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Theorem erclwwlktr 29876
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 ∼ = {βŸ¨π‘’, π‘€βŸ© ∣ (𝑒 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑀 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlktr ((π‘₯ ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) β†’ π‘₯ ∼ 𝑧)
Distinct variable groups:   𝑛,𝐺,𝑒,𝑀   π‘₯,𝑛,𝑒,𝑀,𝑦   𝑧,𝑛,𝑒,𝑀,π‘₯
Allowed substitution hints:   ∼ (π‘₯,𝑦,𝑧,𝑀,𝑒,𝑛)   𝐺(π‘₯,𝑦,𝑧)

Proof of Theorem erclwwlktr
Dummy variables π‘š π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3467 . 2 π‘₯ ∈ V
2 vex 3467 . 2 𝑦 ∈ V
3 vex 3467 . 2 𝑧 ∈ V
4 erclwwlk.r . . . . . 6 ∼ = {βŸ¨π‘’, π‘€βŸ© ∣ (𝑒 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑀 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘€))𝑒 = (𝑀 cyclShift 𝑛))}
54erclwwlkeqlen 29873 . . . . 5 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)))
653adant3 1129 . . . 4 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)))
74erclwwlkeqlen 29873 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (𝑦 ∼ 𝑧 β†’ (β™―β€˜π‘¦) = (β™―β€˜π‘§)))
873adant1 1127 . . . . . 6 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (𝑦 ∼ 𝑧 β†’ (β™―β€˜π‘¦) = (β™―β€˜π‘§)))
94erclwwlkeq 29872 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))))
1093adant1 1127 . . . . . . 7 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (𝑦 ∼ 𝑧 ↔ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))))
114erclwwlkeq 29872 . . . . . . . . . 10 ((π‘₯ ∈ V ∧ 𝑦 ∈ V) β†’ (π‘₯ ∼ 𝑦 ↔ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))))
12113adant3 1129 . . . . . . . . 9 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (π‘₯ ∼ 𝑦 ↔ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))))
13 simpr1 1191 . . . . . . . . . . . . . . 15 (((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))) β†’ π‘₯ ∈ (ClWWalksβ€˜πΊ))
14 simplr2 1213 . . . . . . . . . . . . . . 15 (((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))) β†’ 𝑧 ∈ (ClWWalksβ€˜πΊ))
15 oveq2 7424 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = π‘š β†’ (𝑦 cyclShift 𝑛) = (𝑦 cyclShift π‘š))
1615eqeq2d 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = π‘š β†’ (π‘₯ = (𝑦 cyclShift 𝑛) ↔ π‘₯ = (𝑦 cyclShift π‘š)))
1716cbvrexvw 3226 . . . . . . . . . . . . . . . . . . . . . . . 24 (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) ↔ βˆƒπ‘š ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift π‘š))
18 oveq2 7424 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = π‘˜ β†’ (𝑧 cyclShift 𝑛) = (𝑧 cyclShift π‘˜))
1918eqeq2d 2736 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = π‘˜ β†’ (𝑦 = (𝑧 cyclShift 𝑛) ↔ 𝑦 = (𝑧 cyclShift π‘˜)))
2019cbvrexvw 3226 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛) ↔ βˆƒπ‘˜ ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift π‘˜))
21 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2221clwwlkbp 29839 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 ∈ (ClWWalksβ€˜πΊ) β†’ (𝐺 ∈ V ∧ 𝑧 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑧 β‰  βˆ…))
2322simp2d 1140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 ∈ (ClWWalksβ€˜πΊ) β†’ 𝑧 ∈ Word (Vtxβ€˜πΊ))
2423ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) β†’ 𝑧 ∈ Word (Vtxβ€˜πΊ))
25 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)))
2624, 25cshwcsh2id 14811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) β†’ (((π‘š ∈ (0...(β™―β€˜π‘¦)) ∧ π‘₯ = (𝑦 cyclShift π‘š)) ∧ (π‘˜ ∈ (0...(β™―β€˜π‘§)) ∧ 𝑦 = (𝑧 cyclShift π‘˜))) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛)))
2726exp5l 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) β†’ (π‘š ∈ (0...(β™―β€˜π‘¦)) β†’ (π‘₯ = (𝑦 cyclShift π‘š) β†’ (π‘˜ ∈ (0...(β™―β€˜π‘§)) β†’ (𝑦 = (𝑧 cyclShift π‘˜) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))))
2827imp41 424 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) ∧ π‘š ∈ (0...(β™―β€˜π‘¦))) ∧ π‘₯ = (𝑦 cyclShift π‘š)) ∧ π‘˜ ∈ (0...(β™―β€˜π‘§))) β†’ (𝑦 = (𝑧 cyclShift π‘˜) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛)))
2928rexlimdva 3145 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) ∧ π‘š ∈ (0...(β™―β€˜π‘¦))) ∧ π‘₯ = (𝑦 cyclShift π‘š)) β†’ (βˆƒπ‘˜ ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift π‘˜) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛)))
3029rexlimdva2 3147 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) β†’ (βˆƒπ‘š ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift π‘š) β†’ (βˆƒπ‘˜ ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift π‘˜) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))
3120, 30syl7bi 254 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) β†’ (βˆƒπ‘š ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift π‘š) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))
3217, 31biimtrid 241 . . . . . . . . . . . . . . . . . . . . . . 23 ((((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ)) ∧ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦))) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))
