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Theorem ccatopth 13322
 Description: An opth 4871-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 6556 . . . 4 ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((𝐴 ++ 𝐵) substr ⟨0, (#‘𝐴)⟩) = ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐴)⟩))
2 swrdccat1 13309 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨0, (#‘𝐴)⟩) = 𝐴)
323ad2ant1 1075 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐴 ++ 𝐵) substr ⟨0, (#‘𝐴)⟩) = 𝐴)
4 simp3 1056 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → (#‘𝐴) = (#‘𝐶))
54opeq2d 4347 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘𝐶)⟩)
65oveq2d 6565 . . . . . 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 13309 . . . . . . 7 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐶)⟩) = 𝐶)
873ad2ant2 1076 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐶)⟩) = 𝐶)
96, 8eqtrd 2644 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (#‘𝐴)⟩) = 𝐶)
103, 9eqeq12d 2625 . . . 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 233 . . 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 1059 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (#‘𝐴) = (#‘𝐶))
1412fveq2d 6107 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (#‘(𝐴 ++ 𝐵)) = (#‘(𝐶 ++ 𝐷)))
15 simpl1 1057 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋))
16 ccatlen 13213 . . . . . . . . 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 1058 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋))
19 ccatlen 13213 . . . . . . . . 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 2652 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (#‘𝐴) = (#‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐶) + (#‘𝐷)))
2213, 21opeq12d 4348 . . . . . 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 6567 . . . . 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 13310 . . . . . 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 13310 . . . . . 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 2652 . . . 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 6558 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
3230, 31impbid1 214 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  0cc0 9815   + caddc 9818  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158 This theorem is referenced by:  ccatopth2  13323  ccatlcan  13324  splval2  13359  s2eq2s1eq  13531  efgredleme  17979  efgredlemc  17981
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