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Theorem erclwwlkntr 28480
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 ∌ 𝑊 ∧ 𝑊 ∌ 𝑧) → 𝑥 ∌ 𝑧)
Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥   𝑊,𝑛,𝑡,𝑢,𝑥   𝑛,𝑊   𝑧,𝑛,𝑡,𝑢,𝑊,𝑥
Allowed substitution hints:   ∌ (𝑥,𝑊,𝑧,𝑢,𝑡,𝑛)   𝐺(𝑥,𝑊,𝑧,𝑢,𝑡,𝑛)   𝑁(𝑊,𝑧)   𝑊(𝑥,𝑊,𝑧)

Proof of Theorem erclwwlkntr
Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3441 . 2 𝑥 ∈ V
2 vex 3441 . 2 𝑊 ∈ V
3 vex 3441 . 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))}
64, 5erclwwlkneqlen 28477 . . . . 5 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 → (♯‘𝑥) = (♯‘𝑊)))
763adant3 1132 . . . 4 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∌ 𝑊 → (♯‘𝑥) = (♯‘𝑊)))
84, 5erclwwlkneqlen 28477 . . . . . . 7 ((𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑊 ∌ 𝑧 → (♯‘𝑊) = (♯‘𝑧)))
983adant1 1130 . . . . . 6 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑊 ∌ 𝑧 → (♯‘𝑊) = (♯‘𝑧)))
104, 5erclwwlkneq 28476 . . . . . . . 8 ((𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑊 ∌ 𝑧 ↔ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))))
11103adant1 1130 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑊 ∌ 𝑧 ↔ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))))
124, 5erclwwlkneq 28476 . . . . . . . . . 10 ((𝑥 ∈ V ∧ 𝑊 ∈ V) → (𝑥 ∌ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))))
13123adant3 1132 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∌ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))))
14 simpr1 1194 . . . . . . . . . . . . . . 15 (((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))) → 𝑥 ∈ 𝑊)
15 simplr2 1216 . . . . . . . . . . . . . . 15 (((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))) → 𝑧 ∈ 𝑊)
16 oveq2 7315 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑚 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 𝑚))
1716eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑚 → (𝑥 = (𝑊 cyclShift 𝑛) ↔ 𝑥 = (𝑊 cyclShift 𝑚)))
1817cbvrexvw 3223 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) ↔ ∃𝑚 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑚))
19 oveq2 7315 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift 𝑘))
2019eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (𝑊 = (𝑧 cyclShift 𝑛) ↔ 𝑊 = (𝑧 cyclShift 𝑘)))
2120cbvrexvw 3223 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛) ↔ ∃𝑘 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑘))
22 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Vtx‘𝐺) = (Vtx‘𝐺)
2322clwwlknbp 28444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → (𝑧 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑧) = 𝑁))
24 eqcom 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((♯‘𝑧) = 𝑁 ↔ 𝑁 = (♯‘𝑧))
2524biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((♯‘𝑧) = 𝑁 → 𝑁 = (♯‘𝑧))
2623, 25simpl2im 505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 = (♯‘𝑧))
2726, 4eleq2s 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 ∈ 𝑊 → 𝑁 = (♯‘𝑧))
2827ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → 𝑁 = (♯‘𝑧))
2923simpld 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 ∈ (𝑁 ClWWalksN 𝐺) → 𝑧 ∈ Word (Vtx‘𝐺))
3029, 4eleq2s 2855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑧 ∈ 𝑊 → 𝑧 ∈ Word (Vtx‘𝐺))
3130ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → 𝑧 ∈ Word (Vtx‘𝐺))
3231adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → 𝑧 ∈ Word (Vtx‘𝐺))
33 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))
3432, 33cshwcsh2id 14586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → (((𝑚 ∈ (0...(♯‘𝑊)) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑊 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
35 oveq2 7315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑁 = (♯‘𝑧) → (0...𝑁) = (0...(♯‘𝑧)))
36 oveq2 7315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((♯‘𝑧) = (♯‘𝑊) → (0...(♯‘𝑧)) = (0...(♯‘𝑊)))
3736eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((♯‘𝑊) = (♯‘𝑧) → (0...(♯‘𝑧)) = (0...(♯‘𝑊)))
3837adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) → (0...(♯‘𝑧)) = (0...(♯‘𝑊)))
3938adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → (0...(♯‘𝑧)) = (0...(♯‘𝑊)))
4035, 39sylan9eq 2796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → (0...𝑁) = (0...(♯‘𝑊)))
4140eleq2d 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → (𝑚 ∈ (0...𝑁) ↔ 𝑚 ∈ (0...(♯‘𝑊))))
4241anbi1d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → ((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(♯‘𝑊)) ∧ 𝑥 = (𝑊 cyclShift 𝑚))))
4335eleq2d 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 = (♯‘𝑧) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑧))))
4443anbi1d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑁 = (♯‘𝑧) → ((𝑘 ∈ (0...𝑁) ∧ 𝑊 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑊 = (𝑧 cyclShift 𝑘))))
4544adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → ((𝑘 ∈ (0...𝑁) ∧ 𝑊 = (𝑧 cyclShift 𝑘)) ↔ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑊 = (𝑧 cyclShift 𝑘))))
4642, 45anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑊 = (𝑧 cyclShift 𝑘))) ↔ ((𝑚 ∈ (0...(♯‘𝑊)) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑊 = (𝑧 cyclShift 𝑘)))))
4735rexeqdv 3361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 = (♯‘𝑧) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4847adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
4934, 46, 483imtr4d 294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 = (♯‘𝑧) ∧ (((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑊 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5028, 49mpancom 686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → (((𝑚 ∈ (0...𝑁) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑊 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5150exp5l 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → (𝑚 ∈ (0...𝑁) → (𝑥 = (𝑊 cyclShift 𝑚) → (𝑘 ∈ (0...𝑁) → (𝑊 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