3332exp31 418 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ (𝑧 ∈ (ClWWalksβ€˜πΊ) β†’ (((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))))
3433com15 101 . . . . . . . . . . . . . . . . . . . . 21 (βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛) β†’ (𝑧 ∈ (ClWWalksβ€˜πΊ) β†’ (((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))))
3534impcom 406 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ (((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛)))))
36353adant1 1127 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ (((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛)))))
3736impcom 406 . . . . . . . . . . . . . . . . . 18 ((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))
3837com13 88 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ)) β†’ (βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛) β†’ ((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))
39383impia 1114 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) β†’ ((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛)))
4039impcom 406 . . . . . . . . . . . . . . 15 (((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))) β†’ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))
4113, 14, 403jca 1125 . . . . . . . . . . . . . 14 (((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))) β†’ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛)))
424erclwwlkeq 29872 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ V ∧ 𝑧 ∈ V) β†’ (π‘₯ ∼ 𝑧 ↔ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))
43423adant2 1128 . . . . . . . . . . . . . 14 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (π‘₯ ∼ 𝑧 ↔ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))π‘₯ = (𝑧 cyclShift 𝑛))))
4441, 43syl5ibrcom 246 . . . . . . . . . . . . 13 (((((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) ∧ (𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛))) ∧ (π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛))) β†’ ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ π‘₯ ∼ 𝑧))
4544exp31 418 . . . . . . . . . . . 12 (((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ ((𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) β†’ ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ π‘₯ ∼ 𝑧))))
4645com24 95 . . . . . . . . . . 11 (((β™―β€˜π‘¦) = (β™―β€˜π‘§) ∧ (β™―β€˜π‘₯) = (β™―β€˜π‘¦)) β†’ ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) β†’ ((𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ π‘₯ ∼ 𝑧))))
4746ex 411 . . . . . . . . . 10 ((β™―β€˜π‘¦) = (β™―β€˜π‘§) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) β†’ ((𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ π‘₯ ∼ 𝑧)))))
4847com4t 93 . . . . . . . . 9 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ ((π‘₯ ∈ (ClWWalksβ€˜πΊ) ∧ 𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘¦))π‘₯ = (𝑦 cyclShift 𝑛)) β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ ((𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ π‘₯ ∼ 𝑧)))))
4912, 48sylbid 239 . . . . . . . 8 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ ((𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ π‘₯ ∼ 𝑧)))))
5049com25 99 . . . . . . 7 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ ((𝑦 ∈ (ClWWalksβ€˜πΊ) ∧ 𝑧 ∈ (ClWWalksβ€˜πΊ) ∧ βˆƒπ‘› ∈ (0...(β™―β€˜π‘§))𝑦 = (𝑧 cyclShift 𝑛)) β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ (π‘₯ ∼ 𝑦 β†’ π‘₯ ∼ 𝑧)))))
5110, 50sylbid 239 . . . . . 6 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (𝑦 ∼ 𝑧 β†’ ((β™―β€˜π‘¦) = (β™―β€˜π‘§) β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ (π‘₯ ∼ 𝑦 β†’ π‘₯ ∼ 𝑧)))))
528, 51mpdd 43 . . . . 5 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (𝑦 ∼ 𝑧 β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ (π‘₯ ∼ 𝑦 β†’ π‘₯ ∼ 𝑧))))
5352com24 95 . . . 4 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ ((β™―β€˜π‘₯) = (β™―β€˜π‘¦) β†’ (𝑦 ∼ 𝑧 β†’ π‘₯ ∼ 𝑧))))
546, 53mpdd 43 . . 3 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ (π‘₯ ∼ 𝑦 β†’ (𝑦 ∼ 𝑧 β†’ π‘₯ ∼ 𝑧)))
5554impd 409 . 2 ((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ ((π‘₯ ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) β†’ π‘₯ ∼ 𝑧))
561, 2, 3, 55mp3an 1457 1 ((π‘₯ ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) β†’ π‘₯ ∼ 𝑧)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060  Vcvv 3463  βˆ…c0 4318   class class class wbr 5143  {copab 5205  β€˜cfv 6543  (class class class)co 7416  0cc0 11138  ...cfz 13516  β™―chash 14321  Word cword 14496   cyclShift ccsh 14770  Vtxcvtx 28853  ClWWalkscclwwlk 29835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-sup 9465  df-inf 9466  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-n0 12503  df-z 12589  df-uz 12853  df-rp 13007  df-fz 13517  df-fzo 13660  df-fl 13789  df-mod 13867  df-hash 14322  df-word 14497  df-concat 14553  df-substr 14623  df-pfx 14653  df-csh 14771  df-clwwlk 29836
This theorem is referenced by:  erclwwlk  29877
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