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Theorem 2clwwlk2clwwlklem 30143
Description: Lemma for 2clwwlk2clwwlk 30147. (Contributed by AV, 27-Apr-2022.)
Assertion
Ref Expression
2clwwlk2clwwlklem ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ (π‘Š substr ⟨(𝑁 βˆ’ 2), π‘βŸ©) ∈ (𝑋(ClWWalksNOnβ€˜πΊ)2))

Proof of Theorem 2clwwlk2clwwlklem
Dummy variables 𝑛 𝑣 𝑀 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isclwwlknon 29888 . . . . . . 7 (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋))
2 eqid 2727 . . . . . . . . . 10 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
32clwwlknbp 29832 . . . . . . . . 9 (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁))
4 simpll 766 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ))
5 uzuzle23 12895 . . . . . . . . . . . . . 14 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
6 eluzfz2 13533 . . . . . . . . . . . . . 14 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝑁 ∈ (2...𝑁))
75, 6syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ (2...𝑁))
87adantl 481 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑁 ∈ (2...𝑁))
9 oveq2 7422 . . . . . . . . . . . . . 14 ((β™―β€˜π‘Š) = 𝑁 β†’ (2...(β™―β€˜π‘Š)) = (2...𝑁))
109eleq2d 2814 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = 𝑁 β†’ (𝑁 ∈ (2...(β™―β€˜π‘Š)) ↔ 𝑁 ∈ (2...𝑁)))
1110ad2antlr 726 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑁 ∈ (2...(β™―β€˜π‘Š)) ↔ 𝑁 ∈ (2...𝑁)))
128, 11mpbird 257 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑁 ∈ (2...(β™―β€˜π‘Š)))
134, 12jca 511 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (2...(β™―β€˜π‘Š))))
1413ex 412 . . . . . . . . 9 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (2...(β™―β€˜π‘Š)))))
153, 14syl 17 . . . . . . . 8 (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (2...(β™―β€˜π‘Š)))))
1615adantr 480 . . . . . . 7 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (2...(β™―β€˜π‘Š)))))
171, 16sylbi 216 . . . . . 6 (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (2...(β™―β€˜π‘Š)))))
1817impcom 407 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁)) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (2...(β™―β€˜π‘Š))))
19 swrds2m 14916 . . . . 5 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ 𝑁 ∈ (2...(β™―β€˜π‘Š))) β†’ (π‘Š substr ⟨(𝑁 βˆ’ 2), π‘βŸ©) = βŸ¨β€œ(π‘Šβ€˜(𝑁 βˆ’ 2))(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
2018, 19syl 17 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁)) β†’ (π‘Š substr ⟨(𝑁 βˆ’ 2), π‘βŸ©) = βŸ¨β€œ(π‘Šβ€˜(𝑁 βˆ’ 2))(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
21203adant3 1130 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ (π‘Š substr ⟨(𝑁 βˆ’ 2), π‘βŸ©) = βŸ¨β€œ(π‘Šβ€˜(𝑁 βˆ’ 2))(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
22 simp3 1136 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0))
23 eqidd 2728 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
2422, 23s2eqd 14838 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ βŸ¨β€œ(π‘Šβ€˜(𝑁 βˆ’ 2))(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ© = βŸ¨β€œ(π‘Šβ€˜0)(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
25 simpr 484 . . . . . 6 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ (π‘Šβ€˜0) = 𝑋)
26 eqidd 2728 . . . . . 6 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
2725, 26s2eqd 14838 . . . . 5 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ βŸ¨β€œ(π‘Šβ€˜0)(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ© = βŸ¨β€œπ‘‹(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
281, 27sylbi 216 . . . 4 (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) β†’ βŸ¨β€œ(π‘Šβ€˜0)(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ© = βŸ¨β€œπ‘‹(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
29283ad2ant2 1132 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ βŸ¨β€œ(π‘Šβ€˜0)(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ© = βŸ¨β€œπ‘‹(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
3021, 24, 293eqtrd 2771 . 2 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ (π‘Š substr ⟨(𝑁 βˆ’ 2), π‘βŸ©) = βŸ¨β€œπ‘‹(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ©)
31 clwwlknonmpo 29886 . . . . 5 (ClWWalksNOnβ€˜πΊ) = (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣})
3231elmpocl1 7657 . . . 4 (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) β†’ 𝑋 ∈ (Vtxβ€˜πΊ))
33323ad2ant2 1132 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ 𝑋 ∈ (Vtxβ€˜πΊ))
34 eluzge3nn 12896 . . . . . . . . . . . . 13 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ β„•)
35 fzo0end 13748 . . . . . . . . . . . . 13 (𝑁 ∈ β„• β†’ (𝑁 βˆ’ 1) ∈ (0..^𝑁))
3634, 35syl 17 . . . . . . . . . . . 12 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 1) ∈ (0..^𝑁))
3736adantl 481 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑁 βˆ’ 1) ∈ (0..^𝑁))
38 oveq2 7422 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = 𝑁 β†’ (0..^(β™―β€˜π‘Š)) = (0..^𝑁))
3938ad2antlr 726 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (0..^(β™―β€˜π‘Š)) = (0..^𝑁))
4039eleq2d 2814 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑁 βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š)) ↔ (𝑁 βˆ’ 1) ∈ (0..^𝑁)))
4137, 40mpbird 257 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑁 βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š)))
