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Theorem swrdccat3b 13293
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 471 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 simpr 475 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → 𝑀 ∈ (0...(𝐿 + (#‘𝐵))))
3 elfzubelfz 12179 . . . . 5 (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))
43adantl 480 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿 + (#‘𝐵)) ∈ (0...(𝐿 + (#‘𝐵))))
5 swrdccatin12.l . . . . . 6 𝐿 = (#‘𝐴)
65swrdccat3 13289 . . . . 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 443 . . . 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 1315 . . 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 13292 . . . 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 4041 . . . . . 6 (𝐿𝑀 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩))
11103ad2ant3 1076 . . . . 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 13125 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
1312nn0cnd 11200 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℂ)
14 lencl 13125 . . . . . . . . . . . 12 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
1514nn0cnd 11200 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℂ)
165eqcomi 2618 . . . . . . . . . . . . 13 (#‘𝐴) = 𝐿
1716eleq1i 2678 . . . . . . . . . . . 12 ((#‘𝐴) ∈ ℂ ↔ 𝐿 ∈ ℂ)
18 pncan2 10139 . . . . . . . . . . . 12 ((𝐿 ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
1917, 18sylanb 487 . . . . . . . . . . 11 (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
2013, 15, 19syl2an 492 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
2120eqcomd 2615 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
2221adantr 479 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
23223ad2ant1 1074 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿𝐿𝑀) → (#‘𝐵) = ((𝐿 + (#‘𝐵)) − 𝐿))
2423opeq2d 4341 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿𝐿𝑀) → ⟨(𝑀𝐿), (#‘𝐵)⟩ = ⟨(𝑀𝐿), ((𝐿 + (#‘𝐵)) − 𝐿)⟩)
2524oveq2d 6543 . . . . 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 2643 . . . 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 4044 . . . . . 6 𝐿𝑀 → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))
28273ad2ant3 1076 . . . . 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 479 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
30293ad2ant1 1074 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → ((𝐿 + (#‘𝐵)) − 𝐿) = (#‘𝐵))
3130opeq2d 4341 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩ = ⟨0, (#‘𝐵)⟩)
3231oveq2d 6543 . . . . . . 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 13226 . . . . . . . . . 10 (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
3433adantl 480 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
3534adantr 479 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
36353ad2ant1 1074 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → (𝐵 substr ⟨0, (#‘𝐵)⟩) = 𝐵)
3732, 36eqtr2d 2644 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ ¬ (𝐿 + (#‘𝐵)) ≤ 𝐿 ∧ ¬ 𝐿𝑀) → 𝐵 = (𝐵 substr ⟨0, ((𝐿 + (#‘𝐵)) − 𝐿)⟩))
3837oveq2d 6543 . . . . 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 2643 . . . 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 4085 . . 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 2646 . 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 448 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 382  w3a 1030   = wceq 1474  wcel 1976  ifcif 4035  cop 4130   class class class wbr 4577  cfv 5790  (class class class)co 6527  cc 9790  0cc0 9792   + caddc 9795  cle 9931  cmin 10117  ...cfz 12152  #chash 12934  Word cword 13092   ++ cconcat 13094   substr csubstr 13096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-substr 13104
This theorem is referenced by: (None)
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