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Theorem splcl 14738

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 3480	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ V)
2		otex 5467	. . . 4 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
3		id 22	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		2fveq3 6901	. . . . . . . 8 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1^st ‘(1^st ‘𝑏)) = (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)))
5	3, 4	oveqan12d 7438	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 prefix (1^st ‘(1^st ‘𝑏))) = (𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))))
6		simpr 483	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)
7	6	fveq2d 6900	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2^nd ‘𝑏) = (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩))
8	5, 7	oveq12d 7437	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ((𝑠 prefix (1^st ‘(1^st ‘𝑏))) ++ (2^nd ‘𝑏)) = ((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)))
9		simpl 481	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑠 = 𝑆)
10	6	fveq2d 6900	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1^st ‘𝑏) = (1^st ‘⟨𝐹, 𝑇, 𝑅⟩))
11	10	fveq2d 6900	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2^nd ‘(1^st ‘𝑏)) = (2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)))
12	9	fveq2d 6900	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (♯‘𝑠) = (♯‘𝑆))
13	11, 12	opeq12d 4883	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ⟨(2^nd ‘(1^st ‘𝑏)), (♯‘𝑠)⟩ = ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)
14	9, 13	oveq12d 7437	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨(2^nd ‘(1^st ‘𝑏)), (♯‘𝑠)⟩) = (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩))
15	8, 14	oveq12d 7437	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (((𝑠 prefix (1^st ‘(1^st ‘𝑏))) ++ (2^nd ‘𝑏)) ++ (𝑠 substr ⟨(2^nd ‘(1^st ‘𝑏)), (♯‘𝑠)⟩)) = (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
16		df-splice 14736	. . . . 5 ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1^st ‘(1^st ‘𝑏))) ++ (2^nd ‘𝑏)) ++ (𝑠 substr ⟨(2^nd ‘(1^st ‘𝑏)), (♯‘𝑠)⟩)))
17		ovex 7452	. . . . 5 ⊢ (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)) ∈ V
18	15, 16, 17	ovmpoa 7576	. . . 4 ⊢ ((𝑆 ∈ V ∧ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
19	1, 2, 18	sylancl 584	. . 3 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
20	19	adantr 479	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)))
21		pfxcl 14663	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
22	21	adantr 479	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴)
23		ot3rdg 8010	. . . . . 6 ⊢ (𝑅 ∈ Word 𝐴 → (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
24	23	adantl 480	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
25		simpr 483	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → 𝑅 ∈ Word 𝐴)
26	24, 25	eqeltrd 2825	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
27		ccatcl 14560	. . . 4 ⊢ (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ∈ Word 𝐴 ∧ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴) → ((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
28	22, 26, 27	syl2anc 582	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → ((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
29		swrdcl 14631	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
30	29	adantr 479	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴)
31		ccatcl 14560	. . 3 ⊢ ((((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴 ∧ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩) ∈ Word 𝐴) → (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)) ∈ Word 𝐴)
32	28, 30, 31	syl2anc 582	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (((𝑆 prefix (1^st ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩))) ++ (2^nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2^nd ‘(1^st ‘⟨𝐹, 𝑇, 𝑅⟩)), (♯‘𝑆)⟩)) ∈ Word 𝐴)
33	20, 32	eqeltrd 2825	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ⟨cop 4636 ⟨cotp 4638 ‘cfv 6549 (class class class)co 7419 1^st c1st 7992 2^nd c2nd 7993 ♯chash 14325 Word cword 14500 ++ cconcat 14556 substr csubstr 14626 prefix cpfx 14656 splice csplice 14735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-concat 14557 df-substr 14627 df-pfx 14657 df-splice 14736

This theorem is referenced by: psgnunilem2 19462 efglem 19683 efgtf 19689 frgpuplem 19739 cycpmco2lem4 32942 cycpmco2lem5 32943 cycpmco2lem6 32944 cycpmco2lem7 32945 cycpmco2 32946

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