42 wrdsymbcl 14501 . . . . . . . . . 10 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (𝑁 βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š))) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ))
434, 41, 42syl2anc 583 . . . . . . . . 9 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ))
4443ex 412 . . . . . . . 8 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ)))
453, 44syl 17 . . . . . . 7 (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ)))
4645adantr 480 . . . . . 6 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ)))
471, 46sylbi 216 . . . . 5 (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ)))
4847impcom 407 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁)) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ))
49483adant3 1130 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ))
50 preq1 4733 . . . . . . . . 9 ((π‘Šβ€˜0) = 𝑋 β†’ {(π‘Šβ€˜0), (π‘Šβ€˜(𝑁 βˆ’ 1))} = {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))})
5150adantl 481 . . . . . . . 8 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ {(π‘Šβ€˜0), (π‘Šβ€˜(𝑁 βˆ’ 1))} = {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))})
5251eqcomd 2733 . . . . . . 7 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))} = {(π‘Šβ€˜0), (π‘Šβ€˜(𝑁 βˆ’ 1))})
53523ad2ant2 1132 . . . . . 6 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))} = {(π‘Šβ€˜0), (π‘Šβ€˜(𝑁 βˆ’ 1))})
54 prcom 4732 . . . . . 6 {(π‘Šβ€˜0), (π‘Šβ€˜(𝑁 βˆ’ 1))} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)}
5553, 54eqtrdi 2783 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)})
56 eqid 2727 . . . . . . . . 9 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
572, 56clwwlknp 29834 . . . . . . . 8 (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
5857adantr 480 . . . . . . 7 ((π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) β†’ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
59583ad2ant2 1132 . . . . . 6 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
60 lsw 14538 . . . . . . . . . . . . . . 15 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
61 fvoveq1 7437 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘Š) = 𝑁 β†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
6260, 61sylan9eq 2787 . . . . . . . . . . . . . 14 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
6362adantr 480 . . . . . . . . . . . . 13 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ (𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0))) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
6463preq1d 4739 . . . . . . . . . . . 12 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ (𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0))) β†’ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} = {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)})
6564eleq1d 2813 . . . . . . . . . . 11 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ (𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0))) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
6665biimpd 228 . . . . . . . . . 10 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ (𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0))) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
6766ex 412 . . . . . . . . 9 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))))
6867com23 86 . . . . . . . 8 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) β†’ ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))))
6968a1d 25 . . . . . . 7 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) β†’ ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))))
70693imp 1109 . . . . . 6 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) β†’ ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
7159, 70mpcom 38 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {(π‘Šβ€˜(𝑁 βˆ’ 1)), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))
7255, 71eqeltrd 2828 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Šβ€˜0) = 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))} ∈ (Edgβ€˜πΊ))
731, 72syl3an2b 1402 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))} ∈ (Edgβ€˜πΊ))
74 eqid 2727 . . . 4 (ClWWalksNOnβ€˜πΊ) = (ClWWalksNOnβ€˜πΊ)
7574, 2, 56s2elclwwlknon2 29901 . . 3 ((𝑋 ∈ (Vtxβ€˜πΊ) ∧ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (Vtxβ€˜πΊ) ∧ {𝑋, (π‘Šβ€˜(𝑁 βˆ’ 1))} ∈ (Edgβ€˜πΊ)) β†’ βŸ¨β€œπ‘‹(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ© ∈ (𝑋(ClWWalksNOnβ€˜πΊ)2))
7633, 49, 73, 75syl3anc 1369 . 2 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ βŸ¨β€œπ‘‹(π‘Šβ€˜(𝑁 βˆ’ 1))β€βŸ© ∈ (𝑋(ClWWalksNOnβ€˜πΊ)2))
7730, 76eqeltrd 2828 1 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜0)) β†’ (π‘Š substr ⟨(𝑁 βˆ’ 2), π‘βŸ©) ∈ (𝑋(ClWWalksNOnβ€˜πΊ)2))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  {crab 3427  {cpr 4626  βŸ¨cop 4630  β€˜cfv 6542  (class class class)co 7414  0cc0 11130  1c1 11131   + caddc 11133   βˆ’ cmin 11466  β„•cn 12234  2c2 12289  3c3 12290  β„•0cn0 12494  β„€β‰₯cuz 12844  ...cfz 13508  ..^cfzo 13651  β™―chash 14313  Word cword 14488  lastSclsw 14536   substr csubstr 14614  βŸ¨β€œcs2 14816  Vtxcvtx 28796  Edgcedg 28847   ClWWalksN cclwwlkn 29821  ClWWalksNOncclwwlknon 29884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-s2 14823  df-clwwlk 29779  df-clwwlkn 29822  df-clwwlknon 29885
This theorem is referenced by:  2clwwlk2clwwlk  30147
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