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Theorem ccatopth 14754
Description: An opth 5481-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)
Assertion
Ref Expression
ccatopth (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem ccatopth
StepHypRef Expression
1 oveq1 7438 . . . . 5 ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐴)))
2 pfxccat1 14740 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = 𝐴)
3 oveq2 7439 . . . . . . 7 ((♯‘𝐴) = (♯‘𝐶) → ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐶)))
4 pfxccat1 14740 . . . . . . 7 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) prefix (♯‘𝐶)) = 𝐶)
53, 4sylan9eqr 2799 . . . . . 6 (((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) = 𝐶)
62, 5eqeqan12d 2751 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶))) → (((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) ↔ 𝐴 = 𝐶))
71, 6imbitrid 244 . . . 4 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶))) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
873impb 1115 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
9 simpr 484 . . . . . 6 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
10 simpl3 1194 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘𝐴) = (♯‘𝐶))
119fveq2d 6910 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)))
12 simpl1 1192 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋))
13 ccatlen 14613 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
1412, 13syl 17 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
15 simpl2 1193 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋))
16 ccatlen 14613 . . . . . . . . 9 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
1715, 16syl 17 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
1811, 14, 173eqtr3d 2785 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐶) + (♯‘𝐷)))
1910, 18opeq12d 4881 . . . . . 6 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩ = ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩)
209, 19oveq12d 7449 . . . . 5 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩))
21 swrdccat2 14707 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = 𝐵)
2212, 21syl 17 . . . . 5 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = 𝐵)
23 swrdccat2 14707 . . . . . 6 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩) = 𝐷)
2415, 23syl 17 . . . . 5 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩) = 𝐷)
2520, 22, 243eqtr3d 2785 . . . 4 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → 𝐵 = 𝐷)
2625ex 412 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐵 = 𝐷))
278, 26jcad 512 . 2 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
28 oveq12 7440 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
2927, 28impbid1 225 1 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cop 4632  cfv 6561  (class class class)co 7431   + caddc 11158  chash 14369  Word cword 14552   ++ cconcat 14608   substr csubstr 14678   prefix cpfx 14708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-substr 14679  df-pfx 14709
This theorem is referenced by:  ccatopth2  14755  ccatlcan  14756  splval2  14795  s2eq2s1eq  14975  s3eqs2s1eq  14977  efgredleme  19761  efgredlemc  19763
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