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Theorem pfxwlk 34738
Description: A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)
Assertion
Ref Expression
pfxwlk ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 prefix 𝐿)(Walksβ€˜πΊ)(𝑃 prefix (𝐿 + 1)))
Proof of Theorem pfxwlk
Dummy variables π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2727 . . . . 5 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
21wlkf 29446 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝐹 ∈ Word dom (iEdgβ€˜πΊ))
32adantr 479 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝐹 ∈ Word dom (iEdgβ€˜πΊ))
4 pfxcl 14665 . . 3 (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ (𝐹 prefix 𝐿) ∈ Word dom (iEdgβ€˜πΊ))
53, 4syl 17 . 2 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 prefix 𝐿) ∈ Word dom (iEdgβ€˜πΊ))
6 eqid 2727 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
76wlkp 29448 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
87adantr 479 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
9 elfzuz3 13536 . . . . . . 7 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜πΏ))
10 fzss2 13579 . . . . . . 7 ((β™―β€˜πΉ) ∈ (β„€β‰₯β€˜πΏ) β†’ (0...𝐿) βŠ† (0...(β™―β€˜πΉ)))
119, 10syl 17 . . . . . 6 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (0...𝐿) βŠ† (0...(β™―β€˜πΉ)))
1211adantl 480 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...𝐿) βŠ† (0...(β™―β€˜πΉ)))
138, 12fssresd 6767 . . . 4 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 β†Ύ (0...𝐿)):(0...𝐿)⟢(Vtxβ€˜πΊ))
14 pfxlen 14671 . . . . . . 7 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 prefix 𝐿)) = 𝐿)
152, 14sylan 578 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 prefix 𝐿)) = 𝐿)
1615oveq2d 7440 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...(β™―β€˜(𝐹 prefix 𝐿))) = (0...𝐿))
1716feq2d 6711 . . . 4 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ ((𝑃 β†Ύ (0...𝐿)):(0...(β™―β€˜(𝐹 prefix 𝐿)))⟢(Vtxβ€˜πΊ) ↔ (𝑃 β†Ύ (0...𝐿)):(0...𝐿)⟢(Vtxβ€˜πΊ)))
1813, 17mpbird 256 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 β†Ύ (0...𝐿)):(0...(β™―β€˜(𝐹 prefix 𝐿)))⟢(Vtxβ€˜πΊ))
196wlkpwrd 29449 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
20 fzp1elp1 13592 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (𝐿 + 1) ∈ (0...((β™―β€˜πΉ) + 1)))
2120adantl 480 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐿 + 1) ∈ (0...((β™―β€˜πΉ) + 1)))
22 wlklenvp1 29450 . . . . . . . . 9 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
2322oveq2d 7440 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (0...(β™―β€˜π‘ƒ)) = (0...((β™―β€˜πΉ) + 1)))
2423adantr 479 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...(β™―β€˜π‘ƒ)) = (0...((β™―β€˜πΉ) + 1)))
2521, 24eleqtrrd 2831 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐿 + 1) ∈ (0...(β™―β€˜π‘ƒ)))
26 pfxres 14667 . . . . . 6 ((𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ (𝐿 + 1) ∈ (0...(β™―β€˜π‘ƒ))) β†’ (𝑃 prefix (𝐿 + 1)) = (𝑃 β†Ύ (0..^(𝐿 + 1))))
2719, 25, 26syl2an2r 683 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 prefix (𝐿 + 1)) = (𝑃 β†Ύ (0..^(𝐿 + 1))))
28 elfzelz 13539 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ 𝐿 ∈ β„€)
29 fzval3 13739 . . . . . . . 8 (𝐿 ∈ β„€ β†’ (0...𝐿) = (0..^(𝐿 + 1)))
3028, 29syl 17 . . . . . . 7 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (0...𝐿) = (0..^(𝐿 + 1)))
3130adantl 480 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...𝐿) = (0..^(𝐿 + 1)))
3231reseq2d 5987 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 β†Ύ (0...𝐿)) = (𝑃 β†Ύ (0..^(𝐿 + 1))))
3327, 32eqtr4d 2770 . . . 4 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 prefix (𝐿 + 1)) = (𝑃 β†Ύ (0...𝐿)))
3433feq1d 6710 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ ((𝑃 prefix (𝐿 + 1)):(0...(β™―β€˜(𝐹 prefix 𝐿)))⟢(Vtxβ€˜πΊ) ↔ (𝑃 β†Ύ (0...𝐿)):(0...(β™―β€˜(𝐹 prefix 𝐿)))⟢(Vtxβ€˜πΊ)))
3518, 34mpbird 256 . 2 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 prefix (𝐿 + 1)):(0...(β™―β€˜(𝐹 prefix 𝐿)))⟢(Vtxβ€˜πΊ))
366, 1wlkprop 29443 . . . . . . 7 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)))))
3736simp3d 1141 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ βˆ€π‘₯ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯))))
3837adantr 479 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ βˆ€π‘₯ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯))))
3938adantr 479 . . . 4 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ βˆ€π‘₯ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯))))
4015oveq2d 7440 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^(β™―β€˜(𝐹 prefix 𝐿))) = (0..^𝐿))
4140eleq2d 2814 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿))) ↔ π‘˜ ∈ (0..^𝐿)))
4233fveq1d 6902 . . . . . . . . . . . 12 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 β†Ύ (0...𝐿))β€˜π‘˜))
4342adantr 479 . . . . . . . . . . 11 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 β†Ύ (0...𝐿))β€˜π‘˜))
44 fzossfz 13689 . . . . . . . . . . . . . 14 (0..^𝐿) βŠ† (0...𝐿)
4544a1i 11 . . . . . . . . . . . . 13 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^𝐿) βŠ† (0...𝐿))
