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Theorem splid 13298
Description: Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Assertion
Ref Expression
splid ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)

Proof of Theorem splid
StepHypRef Expression
1 ovex 6552 . . 3 (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ V
2 splval 13296 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)) ∧ (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ V)) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)))
31, 2mp3anr3 1414 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)))
4 simpl 471 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑆 ∈ Word 𝐴)
5 elfzuz 12161 . . . . . . 7 (𝑋 ∈ (0...𝑌) → 𝑋 ∈ (ℤ‘0))
65ad2antrl 759 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (ℤ‘0))
7 eluzfz1 12171 . . . . . 6 (𝑋 ∈ (ℤ‘0) → 0 ∈ (0...𝑋))
86, 7syl 17 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 0 ∈ (0...𝑋))
9 simprl 789 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (0...𝑌))
10 simprr 791 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (0...(#‘𝑆)))
11 ccatswrd 13251 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (0 ∈ (0...𝑋) ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑆 substr ⟨0, 𝑌⟩))
124, 8, 9, 10, 11syl13anc 1319 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑆 substr ⟨0, 𝑌⟩))
1312oveq1d 6539 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)))
14 elfzuz 12161 . . . . . . 7 (𝑌 ∈ (0...(#‘𝑆)) → 𝑌 ∈ (ℤ‘0))
1514ad2antll 760 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (ℤ‘0))
16 eluzfz1 12171 . . . . . 6 (𝑌 ∈ (ℤ‘0) → 0 ∈ (0...𝑌))
1715, 16syl 17 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 0 ∈ (0...𝑌))
18 elfzuz2 12169 . . . . . . 7 (𝑌 ∈ (0...(#‘𝑆)) → (#‘𝑆) ∈ (ℤ‘0))
1918ad2antll 760 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (#‘𝑆) ∈ (ℤ‘0))
20 eluzfz2 12172 . . . . . 6 ((#‘𝑆) ∈ (ℤ‘0) → (#‘𝑆) ∈ (0...(#‘𝑆)))
2119, 20syl 17 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (#‘𝑆) ∈ (0...(#‘𝑆)))
22 ccatswrd 13251 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (0 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)) ∧ (#‘𝑆) ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = (𝑆 substr ⟨0, (#‘𝑆)⟩))
234, 17, 10, 21, 22syl13anc 1319 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = (𝑆 substr ⟨0, (#‘𝑆)⟩))
24 swrdid 13223 . . . . 5 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)
2524adantr 479 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)
2623, 25eqtrd 2640 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = 𝑆)
2713, 26eqtrd 2640 . 2 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = 𝑆)
283, 27eqtrd 2640 1 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3169  cop 4127  cotp 4129  cfv 5787  (class class class)co 6524  0cc0 9789  cuz 11516  ...cfz 12149  #chash 12931  Word cword 13089   ++ cconcat 13091   substr csubstr 13093   splice csplice 13094
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 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-ot 4130  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-fzo 12287  df-hash 12932  df-word 13097  df-concat 13099  df-substr 13101  df-splice 13102
This theorem is referenced by:  psgnunilem2  17681
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