5251imp41 427 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑊 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5352rexlimdva 3149 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) ∧ 𝑚 ∈ (0...𝑁)) ∧ 𝑥 = (𝑊 cyclShift 𝑚)) → (∃𝑘 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
5453ex 414 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑥 = (𝑊 cyclShift 𝑚) → (∃𝑘 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5554rexlimdva 3149 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → (∃𝑚 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑚) → (∃𝑘 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑘) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5621, 55syl7bi 255 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → (∃𝑚 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑚) → (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5718, 56biimtrid 241 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) ∧ 𝑧 ∈ 𝑊) ∧ ((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
5857exp31 421 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → (𝑧 ∈ 𝑊 → (((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
5958com15 101 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛) → (𝑧 ∈ 𝑊 → (((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))))
6059impcom 409 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
61603adant1 1130 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → (((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))))
6261impcom 409 . . . . . . . . . . . . . . . . . 18 ((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
6362com13 88 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛) → ((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
64633impia 1117 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
6564impcom 409 . . . . . . . . . . . . . . 15 (((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))
6614, 15, 653jca 1128 . . . . . . . . . . . . . 14 (((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))) → (𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛)))
674, 5erclwwlkneq 28476 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∌ 𝑧 ↔ (𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
68673adant2 1131 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∌ 𝑧 ↔ (𝑥 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑧 cyclShift 𝑛))))
6966, 68syl5ibrcom 247 . . . . . . . . . . . . 13 (((((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) ∧ (𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛))) ∧ (𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛))) → ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∌ 𝑧))
7069exp31 421 . . . . . . . . . . . 12 (((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) → ((𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → 𝑥 ∌ 𝑧))))
7170com24 95 . . . . . . . . . . 11 (((♯‘𝑊) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑊)) → ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → 𝑥 ∌ 𝑧))))
7271ex 414 . . . . . . . . . 10 ((♯‘𝑊) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑊) → ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → 𝑥 ∌ 𝑧)))))
7372com4t 93 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ 𝑊 ∧ 𝑊 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑊 cyclShift 𝑛)) → ((♯‘𝑊) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑊) → ((𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → 𝑥 ∌ 𝑧)))))
7413, 73sylbid 239 . . . . . . . 8 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∌ 𝑊 → ((♯‘𝑊) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑊) → ((𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → 𝑥 ∌ 𝑧)))))
7574com25 99 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → ((𝑊 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑊 = (𝑧 cyclShift 𝑛)) → ((♯‘𝑊) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑊) → (𝑥 ∌ 𝑊 → 𝑥 ∌ 𝑧)))))
7611, 75sylbid 239 . . . . . 6 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑊 ∌ 𝑧 → ((♯‘𝑊) = (♯‘𝑧) → ((♯‘𝑥) = (♯‘𝑊) → (𝑥 ∌ 𝑊 → 𝑥 ∌ 𝑧)))))
779, 76mpdd 43 . . . . 5 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑊 ∌ 𝑧 → ((♯‘𝑥) = (♯‘𝑊) → (𝑥 ∌ 𝑊 → 𝑥 ∌ 𝑧))))
7877com24 95 . . . 4 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∌ 𝑊 → ((♯‘𝑥) = (♯‘𝑊) → (𝑊 ∌ 𝑧 → 𝑥 ∌ 𝑧))))
797, 78mpdd 43 . . 3 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ∌ 𝑊 → (𝑊 ∌ 𝑧 → 𝑥 ∌ 𝑧)))
8079impd 412 . 2 ((𝑥 ∈ V ∧ 𝑊 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∌ 𝑊 ∧ 𝑊 ∌ 𝑧) → 𝑥 ∌ 𝑧))
811, 2, 3, 80mp3an 1461 1 ((𝑥 ∌ 𝑊 ∧ 𝑊 ∌ 𝑧) → 𝑥 ∌ 𝑧)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  âˆƒwrex 3071  Vcvv 3437   class class class wbr 5081  {copab 5143  â€˜cfv 6458  (class class class)co 7307  0cc0 10917  ...cfz 13285  â™¯chash 14090  Word cword 14262   cyclShift ccsh 14546  Vtxcvtx 27411   ClWWalksN cclwwlkn 28433
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994  ax-pre-sup 10995
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-sup 9245  df-inf 9246  df-card 9741  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-div 11679  df-nn 12020  df-2 12082  df-n0 12280  df-z 12366  df-uz 12629  df-rp 12777  df-fz 13286  df-fzo 13429  df-fl 13558  df-mod 13636  df-hash 14091  df-word 14263  df-concat 14319  df-substr 14399  df-pfx 14429  df-csh 14547  df-clwwlk 28391  df-clwwlkn 28434
This theorem is referenced by:  erclwwlkn  28481
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