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Theorem revpfxsfxrev 34106
Description: The reverse of a prefix of a word is equal to the same-length suffix of the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.)
Assertion
Ref Expression
revpfxsfxrev ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (reverseβ€˜(π‘Š prefix 𝐿)) = ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))

Proof of Theorem revpfxsfxrev
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 pfxcl 14627 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (π‘Š prefix 𝐿) ∈ Word 𝑉)
2 revcl 14711 . . . . 5 ((π‘Š prefix 𝐿) ∈ Word 𝑉 β†’ (reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉)
3 wrdfn 14478 . . . . 5 ((reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉 β†’ (reverseβ€˜(π‘Š prefix 𝐿)) Fn (0..^(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))))
41, 2, 33syl 18 . . . 4 (π‘Š ∈ Word 𝑉 β†’ (reverseβ€˜(π‘Š prefix 𝐿)) Fn (0..^(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))))
54adantr 482 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (reverseβ€˜(π‘Š prefix 𝐿)) Fn (0..^(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))))
6 revlen 14712 . . . . . . . 8 ((π‘Š prefix 𝐿) ∈ Word 𝑉 β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (β™―β€˜(π‘Š prefix 𝐿)))
71, 6syl 17 . . . . . . 7 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (β™―β€˜(π‘Š prefix 𝐿)))
87adantr 482 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (β™―β€˜(π‘Š prefix 𝐿)))
9 pfxlen 14633 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(π‘Š prefix 𝐿)) = 𝐿)
108, 9eqtrd 2773 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = 𝐿)
1110oveq2d 7425 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (0..^(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))) = (0..^𝐿))
1211fneq2d 6644 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((reverseβ€˜(π‘Š prefix 𝐿)) Fn (0..^(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))) ↔ (reverseβ€˜(π‘Š prefix 𝐿)) Fn (0..^𝐿)))
135, 12mpbid 231 . 2 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (reverseβ€˜(π‘Š prefix 𝐿)) Fn (0..^𝐿))
14 revcl 14711 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (reverseβ€˜π‘Š) ∈ Word 𝑉)
15 swrdcl 14595 . . . . 5 ((reverseβ€˜π‘Š) ∈ Word 𝑉 β†’ ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) ∈ Word 𝑉)
16 wrdfn 14478 . . . . 5 (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) ∈ Word 𝑉 β†’ ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) Fn (0..^(β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))))
1714, 15, 163syl 18 . . . 4 (π‘Š ∈ Word 𝑉 β†’ ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) Fn (0..^(β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))))
1817adantr 482 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) Fn (0..^(β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))))
19 fznn0sub2 13608 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)))
20 lencl 14483 . . . . . . . . . 10 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
21 nn0fz0 13599 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 ↔ (β™―β€˜π‘Š) ∈ (0...(β™―β€˜π‘Š)))
2220, 21sylib 217 . . . . . . . . 9 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ (0...(β™―β€˜π‘Š)))
23 revlen 14712 . . . . . . . . . 10 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜(reverseβ€˜π‘Š)) = (β™―β€˜π‘Š))
2423oveq2d 7425 . . . . . . . . 9 (π‘Š ∈ Word 𝑉 β†’ (0...(β™―β€˜(reverseβ€˜π‘Š))) = (0...(β™―β€˜π‘Š)))
2522, 24eleqtrrd 2837 . . . . . . . 8 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ (0...(β™―β€˜(reverseβ€˜π‘Š))))
26 swrdlen 14597 . . . . . . . 8 (((reverseβ€˜π‘Š) ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) ∈ (0...(β™―β€˜(reverseβ€˜π‘Š)))) β†’ (β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)) = ((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))
2714, 19, 25, 26syl3an 1161 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘Š ∈ Word 𝑉) β†’ (β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)) = ((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))
