Theorem splcl 14702

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 3493	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ V)
2		otex 5466	. . . 4 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
3		id 22	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		2fveq3 6897	. . . . . . . 8 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	3, 4	oveqan12d 7428	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 prefix (1st ‘(1st ‘𝑏))) = (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))))
6		simpr 486	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)
7	6	fveq2d 6896	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
8	5, 7	oveq12d 7427	. . . . . 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 484	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑠 = 𝑆)
10	6	fveq2d 6896	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
11	10	fveq2d 6896	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
12	9	fveq2d 6896	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (♯‘𝑠) = (♯‘𝑆))
13	11, 12	opeq12d 4882	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩ = ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)
14	9, 13	oveq12d 7427	. . . . . 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 7427	. . . . 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 14700	. . . . 5 ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
17		ovex 7442	. . . . 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 7563	. . . 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 587	. . 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 482	. 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 14627	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
22	21	adantr 482	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
23		ot3rdg 7991	. . . . . 6 ⊢ (𝑅 ∈ Word 𝐴 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
24	23	adantl 483	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
25		simpr 486	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → 𝑅 ∈ Word 𝐴)
26	24, 25	eqeltrd 2834	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
27		ccatcl 14524	. . . 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 585	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → ((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
29		swrdcl 14595	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
30	29	adantr 482	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
31		ccatcl 14524	. . 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 585	. 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 2834	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ⟨cop 4635  ⟨cotp 4637  ‘cfv 6544  (class class class)co 7409  1^st c1st 7973  2^nd c2nd 7974  ♯chash 14290  Word cword 14464   ++ cconcat 14520   substr csubstr 14590   prefix cpfx 14620   splice csplice 14699

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-concat 14521  df-substr 14591  df-pfx 14621  df-splice 14700

This theorem is referenced by:  psgnunilem2  19363  efglem  19584  efgtf  19590  frgpuplem  19640  cycpmco2lem4  32288  cycpmco2lem5  32289  cycpmco2lem6  32290  cycpmco2lem7  32291  cycpmco2  32292