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Theorem erclwwlkntr 27589
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⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkntr ((𝑥 𝑦𝑦 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.
StepHypRef Expression
1 vex 3412 . 2 𝑥 ∈ V
2 vex 3412 . 2 𝑦 ∈ V
3 vex 3412 . 2 𝑧 ∈ V
4 erclwwlkn.w . . . . . 6 𝑊 = (𝑁 ClWWalksN 𝐺)
5 erclwwlkn.r . . . . . 6 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
64, 5erclwwlkneqlen 27586 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
763adant3 1112 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (♯‘𝑥) = (♯‘𝑦)))
84, 5erclwwlkneqlen 27586 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
983adant1 1110 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → (♯‘𝑦) = (♯‘𝑧)))
104, 5erclwwlkneq 27585 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
11103adant1 1110 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 ↔ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))))
124, 5erclwwlkneq 27585 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
13123adant3 1112 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 ↔ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))))
14 simpr1 1174 . . . . . . . . . . . . . . 15 (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑥𝑊)
15 simplr2 1196 . . . . . . . . . . . . . . 15 (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → 𝑧𝑊)
16 oveq2 6978 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑦 cyclShift 𝑛) = (𝑦 cyclShift 𝑚))
1716eqeq2d 2782 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (𝑥 = (𝑦 cyclShift 𝑛) ↔ 𝑥 = (𝑦 cyclShift 𝑚)))
1817cbvrexv 3378 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑚))
19 oveq2 6978 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
2019eqeq2d 2782 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift 𝑘)))
2120cbvrexv 3378 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑘))
22 eqid 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Vtx‘𝐺) = (Vtx‘𝐺)
2322clwwlknbp 27544 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → (𝑧 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑧) = 𝑁))
24 eqcom 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((♯‘𝑧) = 𝑁𝑁 = (♯‘𝑧))
2524biimpi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((♯‘𝑧) = 𝑁𝑁 = (♯‘𝑧))
2623, 25simpl2im 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑧))
2726, 4eleq2s 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧𝑊𝑁 = (♯‘𝑧))
2827ad2antlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑁 = (♯‘𝑧))
2923simpld 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑧 ∈ Word (Vtx‘𝐺))
3029, 4eleq2s 2878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑧𝑊𝑧 ∈ Word (Vtx‘𝐺))
3130ad2antlr 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → 𝑧 ∈ Word (Vtx‘𝐺))
3231adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → 𝑧 ∈ Word (Vtx‘𝐺))
33 simprr 760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
3432, 33cshwcsh2id 14046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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 6978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 = (♯‘𝑧) → (0...𝑁) = (0...(♯‘𝑧)))
36 oveq2 6978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((♯‘𝑧) = (♯‘𝑦) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
3736eqcoms 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((♯‘𝑦) = (♯‘𝑧) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
3837adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
3938adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦))) → (0...(♯‘𝑧)) = (0...(♯‘𝑦)))
4035, 39sylan9eq 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (0...𝑁) = (0...(♯‘𝑦)))
4140eleq2d 2845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (𝑚 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...(♯‘𝑦))))
4241anbi1d 620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → ((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))))
4335eleq2d 2845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 = (♯‘𝑧) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑧))))
4443anbi1d 620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑁 = (♯‘𝑧) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
4544adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → ((𝑘 ∈ (0...𝑁) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))))
4642, 45anbi12d 621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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)))))
4735rexeqdv 3350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 = (♯‘𝑧) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4847adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4934, 46, 483imtr4d 286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 = (♯‘𝑧) ∧ (((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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)))
5028, 49mpancom 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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)))
5150exp5l 439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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))))))
5251imp41 418 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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)))
5352rexlimdva 3223 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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)))
5453ex 405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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))))
5554rexlimdva 3223 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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))))
5621, 55syl7bi 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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))))
5718, 56syl5bi 234 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑥𝑊𝑦𝑊) ∧ 𝑧𝑊) ∧ ((♯‘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))))
5857exp31 412 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑊𝑦𝑊) → (𝑧𝑊 → (((♯‘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))))))
5958com15 101 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛) → (𝑧𝑊 → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
6059impcom 399 . . . . . . . . . . . . . . . . . . . 20 ((𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
61603adant1 1110 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
6261impcom 399 . . . . . . . . . . . . . . . . . 18 ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((𝑥𝑊𝑦𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
6362com13 88 . . . . . . . . . . . . . . . . 17 ((𝑥𝑊𝑦𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛) → ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
64633impia 1097 . . . . . . . . . . . . . . . 16 ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
6564impcom 399 . . . . . . . . . . . . . . 15 (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))
6614, 15, 653jca 1108 . . . . . . . . . . . . . 14 (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
674, 5erclwwlkneq 27585 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
68673adant2 1111 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑧 ↔ (𝑥𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
6966, 68syl5ibrcom 239 . . . . . . . . . . . . 13 (((((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) ∧ (𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛))) ∧ (𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))
7069exp31 412 . . . . . . . . . . . 12 (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝑥 𝑧))))
7170com24 95 . . . . . . . . . . 11 (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧))))
7271ex 405 . . . . . . . . . 10 ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
7372com4t 93 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥𝑊𝑦𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑦 cyclShift 𝑛)) → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
7413, 73sylbid 232 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → 𝑥 𝑧)))))
7574com25 99 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦𝑊𝑧𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑧 cyclShift 𝑛)) → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
7611, 75sylbid 232 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((♯‘𝑦) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 𝑦𝑥 𝑧)))))
779, 76mpdd 43 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 𝑧 → ((♯‘𝑥) = (♯‘𝑦) → (𝑥 𝑦𝑥 𝑧))))
7877com24 95 . . . 4 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → ((♯‘𝑥) = (♯‘𝑦) → (𝑦 𝑧𝑥 𝑧))))
797, 78mpdd 43 . . 3 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑥 𝑦 → (𝑦 𝑧𝑥 𝑧)))
8079impd 402 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
811, 2, 3, 80mp3an 1440 1 ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3083  Vcvv 3409   class class class wbr 4923  {copab 4985  cfv 6182  (class class class)co 6970  0cc0 10329  ...cfz 12702  chash 13499  Word cword 13666   cyclShift ccsh 14001  Vtxcvtx 26478   ClWWalksN cclwwlkn 27533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406  ax-pre-sup 10407
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7495  df-2nd 7496  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-sup 8695  df-inf 8696  df-card 9156  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-div 11093  df-nn 11434  df-2 11497  df-n0 11702  df-z 11788  df-uz 12053  df-rp 12199  df-fz 12703  df-fzo 12844  df-fl 12971  df-mod 13047  df-hash 13500  df-word 13667  df-concat 13728  df-substr 13798  df-pfx 13847  df-csh 14003  df-clwwlk 27482  df-clwwlkn 27534
This theorem is referenced by:  erclwwlkn  27590
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