Theorem erclwwlktr 30002

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 3448	. 2 ⊢ 𝑥 ∈ V
2		vex 3448	. 2 ⊢ 𝑦 ∈ V
3		vex 3448	. 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 29999	. . . . 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 29999	. . . . . . 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 29998	. . . . . . . 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 29998	. . . . . . . . . 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 7377	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
16	15	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
17	16	cbvrexvw 3214	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑛 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(♯‘𝑦))𝑥 = (𝑦 cyclShift 𝑚))
18		oveq2 7377	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
19	18	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
20	19	cbvrexvw 3214	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃𝑛 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...(♯‘𝑧))𝑦 = (𝑧 cyclShift 𝑘))
21		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
22	21	clwwlkbp 29965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3134	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3136	. . . . . . . . . . . . . . . . . . . . . . . . 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 29998	. . . . . . . . . . . . . . 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 2925  ∃wrex 3053  Vcvv 3444  ∅c0 4292   class class class wbr 5102  {copab 5164  ‘cfv 6499  (class class class)co 7369  0cc0 11046  ...cfz 13446  ♯chash 14273  Word cword 14456   cyclShift ccsh 14730  Vtxcvtx 28977  ClWWalkscclwwlk 29961

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11102  ax-resscn 11103  ax-1cn 11104  ax-icn 11105  ax-addcl 11106  ax-addrcl 11107  ax-mulcl 11108  ax-mulrcl 11109  ax-mulcom 11110  ax-addass 11111  ax-mulass 11112  ax-distr 11113  ax-i2m1 11114  ax-1ne0 11115  ax-1rid 11116  ax-rnegex 11117  ax-rrecex 11118  ax-cnre 11119  ax-pre-lttri 11120  ax-pre-lttrn 11121  ax-pre-ltadd 11122  ax-pre-mulgt0 11123  ax-pre-sup 11124

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-card 9870  df-pnf 11188  df-mnf 11189  df-xr 11190  df-ltxr 11191  df-le 11192  df-sub 11385  df-neg 11386  df-div 11814  df-nn 12165  df-2 12227  df-n0 12421  df-z 12508  df-uz 12772  df-rp 12930  df-fz 13447  df-fzo 13594  df-fl 13732  df-mod 13810  df-hash 14274  df-word 14457  df-concat 14514  df-substr 14584  df-pfx 14614  df-csh 14731  df-clwwlk 29962

This theorem is referenced by:  erclwwlk  30003