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Theorem swrdccatid 13543
Description: A prefix of a concatenation of length of the first concatenated word is the first word itself. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
Assertion
Ref Expression
swrdccatid ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)

Proof of Theorem swrdccatid
StepHypRef Expression
1 3simpa 1078 . . 3 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 lencl 13356 . . . . 5 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
3 lencl 13356 . . . . . 6 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
4 simplr 807 . . . . . . . . 9 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ∈ ℕ0)
5 eleq1 2718 . . . . . . . . . 10 (𝑁 = (#‘𝐴) → (𝑁 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0))
65adantl 481 . . . . . . . . 9 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (𝑁 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0))
74, 6mpbird 247 . . . . . . . 8 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ ℕ0)
8 nn0addcl 11366 . . . . . . . . . 10 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
98ancoms 468 . . . . . . . . 9 (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
109adantr 480 . . . . . . . 8 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0)
11 nn0re 11339 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ)
1211anim1i 591 . . . . . . . . . . . 12 (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0) → ((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0))
1312ancoms 468 . . . . . . . . . . 11 (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → ((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0))
14 nn0addge1 11377 . . . . . . . . . . 11 (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℕ0) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
1513, 14syl 17 . . . . . . . . . 10 (((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
1615adantr 480 . . . . . . . . 9 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵)))
17 breq1 4688 . . . . . . . . . 10 (𝑁 = (#‘𝐴) → (𝑁 ≤ ((#‘𝐴) + (#‘𝐵)) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵))))
1817adantl 481 . . . . . . . . 9 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → (𝑁 ≤ ((#‘𝐴) + (#‘𝐵)) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝐵))))
1916, 18mpbird 247 . . . . . . . 8 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ≤ ((#‘𝐴) + (#‘𝐵)))
20 elfz2nn0 12469 . . . . . . . 8 (𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℕ0𝑁 ≤ ((#‘𝐴) + (#‘𝐵))))
217, 10, 19, 20syl3anbrc 1265 . . . . . . 7 ((((#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁 = (#‘𝐴)) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))
2221exp31 629 . . . . . 6 ((#‘𝐵) ∈ ℕ0 → ((#‘𝐴) ∈ ℕ0 → (𝑁 = (#‘𝐴) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
233, 22syl 17 . . . . 5 (𝐵 ∈ Word 𝑉 → ((#‘𝐴) ∈ ℕ0 → (𝑁 = (#‘𝐴) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
242, 23syl5com 31 . . . 4 (𝐴 ∈ Word 𝑉 → (𝐵 ∈ Word 𝑉 → (𝑁 = (#‘𝐴) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))))
25243imp 1275 . . 3 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → 𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))))
26 eqid 2651 . . . 4 (#‘𝐴) = (#‘𝐴)
2726swrdccat3a 13540 . . 3 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...((#‘𝐴) + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ (#‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (#‘𝐴))⟩)))))
281, 25, 27sylc 65 . 2 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ (#‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (#‘𝐴))⟩))))
292, 11syl 17 . . . . . 6 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℝ)
3029leidd 10632 . . . . 5 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ≤ (#‘𝐴))
31303ad2ant1 1102 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → (#‘𝐴) ≤ (#‘𝐴))
32 breq1 4688 . . . . 5 (𝑁 = (#‘𝐴) → (𝑁 ≤ (#‘𝐴) ↔ (#‘𝐴) ≤ (#‘𝐴)))
33323ad2ant3 1104 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → (𝑁 ≤ (#‘𝐴) ↔ (#‘𝐴) ≤ (#‘𝐴)))
3431, 33mpbird 247 . . 3 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → 𝑁 ≤ (#‘𝐴))
3534iftrued 4127 . 2 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → if(𝑁 ≤ (#‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (#‘𝐴))⟩))) = (𝐴 substr ⟨0, 𝑁⟩))
36 swrdid 13474 . . . 4 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
37363ad2ant1 1102 . . 3 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴)
38 opeq2 4434 . . . . . 6 (𝑁 = (#‘𝐴) → ⟨0, 𝑁⟩ = ⟨0, (#‘𝐴)⟩)
3938oveq2d 6706 . . . . 5 (𝑁 = (#‘𝐴) → (𝐴 substr ⟨0, 𝑁⟩) = (𝐴 substr ⟨0, (#‘𝐴)⟩))
4039eqeq1d 2653 . . . 4 (𝑁 = (#‘𝐴) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴))
41403ad2ant3 1104 . . 3 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (#‘𝐴)⟩) = 𝐴))
4237, 41mpbird 247 . 2 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → (𝐴 substr ⟨0, 𝑁⟩) = 𝐴)
4328, 35, 423eqtrd 2689 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  ifcif 4119  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974   + caddc 9977  cle 10113  cmin 10304  0cn0 11330  ...cfz 12364  #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:  ccats1swrdeqbi  13544  clwlkclwwlk2  26969  clwlksfoclwwlk  27050  numclwlk1lem2foalem  27341  numclwlk1lem2fo  27348
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