Theorem splcl 14465

Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)

Assertion

Ref	Expression
splcl	⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ 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.

Step	Hyp	Ref	Expression
1		elex 3450	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ V)
2		otex 5380	. . . 4 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
3		id 22	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		2fveq3 6779	. . . . . . . 8 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	3, 4	oveqan12d 7294	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 prefix (1st ‘(1st ‘𝑏))) = (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))))
6		simpr 485	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)
7	6	fveq2d 6778	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
8	5, 7	oveq12d 7293	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) = ((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)))
9		simpl 483	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑠 = 𝑆)
10	6	fveq2d 6778	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
11	10	fveq2d 6778	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
12	9	fveq2d 6778	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (♯‘𝑠) = (♯‘𝑆))
13	11, 12	opeq12d 4812	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩ = ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)
14	9, 13	oveq12d 7293	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩) = (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩))
15	8, 14	oveq12d 7293	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)) = (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
16		df-splice 14463	. . . . 5 ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
17		ovex 7308	. . . . 5 ⊢ (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)) ∈ V
18	15, 16, 17	ovmpoa 7428	. . . 4 ⊢ ((𝑆 ∈ V ∧ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
19	1, 2, 18	sylancl 586	. . 3 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
20	19	adantr 481	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
21		pfxcl 14390	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
22	21	adantr 481	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
23		ot3rdg 7847	. . . . . 6 ⊢ (𝑅 ∈ Word 𝐴 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
24	23	adantl 482	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
25		simpr 485	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → 𝑅 ∈ Word 𝐴)
26	24, 25	eqeltrd 2839	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
27		ccatcl 14277	. . . 4 ⊢ (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴 ∧ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴) → ((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
28	22, 26, 27	syl2anc 584	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → ((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
29		swrdcl 14358	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
30	29	adantr 481	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
31		ccatcl 14277	. . 3 ⊢ ((((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴 ∧ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴) → (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)) ∈ Word 𝐴)
32	28, 30, 31	syl2anc 584	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)) ∈ Word 𝐴)
33	20, 32	eqeltrd 2839	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567  ⟨cotp 4569  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  ♯chash 14044  Word cword 14217   ++ cconcat 14273   substr csubstr 14353   prefix cpfx 14383   splice csplice 14462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-substr 14354  df-pfx 14384  df-splice 14463

This theorem is referenced by:  psgnunilem2  19103  efglem  19322  efgtf  19328  frgpuplem  19378  cycpmco2lem4  31396  cycpmco2lem5  31397  cycpmco2lem6  31398  cycpmco2lem7  31399  cycpmco2  31400