Theorem erclwwlkntr 30363

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.)

Hypotheses

Ref	Expression
erclwwlkn.w	⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.r	⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

Assertion

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

Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥   𝑦,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊   𝑧,𝑛,𝑡,𝑢,𝑦,𝑥

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑧,𝑢,𝑡,𝑛)   𝐺(𝑥,𝑦,𝑧,𝑢,𝑡,𝑛)   𝑁(𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem erclwwlkntr

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

Step	Hyp	Ref	Expression
1		vex 3465	. 2 ⊢ 𝑥 ∈ V
2		vex 3465	. 2 ⊢ 𝑦 ∈ V
3		vex 3465	. 2 ⊢ 𝑧 ∈ V
4		erclwwlkn.w	. . . . . 6 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
5		erclwwlkn.r	. . . . . 6 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
6	4, 5	erclwwlkneqlen 30360	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
7	6	3adant3 1148	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
8	4, 5	erclwwlkneqlen 30360	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
9	8	3adant1 1146	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
10	4, 5	erclwwlkneq 30359	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
11	10	3adant1 1146	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
12	4, 5	erclwwlkneq 30359	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
13	12	3adant3 1148	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 ↔ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
14		simpr1 1211	. . . . . . . . . . . . . . 15 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥 ∈ 𝑊)
15		simplr2 1233	. . . . . . . . . . . . . . 15 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧 ∈ 𝑊)
16		oveq2 7419	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
17	16	eqeq2d 2780	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
18	17	cbvrexvw 3250	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚))
19		oveq2 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
20	19	eqeq2d 2780	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
21	20	cbvrexvw 3250	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘))
22		eqid 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
23	22	clwwlknbp 30327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → (𝑧 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑧) = 𝑁))
24		eqcom 2776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((♯‘𝑧) = 𝑁 ↔ 𝑁 = (♯‘𝑧))
25	24	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((♯‘𝑧) = 𝑁 → 𝑁 = (♯‘𝑧))
26	23, 25	simpl2im 512	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑧))
27	26, 4	eleq2s 2887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ 𝑊 → 𝑁 = (♯‘𝑧))
28	27	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑁 = (♯‘𝑧))
29	23	simpld 499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑧 ∈ Word (Vtx‘𝐺))
30	29, 4	eleq2s 2887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ 𝑊 → 𝑧 ∈ Word (Vtx‘𝐺))
31	30	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑧 ∈ Word (Vtx‘𝐺))
32	31	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → 𝑧 ∈ Word (Vtx‘𝐺))
33		simprr 784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
34	32, 33	cshwcsh2id 14865	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘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		oveq2 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 = (♯‘𝑧) → (0...𝑁) = (0...(♯‘𝑧)))
36		oveq2 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((♯‘𝑧) = (♯‘𝑦) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
37	36	eqcoms 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((♯‘𝑦) = (♯‘𝑧) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
38	37	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
39	38	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
40	35, 39	sylan9eq 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (0...𝑁) = (0...(♯‘𝑦)))
41	40	eleq2d 2855	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (𝑚 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...(♯‘𝑦))))
42	41	anbi1d 642	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → ((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))))
43	35	eleq2d 2855	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 = (♯‘𝑧) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑧))))
44	43	anbi1d 642	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 = (♯‘𝑧) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
45	44	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
46	42, 45	anbi12d 643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) ↔ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)))))
47	35	rexeqdv 3330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 = (♯‘𝑧) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
48	47	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
49	34, 46, 48	3imtr4d 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
50	28, 49	mpancom 700	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
51	50	exp5l 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (𝑚 ∈ (0...𝑁) → (𝑥 = (𝑦 cyclShift 𝑚) → (𝑘 ∈ (0...𝑁) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
52	51	imp41 430	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
53	52	rexlimdva 3172	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
54	53	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
55	54	rexlimdva 3172	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
56	21, 55	syl7bi 258	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
57	18, 56	biimtrid 245	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
58	57	exp31 424	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → (𝑧 ∈ 𝑊 → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
59	58	com15 102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → (𝑧 ∈ 𝑊 → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
60	59	impcom 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
61	60	3adant1 1146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
62	61	impcom 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
63	62	com13 89	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
64	63	3impia 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
65	64	impcom 412	. . . . . . . . . . . . . . 15 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))
66	14, 15, 65	3jca 1144	. . . . . . . . . . . . . 14 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → (𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
67	4, 5	erclwwlkneq 30359	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
68	67	3adant2 1147	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑧 ↔ (𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
69	66, 68	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∼ 𝑧))
70	69	exp31 424	. . . . . . . . . . . 12 ⊢ (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∼ 𝑧))))
71	70	com24 96	. . . . . . . . . . 11 ⊢ (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧))))
72	71	ex 417	. . . . . . . . . 10 ⊢ ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
73	72	com4t 94	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
74	13, 73	sylbid 243	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 ∼ 𝑧)))))
75	74	com25 100	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧)))))
76	11, 75	sylbid 243	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧)))))
77	9, 76	mpdd 44	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∼ 𝑧 → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 ∼ 𝑦 → 𝑥 ∼ 𝑧))))
78	77	com24 96	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧))))
79	7, 78	mpdd 44	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∼ 𝑦 → (𝑦 ∼ 𝑧 → 𝑥 ∼ 𝑧)))
80	79	impd 415	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))
81	1, 2, 3, 80	mp3an 1487	1 ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3461   class class class wbr 5111  {copab 5175  ‘cfv 6537  (class class class)co 7411  0cc0 11100  ...cfz 13535  ♯chash 14366  Word cword 14550   cyclShift ccsh 14825  Vtxcvtx 29287   ClWWalksN cclwwlkn 30316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-inf 9403  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-rp 13017  df-fz 13536  df-fzo 13683  df-fl 13825  df-mod 13903  df-hash 14367  df-word 14551  df-concat 14608  df-substr 14679  df-pfx 14709  df-csh 14826  df-clwwlk 30274  df-clwwlkn 30317

This theorem is referenced by:  erclwwlkn  30364