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Theorem ccatopth 13516
Description: An opth 4974-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
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 6697 . . . 4 ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((𝐴 ++ 𝐵) substr ⟨0, (#‘𝐴)⟩) = ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐴)⟩))
2 swrdccat1 13503 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨0, (#‘𝐴)⟩) = 𝐴)
323ad2ant1 1102 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐴 ++ 𝐵) substr ⟨0, (#‘𝐴)⟩) = 𝐴)
4 simp3 1083 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → (#‘𝐴) = (#‘𝐶))
54opeq2d 4440 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘𝐶)⟩)
65oveq2d 6706 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐴)⟩) = ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐶)⟩))
7 swrdccat1 13503 . . . . . . 7 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐶)⟩) = 𝐶)
873ad2ant2 1103 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐶)⟩) = 𝐶)
96, 8eqtrd 2685 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐴)⟩) = 𝐶)
103, 9eqeq12d 2666 . . . 4 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → (((𝐴 ++ 𝐵) substr ⟨0, (#‘𝐴)⟩) = ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐴)⟩) ↔ 𝐴 = 𝐶))
111, 10syl5ib 234 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
12 simpr 476 . . . . . 6 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
13 simpl3 1086 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (#‘𝐴) = (#‘𝐶))
1412fveq2d 6233 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (#‘(𝐴 ++ 𝐵)) = (#‘(𝐶 ++ 𝐷)))
15 simpl1 1084 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋))
16 ccatlen 13393 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
1715, 16syl 17 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
18 simpl2 1085 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋))
19 ccatlen 13393 . . . . . . . . 9 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → (#‘(𝐶 ++ 𝐷)) = ((#‘𝐶) + (#‘𝐷)))
2018, 19syl 17 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (#‘(𝐶 ++ 𝐷)) = ((#‘𝐶) + (#‘𝐷)))
2114, 17, 203eqtr3d 2693 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐶) + (#‘𝐷)))
2213, 21opeq12d 4441 . . . . . 6 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ⟨(#‘𝐴), ((#‘𝐴) + (#‘𝐵))⟩ = ⟨(#‘𝐶), ((#‘𝐶) + (#‘𝐷))⟩)
2312, 22oveq12d 6708 . . . . 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))⟩))
24 swrdccat2 13504 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨(#‘𝐴), ((#‘𝐴) + (#‘𝐵))⟩) = 𝐵)
2515, 24syl 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)
26 swrdccat2 13504 . . . . . 6 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨(#‘𝐶), ((#‘𝐶) + (#‘𝐷))⟩) = 𝐷)
2718, 26syl 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)
2823, 25, 273eqtr3d 2693 . . . 4 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → 𝐵 = 𝐷)
2928ex 449 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐵 = 𝐷))
3011, 29jcad 554 . 2 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
31 oveq12 6699 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
3230, 31impbid1 215 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  cop 4216  cfv 5926  (class class class)co 6690  0cc0 9974   + caddc 9977  #chash 13157  Word cword 13323   ++ cconcat 13325   substr csubstr 13327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335
This theorem is referenced by:  ccatopth2  13517  ccatlcan  13518  splval2  13554  s2eq2s1eq  13727  s3eqs2s1eq  13729  efgredleme  18202  efgredlemc  18204
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