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Theorem splcl 13456
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 3202 . . . 4 (𝑆 ∈ Word 𝐴𝑆 ∈ V)
2 otex 4904 . . . 4 𝐹, 𝑇, 𝑅⟩ ∈ V
3 id 22 . . . . . . . 8 (𝑠 = 𝑆𝑠 = 𝑆)
4 fveq2 6158 . . . . . . . . . 10 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
54fveq2d 6162 . . . . . . . . 9 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
65opeq2d 4384 . . . . . . . 8 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → ⟨0, (1st ‘(1st𝑏))⟩ = ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩)
73, 6oveqan12d 6634 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) = (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩))
8 simpr 477 . . . . . . . 8 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)
98fveq2d 6162 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
107, 9oveq12d 6633 . . . . . 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 473 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑠 = 𝑆)
128fveq2d 6162 . . . . . . . . 9 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
1312fveq2d 6162 . . . . . . . 8 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
1411fveq2d 6162 . . . . . . . 8 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (#‘𝑠) = (#‘𝑆))
1513, 14opeq12d 4385 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩ = ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)
1611, 15oveq12d 6633 . . . . . 6 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩) = (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩))
1710, 16oveq12d 6633 . . . . 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 13259 . . . . 5 splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)))
19 ovex 6643 . . . . 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 6756 . . . 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 481 . 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 13373 . . . . 5 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ∈ Word 𝐴)
2423adantr 481 . . . 4 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ∈ Word 𝐴)
25 ot3rdg 7144 . . . . . 6 (𝑅 ∈ Word 𝐴 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
2625adantl 482 . . . . 5 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
27 simpr 477 . . . . 5 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → 𝑅 ∈ Word 𝐴)
2826, 27eqeltrd 2698 . . . 4 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
29 ccatcl 13314 . . . 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 692 . . 3 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → ((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
31 swrdcl 13373 . . . 4 (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩) ∈ Word 𝐴)
3231adantr 481 . . 3 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩) ∈ Word 𝐴)
33 ccatcl 13314 . . 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 692 . 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 2698 1 ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cop 4161  cotp 4163  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  0cc0 9896  #chash 13073  Word cword 13246   ++ cconcat 13248   substr csubstr 13250   splice csplice 13251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-ot 4164  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-concat 13256  df-substr 13258  df-splice 13259
This theorem is referenced by:  psgnunilem2  17855  efglem  18069  efgtf  18075  frgpuplem  18125
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