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Theorem pfxwlk 34102
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 2732 . . . . 5 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
21wlkf 28860 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝐹 ∈ Word dom (iEdgβ€˜πΊ))
32adantr 481 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝐹 ∈ Word dom (iEdgβ€˜πΊ))
4 pfxcl 14623 . . 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 2732 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
76wlkp 28862 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
87adantr 481 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
9 elfzuz3 13494 . . . . . . 7 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜πΏ))
10 fzss2 13537 . . . . . . 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 482 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...𝐿) βŠ† (0...(β™―β€˜πΉ)))
138, 12fssresd 6755 . . . 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 14629 . . . . . . 7 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 prefix 𝐿)) = 𝐿)
152, 14sylan 580 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 prefix 𝐿)) = 𝐿)
1615oveq2d 7421 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...(β™―β€˜(𝐹 prefix 𝐿))) = (0...𝐿))
1716feq2d 6700 . . . 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 28863 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
20 fzp1elp1 13550 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (𝐿 + 1) ∈ (0...((β™―β€˜πΉ) + 1)))
2120adantl 482 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐿 + 1) ∈ (0...((β™―β€˜πΉ) + 1)))
22 wlklenvp1 28864 . . . . . . . . 9 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
2322oveq2d 7421 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (0...(β™―β€˜π‘ƒ)) = (0...((β™―β€˜πΉ) + 1)))
2423adantr 481 . . . . . . 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 2836 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐿 + 1) ∈ (0...(β™―β€˜π‘ƒ)))
26 pfxres 14625 . . . . . 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 13497 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ 𝐿 ∈ β„€)
29 fzval3 13697 . . . . . . . 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 482 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...𝐿) = (0..^(𝐿 + 1)))
3231reseq2d 5979 . . . . 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 2775 . . . 4 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 prefix (𝐿 + 1)) = (𝑃 β†Ύ (0...𝐿)))
3433feq1d 6699 . . 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 28857 . . . . . . 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 1144 . . . . . 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 481 . . . . 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 481 . . . 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 7421 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^(β™―β€˜(𝐹 prefix 𝐿))) = (0..^𝐿))
4140eleq2d 2819 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿))) ↔ π‘˜ ∈ (0..^𝐿)))
4233fveq1d 6890 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 13647 . . . . . . . . . . . . . 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 3981 . . . . . . . . . . . 12 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ π‘˜ ∈ (0...𝐿))
4746fvresd 6908 . . . . . . . . . . 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 2773 . . . . . . . . . 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 6890 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 13725 . . . . . . . . . . . . 13 (π‘˜ ∈ (0..^𝐿) β†’ (π‘˜ + 1) ∈ (0...𝐿))
5251adantl 482 . . . . . . . . . . . 12 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ (π‘˜ + 1) ∈ (0...𝐿))
5352fvresd 6908 . . . . . . . . . . 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 2773 . . . . . . . . . 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 512 . . . . . . . . 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 413 . . . . . . . 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 407 . . . . . 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 549 . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 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 6908 . . . . . . . . . . . . 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 6892 . . . . . . . . . . . 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 580 . . . . . . . . . . 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 2738 . . . . . . . . . 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 413 . . . . . . . . 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 407 . . . . . . 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 14625 . . . . . . . . . 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 6890 . . . . . . . 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 6892 . . . . . . 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 2775 . . . . . 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 512 . . . . 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 482 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜πΏ))
7615fveq2d 6892 . . . . . . . . 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 2836 . . . . . . . 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 13656 . . . . . . . 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 3981 . . . . . 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 28853 . . . . . . 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 3608 . . . . . 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 2749 . . . . . . . 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 481 . . . . . . 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 485 . . . . . . . 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 4637 . . . . . . . . . 10 ((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) β†’ {(π‘ƒβ€˜π‘˜)} = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)})
8887adantr 481 . . . . . . . . 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 481 . . . . . . . 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 2748 . . . . . . 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 4738 . . . . . . . . 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 481 . . . . . . . 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 4014 . . . . . . 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 1078 . . . . . 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 3146 . 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 28858 . . . . 5 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
10099simp1d 1142 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝐺 ∈ V)
101100adantr 481 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝐺 ∈ V)
1026, 1iswlkg 28859 . . 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 1342 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 396  if-wif 1061   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βŠ† wss 3947  {csn 4627  {cpr 4629   class class class wbr 5147  dom cdm 5675   β†Ύ cres 5677  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107   + caddc 11109  β„€cz 12554  β„€β‰₯cuz 12818  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460   prefix cpfx 14616  Vtxcvtx 28245  iEdgciedg 28246  Walkscwlks 28842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-substr 14587  df-pfx 14617  df-wlks 28845
This theorem is referenced by:  swrdwlk  34105
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