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Theorem splid 13355
 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 6577 . . 3 (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ V
2 splval 13353 . . 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 1415 . 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 472 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑆 ∈ Word 𝐴)
5 elfzuz 12209 . . . . . . 7 (𝑋 ∈ (0...𝑌) → 𝑋 ∈ (ℤ‘0))
65ad2antrl 760 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (ℤ‘0))
7 eluzfz1 12219 . . . . . 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 790 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (0...𝑌))
10 simprr 792 . . . . 5 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (0...(#‘𝑆)))
11 ccatswrd 13308 . . . . 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 1320 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑆 substr ⟨0, 𝑌⟩))
1312oveq1d 6564 . . 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 12209 . . . . . . 7 (𝑌 ∈ (0...(#‘𝑆)) → 𝑌 ∈ (ℤ‘0))
1514ad2antll 761 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (ℤ‘0))
16 eluzfz1 12219 . . . . . 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 12217 . . . . . . 7 (𝑌 ∈ (0...(#‘𝑆)) → (#‘𝑆) ∈ (ℤ‘0))
1918ad2antll 761 . . . . . 6 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (#‘𝑆) ∈ (ℤ‘0))
20 eluzfz2 12220 . . . . . 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 13308 . . . . 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 1320 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = (𝑆 substr ⟨0, (#‘𝑆)⟩))
24 swrdid 13280 . . . . 5 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)
2524adantr 480 . . . 4 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)
2623, 25eqtrd 2644 . . 3 ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = 𝑆)
2713, 26eqtrd 2644 . 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 2644 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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℤ≥cuz 11563  ...cfz 12197  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150   splice csplice 13151 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-ot 4134  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  df-splice 13159 This theorem is referenced by:  psgnunilem2  17738
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