) ∧ 𝑧 ≠ ∅))
23	22	simp2d 1143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (ClWWalks‘𝐺) → 𝑧 ∈ Word (Vtx‘𝐺))
24	23	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑧 ∈ Word (Vtx‘𝐺))
25		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑦 ∈ (ClWWalks‘𝐺)) ∧ 𝑧 ∈ (ClWWalks‘𝐺)) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
26	24, 25	cshwcsh2id 14801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 446	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3135	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3137	. . . . . . . . . . . . . . . . . . . . . . . . 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 255	. . . . . . . . . . . . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . 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 29954	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ (ClWWalks‘𝐺) ∧ 𝑧 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
43	42	3adant2 1131	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 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 419	. . . . . . . . . . . 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 412	. . . . . . . . . 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 240	. . . . . . . 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 240	. . . . . 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 410	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))
56	1, 2, 3, 55	mp3an 1463	1 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  Vcvv 3450  ∅c0 4299   class class class wbr 5110  {copab 5172  ‘cfv 6514  (class class class)co 7390  0cc0 11075  ...cfz 13475  ♯chash 14302  Word cword 14485   cyclShift ccsh 14760  Vtxcvtx 28930  ClWWalkscclwwlk 29917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-fl 13761  df-mod 13839  df-hash 14303  df-word 14486  df-concat 14543  df-substr 14613  df-pfx 14643  df-csh 14761  df-clwwlk 29918

This theorem is referenced by:  erclwwlk  29959