28273anidm13 1421 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)) = ((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))
2920nn0cnd 12534 . . . . . . . 8 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„‚)
3029adantr 482 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜π‘Š) ∈ β„‚)
31 elfzelz 13501 . . . . . . . . 9 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ 𝐿 ∈ β„€)
3231zcnd 12667 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ 𝐿 ∈ β„‚)
3332adantl 483 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ 𝐿 ∈ β„‚)
3430, 33nncand 11576 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) = 𝐿)
3528, 34eqtrd 2773 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)) = 𝐿)
3635oveq2d 7425 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (0..^(β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))) = (0..^𝐿))
3736fneq2d 6644 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) Fn (0..^(β™―β€˜((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))) ↔ ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) Fn (0..^𝐿)))
3818, 37mpbid 231 . 2 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩) Fn (0..^𝐿))
39 simp1 1137 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ π‘Š ∈ Word 𝑉)
40 simp3 1139 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ π‘₯ ∈ (0..^𝐿))
419oveq2d 7425 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (0..^(β™―β€˜(π‘Š prefix 𝐿))) = (0..^𝐿))
42413adant3 1133 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (0..^(β™―β€˜(π‘Š prefix 𝐿))) = (0..^𝐿))
4340, 42eleqtrrd 2837 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ π‘₯ ∈ (0..^(β™―β€˜(π‘Š prefix 𝐿))))
44 revfv 14713 . . . . . . 7 (((π‘Š prefix 𝐿) ∈ Word 𝑉 ∧ π‘₯ ∈ (0..^(β™―β€˜(π‘Š prefix 𝐿)))) β†’ ((reverseβ€˜(π‘Š prefix 𝐿))β€˜π‘₯) = ((π‘Š prefix 𝐿)β€˜(((β™―β€˜(π‘Š prefix 𝐿)) βˆ’ 1) βˆ’ π‘₯)))
451, 44sylan 581 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ π‘₯ ∈ (0..^(β™―β€˜(π‘Š prefix 𝐿)))) β†’ ((reverseβ€˜(π‘Š prefix 𝐿))β€˜π‘₯) = ((π‘Š prefix 𝐿)β€˜(((β™―β€˜(π‘Š prefix 𝐿)) βˆ’ 1) βˆ’ π‘₯)))
4639, 43, 45syl2anc 585 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((reverseβ€˜(π‘Š prefix 𝐿))β€˜π‘₯) = ((π‘Š prefix 𝐿)β€˜(((β™―β€˜(π‘Š prefix 𝐿)) βˆ’ 1) βˆ’ π‘₯)))
479oveq1d 7424 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((β™―β€˜(π‘Š prefix 𝐿)) βˆ’ 1) = (𝐿 βˆ’ 1))
4847oveq1d 7424 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (((β™―β€˜(π‘Š prefix 𝐿)) βˆ’ 1) βˆ’ π‘₯) = ((𝐿 βˆ’ 1) βˆ’ π‘₯))
4948fveq2d 6896 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝐿)β€˜(((β™―β€˜(π‘Š prefix 𝐿)) βˆ’ 1) βˆ’ π‘₯)) = ((π‘Š prefix 𝐿)β€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)))
50493adant3 1133 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((π‘Š prefix 𝐿)β€˜(((β™―β€˜(π‘Š prefix 𝐿)) βˆ’ 1) βˆ’ π‘₯)) = ((π‘Š prefix 𝐿)β€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)))
51323ad2ant2 1135 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ 𝐿 ∈ β„‚)
52 elfzoelz 13632 . . . . . . . . . 10 (π‘₯ ∈ (0..^𝐿) β†’ π‘₯ ∈ β„€)
5352zcnd 12667 . . . . . . . . 9 (π‘₯ ∈ (0..^𝐿) β†’ π‘₯ ∈ β„‚)
54533ad2ant3 1136 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ π‘₯ ∈ β„‚)
55 1cnd 11209 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ 1 ∈ β„‚)
5651, 54, 55sub32d 11603 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((𝐿 βˆ’ π‘₯) βˆ’ 1) = ((𝐿 βˆ’ 1) βˆ’ π‘₯))
57 ubmelm1fzo 13728 . . . . . . . 8 (π‘₯ ∈ (0..^𝐿) β†’ ((𝐿 βˆ’ π‘₯) βˆ’ 1) ∈ (0..^𝐿))
58573ad2ant3 1136 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((𝐿 βˆ’ π‘₯) βˆ’ 1) ∈ (0..^𝐿))
5956, 58eqeltrrd 2835 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((𝐿 βˆ’ 1) βˆ’ π‘₯) ∈ (0..^𝐿))
60 pfxfv 14632 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ ((𝐿 βˆ’ 1) βˆ’ π‘₯) ∈ (0..^𝐿)) β†’ ((π‘Š prefix 𝐿)β€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)) = (π‘Šβ€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)))
6159, 60syld3an3 1410 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((π‘Š prefix 𝐿)β€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)) = (π‘Šβ€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)))
