Theorem erclwwlktr 26343

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 Alexander van der Vekens, 11-Jun-2018.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑛,𝐸,𝑢,𝑤   𝑛,𝑉,𝑢,𝑤   𝑥,𝑛,𝑢,𝑤,𝑦   𝑧,𝑛,𝑢,𝑤,𝑥

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑧,𝑤,𝑢,𝑛)   𝐸(𝑥,𝑦,𝑧)   𝑉(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 3176	. 2 ⊢ 𝑥 ∈ V
2		vex 3176	. 2 ⊢ 𝑦 ∈ V
3		vex 3176	. 2 ⊢ 𝑧 ∈ V
4		erclwwlk.r	. . . . . 6 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
5	4	erclwwlkeqlen 26340	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (#‘𝑥) = (#‘𝑦)))
6	5	3adant3 1074	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → (#‘𝑥) = (#‘𝑦)))
7	4	erclwwlkeqlen 26340	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (#‘𝑦) = (#‘𝑧)))
8	7	3adant1 1072	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (#‘𝑦) = (#‘𝑧)))
9	4	erclwwlkeq 26339	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
10	9	3adant1 1072	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))))
11	4	erclwwlkeq 26339	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
12	11	3adant3 1074	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))))
13		simpr1 1060	. . . . . . . . . . . . . . 15 ⊢ (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥 ∈ (𝑉 ClWWalks 𝐸))
14		simplr2 1097	. . . . . . . . . . . . . . 15 ⊢ (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧 ∈ (𝑉 ClWWalks 𝐸))
15		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
16	15	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
17	16	cbvrexv 3148	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑚))
18		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
19	18	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
20	19	cbvrexv 3148	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑘))
21		clwwlkprop 26298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑧 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑧 ∈ Word 𝑉))
22	21	simp3d 1068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 ∈ (𝑉 ClWWalks 𝐸) → 𝑧 ∈ Word 𝑉)
23	22	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → 𝑧 ∈ Word 𝑉)
24		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
25	23, 24	cshwcsh2id 13425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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)))
26	25	expdcom 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ((𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
27	26	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...(#‘𝑦))) → ((𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
28	27	expdcom 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...(#‘𝑦))) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
29	28	com4t 91	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑥 = (𝑦 cyclShift 𝑚) ∧ 𝑚 ∈ (0...(#‘𝑦))) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑘 ∈ (0...(#‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
30	29	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑚 ∈ (0...(#‘𝑦)) → ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦))) → (𝑘 ∈ (0...(#‘𝑧)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
31	30	com13 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))))
32	31	imp41 617	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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)))
33	32	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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)))
34	33	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
35	34	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
36	20, 35	syl7bi 244	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
37	17, 36	syl5bi 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸)) ∧ ((#‘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))))
38	37	exp31 628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → (𝑧 ∈ (𝑉 ClWWalks 𝐸) → (((#‘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))))))
39	38	com15 99	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛) → (𝑧 ∈ (𝑉 ClWWalks 𝐸) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))))
40	39	impcom 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
41	40	3adant1 1072	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))))
42	41	impcom 445	. . . . . . . . . . . . . . . . . 18 ⊢ ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
43	42	com13 86	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸)) → (∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
44	43	3impia 1253	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
45	44	impcom 445	. . . . . . . . . . . . . . 15 ⊢ (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
46	13, 14, 45	3jca 1235	. . . . . . . . . . . . . 14 ⊢ (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
47	4	erclwwlkeq 26339	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
48	47	3adant2 1073	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))))
49	46, 48	syl5ibrcom 236	. . . . . . . . . . . . 13 ⊢ (((((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) ∧ (𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∼ 𝑧))
50	49	exp31 628	. . . . . . . . . . . 12 ⊢ (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∼ 𝑧))))
51	50	com24 93	. . . . . . . . . . 11 ⊢ (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧))))
52	51	ex 449	. . . . . . . . . 10 ⊢ ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
53	52	com4t 91	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑦))𝑥 = (𝑦 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
54	12, 53	sylbid 229	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
55	54	com25 97	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑧 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑧))𝑦 = (𝑧 cyclShift 𝑛)) → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧)))))
56	10, 55	sylbid 229	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → ((#‘𝑦) = (#‘𝑧) → ((#‘𝑥) = (#‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧)))))
57	8, 56	mpdd 42	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → ((#‘𝑥) = (#‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧))))
58	57	com24 93	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → ((#‘𝑥) = (#‘𝑦) → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧))))
59	6, 58	mpdd 42	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧)))
60	59	impd 446	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))
61	1, 2, 3, 60	mp3an 1416	1 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalks cclwwlk 26276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279

This theorem is referenced by:  erclwwlk  26344