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Theorem pfxsuff1eqwrdeq 14594
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 14272 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ π‘Š β‰  βˆ…)
2 lennncl 14429 . . . . . 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 13671 . . . 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 14593 . . 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 14271 . . . . . . . . . . 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 14562 . . . . . . 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 5114 . . . . . . . . . 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 14271 . . . . . . . . . . . . 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 14562 . . . . . . . . . 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 7369 . . . . . . . . . . 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 4843 . . . . . . . . . 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 7378 . . . . . . . . 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 2739 . . . . . . . 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 2753 . . . . 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 6862 . . . . . 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 6860 . . . . . 6 (lastSβ€˜π‘ˆ) ∈ V
36 s111 14510 . . . . . 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 2944  Vcvv 3448  βˆ…c0 4287  βŸ¨cop 4597   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  1c1 11059   < clt 11196   βˆ’ cmin 11392  β„•cn 12160  ..^cfzo 13574  β™―chash 14237  Word cword 14409  lastSclsw 14457  βŸ¨β€œcs1 14490   substr csubstr 14535   prefix cpfx 14565
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458  df-s1 14491  df-substr 14536  df-pfx 14566
This theorem is referenced by:  wwlksnextinj  28886  clwwlkf1  29035
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