6246, 50, 613eqtrd 2777 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((reverseβ€˜(π‘Š prefix 𝐿))β€˜π‘₯) = (π‘Šβ€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)))
6334oveq2d 7425 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿))) = (0..^𝐿))
6463eleq2d 2820 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿))) ↔ π‘₯ ∈ (0..^𝐿)))
6564biimp3ar 1471 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ π‘₯ ∈ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿))))
66 id 22 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ π‘Š ∈ Word 𝑉) β†’ (π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ π‘Š ∈ Word 𝑉))
67663anidm13 1421 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š))) β†’ (π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ π‘Š ∈ Word 𝑉))
68 swrdfv 14598 . . . . . . . . . 10 ((((reverseβ€˜π‘Š) ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) ∈ (0...(β™―β€˜(reverseβ€˜π‘Š)))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯) = ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
6914, 68syl3anl1 1413 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) ∈ (0...(β™―β€˜(reverseβ€˜π‘Š)))) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯) = ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
7025, 69syl3anl3 1415 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ π‘Š ∈ Word 𝑉) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯) = ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
7167, 70stoic3 1779 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯) = ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
7219, 71syl3an2 1165 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)))) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯) = ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
7365, 72syld3an3 1410 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯) = ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
74 0z 12569 . . . . . . . . . 10 0 ∈ β„€
75 elfzuz3 13498 . . . . . . . . . . 11 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜πΏ))
7632addlidd 11415 . . . . . . . . . . . 12 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ (0 + 𝐿) = 𝐿)
7776fveq2d 6896 . . . . . . . . . . 11 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ (β„€β‰₯β€˜(0 + 𝐿)) = (β„€β‰₯β€˜πΏ))
7875, 77eleqtrrd 2837 . . . . . . . . . 10 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜(0 + 𝐿)))
79 eluzsub 12852 . . . . . . . . . 10 ((0 ∈ β„€ ∧ 𝐿 ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ (β„€β‰₯β€˜(0 + 𝐿))) β†’ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (β„€β‰₯β€˜0))
8074, 31, 78, 79mp3an2i 1467 . . . . . . . . 9 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (β„€β‰₯β€˜0))
81 fzoss1 13659 . . . . . . . . 9 (((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ (β„€β‰₯β€˜0) β†’ (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(β™―β€˜π‘Š)) βŠ† (0..^(β™―β€˜π‘Š)))
8280, 81syl 17 . . . . . . . 8 (𝐿 ∈ (0...(β™―β€˜π‘Š)) β†’ (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(β™―β€˜π‘Š)) βŠ† (0..^(β™―β€˜π‘Š)))
83823ad2ant2 1135 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(β™―β€˜π‘Š)) βŠ† (0..^(β™―β€˜π‘Š)))
8420nn0zd 12584 . . . . . . . . . . 11 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„€)
85843ad2ant1 1134 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (β™―β€˜π‘Š) ∈ β„€)
86313ad2ant2 1135 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ 𝐿 ∈ β„€)
8785, 86zsubcld 12671 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ β„€)
88 fzo0addel 13686 . . . . . . . . 9 ((π‘₯ ∈ (0..^𝐿) ∧ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ β„€) β†’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)) ∈ (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(𝐿 + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
8940, 87, 88syl2anc 585 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)) ∈ (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(𝐿 + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
