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Theorem swrdccat3b 13542
Description: A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l 𝐿 = (#‘𝐴)
Assertion
Ref Expression
swrdccat3b ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))))

Proof of Theorem swrdccat3b
StepHypRef Expression
1 simpl 472 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 simpr 476 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → 𝑀 ∈ (0...(𝐿 + (#‘𝐵))))
3 elfzubelfz 12391 . . . . 5 (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))
43adantl 481 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))
5 swrdccatin12.l . . . . . 6 𝐿 = (#‘𝐴)
65swrdccat3 13538 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩))))))
76imp 444 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))))
81, 2, 4, 7syl12anc 1364 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))))
95swrdccat3blem 13541 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ (𝐿 + (#‘𝐵)) ≤ 𝐿) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))
10 iftrue 4125 . . . . . 6 (𝐿𝑀 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩))
11103ad2ant3 1104 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿𝐿𝑀) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩))
12 lencl 13356 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
1312nn0cnd 11391 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℂ)
14 lencl 13356 . . . . . . . . . . . 12 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
1514nn0cnd 11391 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℂ)
165eqcomi 2660 . . . . . . . . . . . . 13 (#‘𝐴) = 𝐿
1716eleq1i 2721 . . . . . . . . . . . 12 ((#‘𝐴) ∈ ℂ ↔ 𝐿 ∈ ℂ)
18 pncan2 10326 . . . . . . . . . . . 12 ((𝐿 ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
1917, 18sylanb 488 . . . . . . . . . . 11 (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
2013, 15, 19syl2an 493 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
2120eqcomd 2657 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
2221adantr 480 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
23223ad2ant1 1102 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿𝐿𝑀) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
2423opeq2d 4440 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿𝐿𝑀) → ⟨(𝑀𝐿), (#‘𝐵)⟩ = ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩)
2524oveq2d 6706 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿𝐿𝑀) → (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩) = (𝐵 substr ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩))
2611, 25eqtrd 2685 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿𝐿𝑀) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐵 substr ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩))
27 iffalse 4128 . . . . . 6 𝐿𝑀 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))
28273ad2ant3 1104 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))
2920adantr 480 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
30293ad2ant1 1102 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
3130opeq2d 4440 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩ = ⟨0, (#‘𝐵)⟩)
3231oveq2d 6706 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩) = (𝐵 substr ⟨0, (#‘𝐵)⟩))
33 swrdid 13474 . . . . . . . . . 10 (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
3433adantl 481 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
3534adantr 480 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
36353ad2ant1 1102 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
3732, 36eqtr2d 2686 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → 𝐵 = (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩))
3837oveq2d 6706 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))
3928, 38eqtrd 2685 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))
409, 26, 392if2 4169 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = if((𝐿 + (#‘𝐵)) ≤ 𝐿, (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩)))))
418, 40eqtr4d 2688 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)))
4241ex 449 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  ifcif 4119  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974   + caddc 9977  cle 10113  cmin 10304  ...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: (None)
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