4645sselda 3980 . . . . . . . . . . . 12 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ π‘˜ ∈ (0...𝐿))
4746fvresd 6920 . . . . . . . . . . 11 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((𝑃 β†Ύ (0...𝐿))β€˜π‘˜) = (π‘ƒβ€˜π‘˜))
4843, 47eqtr2d 2768 . . . . . . . . . 10 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ (π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜))
4933fveq1d 6902 . . . . . . . . . . . 12 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)) = ((𝑃 β†Ύ (0...𝐿))β€˜(π‘˜ + 1)))
5049adantr 479 . . . . . . . . . . 11 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)) = ((𝑃 β†Ύ (0...𝐿))β€˜(π‘˜ + 1)))
51 fzofzp1 13767 . . . . . . . . . . . . 13 (π‘˜ ∈ (0..^𝐿) β†’ (π‘˜ + 1) ∈ (0...𝐿))
5251adantl 480 . . . . . . . . . . . 12 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ (π‘˜ + 1) ∈ (0...𝐿))
5352fvresd 6920 . . . . . . . . . . 11 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((𝑃 β†Ύ (0...𝐿))β€˜(π‘˜ + 1)) = (π‘ƒβ€˜(π‘˜ + 1)))
5450, 53eqtr2d 2768 . . . . . . . . . 10 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)))
5548, 54jca 510 . . . . . . . . 9 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))))
5655ex 411 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘˜ ∈ (0..^𝐿) β†’ ((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)))))
5741, 56sylbid 239 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿))) β†’ ((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)))))
5857imp 405 . . . . . 6 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ ((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))))
593ancli 547 . . . . . . . . . . . 12 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)))
60 simpr 483 . . . . . . . . . . . . . 14 ((((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) ∧ π‘˜ ∈ (0..^𝐿)) β†’ π‘˜ ∈ (0..^𝐿))
6160fvresd 6920 . . . . . . . . . . . . 13 ((((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜) = (πΉβ€˜π‘˜))
6261fveq2d 6904 . . . . . . . . . . . 12 ((((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((iEdgβ€˜πΊ)β€˜((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)))
6359, 62sylan 578 . . . . . . . . . . 11 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((iEdgβ€˜πΊ)β€˜((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)))
6463eqcomd 2733 . . . . . . . . . 10 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜)))
6564ex 411 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘˜ ∈ (0..^𝐿) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜))))
6641, 65sylbid 239 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜))))
6766imp 405 . . . . . . 7 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜)))
68 simplr 767 . . . . . . . . . 10 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ 𝐿 ∈ (0...(β™―β€˜πΉ)))
69 pfxres 14667 . . . . . . . . . 10 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 prefix 𝐿) = (𝐹 β†Ύ (0..^𝐿)))
703, 68, 69syl2an2r 683 . . . . . . . . 9 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ (𝐹 prefix 𝐿) = (𝐹 β†Ύ (0..^𝐿)))
7170fveq1d 6902 . . . . . . . 8 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ ((𝐹 prefix 𝐿)β€˜π‘˜) = ((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜))
7271fveq2d 6904 . . . . . . 7 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 β†Ύ (0..^𝐿))β€˜π‘˜)))
7367, 72eqtr4d 2770 . . . . . 6 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)))
7458, 73jca 510 . . . . 5 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ (((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))))
759adantl 480 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜πΏ))
7615fveq2d 6904 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β„€β‰₯β€˜(β™―β€˜(𝐹 prefix 𝐿))) = (β„€β‰₯β€˜πΏ))
7775, 76eleqtrrd 2831 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜(β™―β€˜(𝐹 prefix 𝐿))))
78 fzoss2 13698 . . . . . . . 8 ((β™―β€˜πΉ) ∈ (β„€β‰₯β€˜(β™―β€˜(𝐹 prefix 𝐿))) β†’ (0..^(β™―β€˜(𝐹 prefix 𝐿))) βŠ† (0..^(β™―β€˜πΉ)))
7977, 78syl 17 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^(β™―β€˜(𝐹 prefix 𝐿))) βŠ† (0..^(β™―β€˜πΉ)))
8079sselda 3980 . . . . . 6 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ π‘˜ ∈ (0..^(β™―β€˜πΉ)))
81 wkslem1 29439 . . . . . . 7 (π‘₯ = π‘˜ β†’ (if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯))) ↔ if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)))))
8281rspcv 3605 . . . . . 6 (π‘˜ ∈ (0..^(β™―β€˜πΉ)) β†’ (βˆ€π‘₯ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯))) β†’ if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)))))
8380, 82syl 17 . . . . 5 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ (βˆ€π‘₯ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯))) β†’ if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)))))
84 eqeq12 2744 . . . . . . . 8 (((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) β†’ ((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)) ↔ ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))))