90303adant3 1133 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (β™―β€˜π‘Š) ∈ β„‚)
9151, 90pncan3d 11574 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (𝐿 + ((β™―β€˜π‘Š) βˆ’ 𝐿)) = (β™―β€˜π‘Š))
9291oveq2d 7425 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(𝐿 + ((β™―β€˜π‘Š) βˆ’ 𝐿))) = (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(β™―β€˜π‘Š)))
9389, 92eleqtrd 2836 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)) ∈ (((β™―β€˜π‘Š) βˆ’ 𝐿)..^(β™―β€˜π‘Š)))
9483, 93sseldd 3984 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)) ∈ (0..^(β™―β€˜π‘Š)))
95 revfv 14713 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)) ∈ (0..^(β™―β€˜π‘Š))) β†’ ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)))))
9639, 94, 95syl2anc 585 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((reverseβ€˜π‘Š)β€˜(π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)))))
9790, 55subcld 11571 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((β™―β€˜π‘Š) βˆ’ 1) ∈ β„‚)
9887zcnd 12667 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((β™―β€˜π‘Š) βˆ’ 𝐿) ∈ β„‚)
9997, 54, 98sub32d 11603 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) = ((((β™―β€˜π‘Š) βˆ’ 1) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) βˆ’ π‘₯))
10097, 54, 98subsub4d 11602 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((((β™―β€˜π‘Š) βˆ’ 1) βˆ’ π‘₯) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) = (((β™―β€˜π‘Š) βˆ’ 1) βˆ’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))))
10190, 55, 98sub32d 11603 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((β™―β€˜π‘Š) βˆ’ 1) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) = (((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) βˆ’ 1))
102101oveq1d 7424 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((((β™―β€˜π‘Š) βˆ’ 1) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) βˆ’ π‘₯) = ((((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) βˆ’ 1) βˆ’ π‘₯))
10399, 100, 1023eqtr3d 2781 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((β™―β€˜π‘Š) βˆ’ 1) βˆ’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))) = ((((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) βˆ’ 1) βˆ’ π‘₯))
104343adant3 1133 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) = 𝐿)
105104oveq1d 7424 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) βˆ’ 1) = (𝐿 βˆ’ 1))
106105oveq1d 7424 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((((β™―β€˜π‘Š) βˆ’ ((β™―β€˜π‘Š) βˆ’ 𝐿)) βˆ’ 1) βˆ’ π‘₯) = ((𝐿 βˆ’ 1) βˆ’ π‘₯))
107103, 106eqtrd 2773 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((β™―β€˜π‘Š) βˆ’ 1) βˆ’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿))) = ((𝐿 βˆ’ 1) βˆ’ π‘₯))
108107fveq2d 6896 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 1) βˆ’ (π‘₯ + ((β™―β€˜π‘Š) βˆ’ 𝐿)))) = (π‘Šβ€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)))
10973, 96, 1083eqtrd 2777 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯) = (π‘Šβ€˜((𝐿 βˆ’ 1) βˆ’ π‘₯)))
11062, 109eqtr4d 2776 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((reverseβ€˜(π‘Š prefix 𝐿))β€˜π‘₯) = (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯))
1111103expa 1119 . 2 (((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) ∧ π‘₯ ∈ (0..^𝐿)) β†’ ((reverseβ€˜(π‘Š prefix 𝐿))β€˜π‘₯) = (((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩)β€˜π‘₯))
11213, 38, 111eqfnfvd 7036 1 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (reverseβ€˜(π‘Š prefix 𝐿)) = ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  βŸ¨cop 4635   Fn wfn 6539  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  0cc0 11110  1c1 11111   + caddc 11113   βˆ’ cmin 11444  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464   substr csubstr 14590   prefix cpfx 14620  reversecreverse 14708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-substr 14591  df-pfx 14621  df-reverse 14709
This theorem is referenced by:  swrdrevpfx  34107
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