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Theorem pfxwlk 34642
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 2726 . . . . 5 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
21wlkf 29376 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝐹 ∈ Word dom (iEdgβ€˜πΊ))
32adantr 480 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝐹 ∈ Word dom (iEdgβ€˜πΊ))
4 pfxcl 14631 . . 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 2726 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
76wlkp 29378 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
87adantr 480 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
9 elfzuz3 13501 . . . . . . 7 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜πΏ))
10 fzss2 13544 . . . . . . 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 481 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...𝐿) βŠ† (0...(β™―β€˜πΉ)))
138, 12fssresd 6751 . . . 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 14637 . . . . . . 7 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 prefix 𝐿)) = 𝐿)
152, 14sylan 579 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜(𝐹 prefix 𝐿)) = 𝐿)
1615oveq2d 7420 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...(β™―β€˜(𝐹 prefix 𝐿))) = (0...𝐿))
1716feq2d 6696 . . . 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 257 . . 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 29379 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
20 fzp1elp1 13557 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ (𝐿 + 1) ∈ (0...((β™―β€˜πΉ) + 1)))
2120adantl 481 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐿 + 1) ∈ (0...((β™―β€˜πΉ) + 1)))
22 wlklenvp1 29380 . . . . . . . . 9 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
2322oveq2d 7420 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (0...(β™―β€˜π‘ƒ)) = (0...((β™―β€˜πΉ) + 1)))
2423adantr 480 . . . . . . 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 2830 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐿 + 1) ∈ (0...(β™―β€˜π‘ƒ)))
26 pfxres 14633 . . . . . 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 682 . . . . 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 13504 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜πΉ)) β†’ 𝐿 ∈ β„€)
29 fzval3 13704 . . . . . . . 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 481 . . . . . 6 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0...𝐿) = (0..^(𝐿 + 1)))
3231reseq2d 5974 . . . . 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 2769 . . . 4 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝑃 prefix (𝐿 + 1)) = (𝑃 β†Ύ (0...𝐿)))
3433feq1d 6695 . . 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 257 . 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 29373 . . . . . . 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 480 . . . . 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 480 . . . 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 7420 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (0..^(β™―β€˜(𝐹 prefix 𝐿))) = (0..^𝐿))
4140eleq2d 2813 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (π‘˜ ∈ (0..^(β™―β€˜(𝐹 prefix 𝐿))) ↔ π‘˜ ∈ (0..^𝐿)))
4233fveq1d 6886 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 13654 . . . . . . . . . . . . . 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 3977 . . . . . . . . . . . 12 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ π‘˜ ∈ (0...𝐿))
4746fvresd 6904 . . . . . . . . . . 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 2767 . . . . . . . . . 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 6886 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 13732 . . . . . . . . . . . . 13 (π‘˜ ∈ (0..^𝐿) β†’ (π‘˜ + 1) ∈ (0...𝐿))
5251adantl 481 . . . . . . . . . . . 12 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) ∧ π‘˜ ∈ (0..^𝐿)) β†’ (π‘˜ + 1) ∈ (0...𝐿))
5352fvresd 6904 . . . . . . . . . . 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 2767 . . . . . . . . . 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 511 . . . . . . . . 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 412 . . . . . . . 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 406 . . . . . 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 548 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . 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 6888 . . . . . . . . . . . 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 579 . . . . . . . . . . 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 2732 . . . . . . . . . 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 412 . . . . . . . . 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 406 . . . . . . 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 766 . . . . . . . . . 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 14633 . . . . . . . . . 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 682 . . . . . . . . 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 6886 . . . . . . . 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 6888 . . . . . . 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 2769 . . . . . 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 511 . . . . 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 481 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜πΏ))
7615fveq2d 6888 . . . . . . . . 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 2830 . . . . . . . 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 13663 . . . . . . . 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 3977 . . . . . 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 29369 . . . . . . 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 3602 . . . . . 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 2743 . . . . . . . 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 480 . . . . . . 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 484 . . . . . . . 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 4633 . . . . . . . . . 10 ((π‘ƒβ€˜π‘˜) = ((𝑃 prefix (𝐿 + 1))β€˜π‘˜) β†’ {(π‘ƒβ€˜π‘˜)} = {((𝑃 prefix (𝐿 + 1))β€˜π‘˜)})
8887adantr 480 . . . . . . . . 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 480 . . . . . . . 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 2742 . . . . . . 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 4734 . . . . . . . . 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 480 . . . . . . . 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 4010 . . . . . . 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 3140 . 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 29374 . . . . 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 480 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ 𝐺 ∈ V)
1026, 1iswlkg 29375 . . 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 395  if-wif 1059   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βŠ† wss 3943  {csn 4623  {cpr 4625   class class class wbr 5141  dom cdm 5669   β†Ύ cres 5671  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  0cc0 11109  1c1 11110   + caddc 11112  β„€cz 12559  β„€β‰₯cuz 12823  ...cfz 13487  ..^cfzo 13630  β™―chash 14293  Word cword 14468   prefix cpfx 14624  Vtxcvtx 28760  iEdgciedg 28761  Walkscwlks 29358
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-fzo 13631  df-hash 14294  df-word 14469  df-substr 14595  df-pfx 14625  df-wlks 29361
This theorem is referenced by:  swrdwlk  34645
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