8584adantr 479 . . . . . . 7 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ ((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)) ↔ ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))))
86 simpr 483 . . . . . . . 8 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)))
87 sneq 4640 . . . . . . . . . 10 ((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) β†’ {(π‘ƒβ€˜π‘˜)} = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)})
8887adantr 479 . . . . . . . . 9 (((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) β†’ {(π‘ƒβ€˜π‘˜)} = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)})
8988adantr 479 . . . . . . . 8 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ {(π‘ƒβ€˜π‘˜)} = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)})
9086, 89eqeq12d 2743 . . . . . . 7 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)} ↔ ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}))
91 preq12 4742 . . . . . . . . 9 (((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) β†’ {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))})
9291adantr 479 . . . . . . . 8 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))})
9392, 86sseq12d 4013 . . . . . . 7 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ ({(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) ↔ {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))))
9485, 90, 93ifpbi123d 1076 . . . . . 6 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ (if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) ↔ if-(((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}, {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)))))
9594biimpd 228 . . . . 5 ((((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) ∧ (π‘ƒβ€˜(π‘˜ + 1)) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))) ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))) β†’ (if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜))) β†’ if-(((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}, {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)))))
9674, 83, 95sylsyld 61 . . . 4 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ (βˆ€π‘₯ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜(π‘₯ + 1)), ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯)) = {(π‘ƒβ€˜π‘₯)}, {(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘₯))) β†’ if-(((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}, {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)))))
9739, 96mpd 15 . . 3 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))) β†’ if-(((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}, {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))))
9897ralrimiva 3142 . 2 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))if-(((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}, {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))))
99 wlkv 29444 . . . . 5 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
10099simp1d 1139 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝐺 ∈ V)
101100adantr 479 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝐺 ∈ V)
1026, 1iswlkg 29445 . . 3 (𝐺 ∈ V β†’ ((𝐹 prefix 𝐿)(Walksβ€˜πΊ)(𝑃 prefix (𝐿 + 1)) ↔ ((𝐹 prefix 𝐿) ∈ Word dom (iEdgβ€˜πΊ) ∧ (𝑃 prefix (𝐿 + 1)):(0...(β™―β€˜(𝐹 prefix 𝐿)))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))if-(((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}, {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))))))
103101, 102syl 17 . 2 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ ((𝐹 prefix 𝐿)(Walksβ€˜πΊ)(𝑃 prefix (𝐿 + 1)) ↔ ((𝐹 prefix 𝐿) ∈ Word dom (iEdgβ€˜πΊ) ∧ (𝑃 prefix (𝐿 + 1)):(0...(β™―β€˜(𝐹 prefix 𝐿)))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿)))if-(((𝑃 prefix (𝐿 + 1))β€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1)), ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜)) = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)}, {((𝑃 prefix (𝐿 + 1))β€˜π‘˜), ((𝑃 prefix (𝐿 + 1))β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜πΊ)β€˜((𝐹 prefix 𝐿)β€˜π‘˜))))))
1045, 35, 98, 103mpbir3and 1339 1 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 prefix 𝐿)(Walksβ€˜πΊ)(𝑃 prefix (𝐿 + 1)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394  if-wif 1060   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3057  Vcvv 3471   βŠ† wss 3947  {csn 4630  {cpr 4632   class class class wbr 5150  dom cdm 5680   β†Ύ cres 5682  βŸΆwf 6547  β€˜cfv 6551  (class class class)co 7424  0cc0 11144  1c1 11145   + caddc 11147  β„€cz 12594  β„€β‰₯cuz 12858  ...cfz 13522  ..^cfzo 13665  β™―chash 14327  Word cword 14502   prefix cpfx 14658  Vtxcvtx 28827  iEdgciedg 28828  Walkscwlks 29428
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-er 8729  df-map 8851  df-pm 8852  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-card 9968  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-n0 12509  df-z 12595  df-uz 12859  df-fz 13523  df-fzo 13666  df-hash 14328  df-word 14503  df-substr 14629  df-pfx 14659  df-wlks 29431
This theorem is referenced by:  swrdwlk  34741
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