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Theorem pfxsuff1eqwrdeq 14649
Description: Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by AV, 6-May-2020.)
Assertion
Ref Expression
pfxsuff1eqwrdeq ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
Proof of Theorem pfxsuff1eqwrdeq
StepHypRef Expression
1 hashgt0n0 14325 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ π‘Š β‰  βˆ…)
2 lennncl 14484 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…) β†’ (β™―β€˜π‘Š) ∈ β„•)
31, 2syldan 592 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
433adant2 1132 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
5 fzo0end 13724 . . . 4 ((β™―β€˜π‘Š) ∈ β„• β†’ ((β™―β€˜π‘Š) βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š)))
64, 5syl 17 . . 3 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š)))
7 pfxsuffeqwrdeq 14648 . . 3 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š))) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩)))))
86, 7syld3an3 1410 . 2 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩)))))
9 hashneq0 14324 . . . . . . . . . . 11 (π‘Š ∈ Word 𝑉 β†’ (0 < (β™―β€˜π‘Š) ↔ π‘Š β‰  βˆ…))
109biimpd 228 . . . . . . . . . 10 (π‘Š ∈ Word 𝑉 β†’ (0 < (β™―β€˜π‘Š) β†’ π‘Š β‰  βˆ…))
1110imdistani 570 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…))
12113adant2 1132 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…))
1312adantr 482 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…))
14 swrdlsw 14617 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ©)
1513, 14syl 17 . . . . . 6 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ©)
16 breq2 5153 . . . . . . . . . 10 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ (0 < (β™―β€˜π‘Š) ↔ 0 < (β™―β€˜π‘ˆ)))
17163anbi3d 1443 . . . . . . . . 9 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ↔ (π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘ˆ))))
18 hashneq0 14324 . . . . . . . . . . . . 13 (π‘ˆ ∈ Word 𝑉 β†’ (0 < (β™―β€˜π‘ˆ) ↔ π‘ˆ β‰  βˆ…))
1918biimpd 228 . . . . . . . . . . . 12 (π‘ˆ ∈ Word 𝑉 β†’ (0 < (β™―β€˜π‘ˆ) β†’ π‘ˆ β‰  βˆ…))
2019imdistani 570 . . . . . . . . . . 11 ((π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘ˆ)) β†’ (π‘ˆ ∈ Word 𝑉 ∧ π‘ˆ β‰  βˆ…))
21203adant1 1131 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘ˆ)) β†’ (π‘ˆ ∈ Word 𝑉 ∧ π‘ˆ β‰  βˆ…))
22 swrdlsw 14617 . . . . . . . . . 10 ((π‘ˆ ∈ Word 𝑉 ∧ π‘ˆ β‰  βˆ…) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)
2321, 22syl 17 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘ˆ)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)
2417, 23syl6bi 253 . . . . . . . 8 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©))
2524impcom 409 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)
26 oveq1 7416 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = ((β™―β€˜π‘ˆ) βˆ’ 1))
27 id 22 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ))
2826, 27opeq12d 4882 . . . . . . . . . 10 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩ = ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩)
2928oveq2d 7425 . . . . . . . . 9 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩))
3029eqeq1d 2735 . . . . . . . 8 ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) β†’ ((π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ© ↔ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©))
3130adantl 483 . . . . . . 7 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ© ↔ (π‘ˆ substr ⟨((β™―β€˜π‘ˆ) βˆ’ 1), (β™―β€˜π‘ˆ)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©))
3225, 31mpbird 257 . . . . . 6 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©)
3315, 32eqeq12d 2749 . . . . 5 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) ↔ βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ©))
34 fvexd 6907 . . . . . 6 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (lastSβ€˜π‘Š) ∈ V)
35 fvex 6905 . . . . . 6 (lastSβ€˜π‘ˆ) ∈ V
36 s111 14565 . . . . . 6 (((lastSβ€˜π‘Š) ∈ V ∧ (lastSβ€˜π‘ˆ) ∈ V) β†’ (βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ© ↔ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))
3734, 35, 36sylancl 587 . . . . 5 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(lastSβ€˜π‘ˆ)β€βŸ© ↔ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))
3833, 37bitrd 279 . . . 4 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ ((π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) ↔ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))
3938anbi2d 630 . . 3 (((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘ˆ)) β†’ (((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩)) ↔ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ))))
4039pm5.32da 580 . 2 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = (π‘ˆ substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩))) ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
418, 40bitrd 279 1 ((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (π‘Š = π‘ˆ ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘ˆ) ∧ ((π‘Š prefix ((β™―β€˜π‘Š) βˆ’ 1)) = (π‘ˆ prefix ((β™―β€˜π‘Š) βˆ’ 1)) ∧ (lastSβ€˜π‘Š) = (lastSβ€˜π‘ˆ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475  βˆ…c0 4323  βŸ¨cop 4635   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   < clt 11248   βˆ’ cmin 11444  β„•cn 12212  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512  βŸ¨β€œcs1 14545   substr csubstr 14590   prefix cpfx 14620
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-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-s1 14546  df-substr 14591  df-pfx 14621
This theorem is referenced by:  wwlksnextinj  29153  clwwlkf1  29302
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