Theorem splcl 14785

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 3476	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ V)
2		otex 5447	. . . 4 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
3		id 23	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		2fveq3 6886	. . . . . . . 8 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	3, 4	oveqan12d 7429	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 prefix (1st ‘(1st ‘𝑏))) = (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))))
6		simpr 489	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)
7	6	fveq2d 6885	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
8	5, 7	oveq12d 7428	. . . . . 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 487	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑠 = 𝑆)
10	6	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
11	10	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
12	9	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (♯‘𝑠) = (♯‘𝑆))
13	11, 12	opeq12d 4846	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩ = ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)
14	9, 13	oveq12d 7428	. . . . . 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 7428	. . . . 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 14783	. . . . 5 ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
17		ovex 7443	. . . . 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 7565	. . . 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 597	. . 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 485	. 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 14711	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
22	21	adantr 485	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
23		ot3rdg 7998	. . . . . 6 ⊢ (𝑅 ∈ Word 𝐴 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
24	23	adantl 486	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
25		simpr 489	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → 𝑅 ∈ Word 𝐴)
26	24, 25	eqeltrd 2863	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
27		ccatcl 14607	. . . 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 595	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → ((𝑆 prefix (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
29		swrdcl 14679	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
30	29	adantr 485	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
31		ccatcl 14607	. . 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 595	. 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 2863	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595  ⟨cotp 4597  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  ♯chash 14362  Word cword 14546   ++ cconcat 14603   substr csubstr 14674   prefix cpfx 14704   splice csplice 14782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-hash 14363  df-word 14547  df-concat 14604  df-substr 14675  df-pfx 14705  df-splice 14783

This theorem is referenced by:  psgnunilem2  19560  efglem  19781  efgtf  19787  frgpuplem  19837  cycpmco2lem4  33449  cycpmco2lem5  33450  cycpmco2lem6  33451  cycpmco2lem7  33452  cycpmco2  33453