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Theorem splcl 13354
Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Assertion
Ref Expression
splcl ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Proof of Theorem splcl
Dummy variables 𝑠 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . . . 4 (𝑆 ∈ Word 𝐴𝑆 ∈ V)
2 otex 4860 . . . 4 𝐹, 𝑇, 𝑅⟩ ∈ V
3 id 22 . . . . . . . 8 (𝑠 = 𝑆𝑠 = 𝑆)
4 fveq2 6103 . . . . . . . . . 10 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
54fveq2d 6107 . . . . . . . . 9 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
65opeq2d 4347 . . . . . . . 8 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → ⟨0, (1st ‘(1st𝑏))⟩ = ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩)
73, 6oveqan12d 6568 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) = (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩))
8 simpr 476 . . . . . . . 8 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)
98fveq2d 6107 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
107, 9oveq12d 6567 . . . . . 6 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) = ((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)))
11 simpl 472 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑠 = 𝑆)
128fveq2d 6107 . . . . . . . . 9 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
1312fveq2d 6107 . . . . . . . 8 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
1411fveq2d 6107 . . . . . . . 8 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (#‘𝑠) = (#‘𝑆))
1513, 14opeq12d 4348 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩ = ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)
1611, 15oveq12d 6567 . . . . . 6 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩) = (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩))
1710, 16oveq12d 6567 . . . . 5 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)) = (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)))
18 df-splice 13159 . . . . 5 splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)))
19 ovex 6577 . . . . 5 (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)) ∈ V
2017, 18, 19ovmpt2a 6689 . . . 4 ((𝑆 ∈ V ∧ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)))
211, 2, 20sylancl 693 . . 3 (𝑆 ∈ Word 𝐴 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)))
2221adantr 480 . 2 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)))
23 swrdcl 13271 . . . . 5 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ∈ Word 𝐴)
2423adantr 480 . . . 4 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ∈ Word 𝐴)
25 ot3rdg 7075 . . . . . 6 (𝑅 ∈ Word 𝐴 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
2625adantl 481 . . . . 5 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
27 simpr 476 . . . . 5 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → 𝑅 ∈ Word 𝐴)
2826, 27eqeltrd 2688 . . . 4 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
29 ccatcl 13212 . . . 4 (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ∈ Word 𝐴 ∧ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴) → ((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
3024, 28, 29syl2anc 691 . . 3 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → ((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
31 swrdcl 13271 . . . 4 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩) ∈ Word 𝐴)
3231adantr 480 . . 3 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩) ∈ Word 𝐴)
33 ccatcl 13212 . . 3 ((((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴 ∧ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩) ∈ Word 𝐴) → (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)) ∈ Word 𝐴)
3430, 32, 33syl2anc 691 . 2 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)) ∈ Word 𝐴)
3522, 34eqeltrd 2688 1 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
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  1st c1st 7057  2nd c2nd 7058  0cc0 9815  #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  efglem  17952  efgtf  17958  frgpuplem  18008
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