Theorem splcl 13503

Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)

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 3212	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ V)
2		otex 4933	. . . 4 ⊢ ⟨𝐹, 𝑇, 𝑅⟩ ∈ V
3		id 22	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
5	4	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st ‘𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
6	5	opeq2d 4409	. . . . . . . 8 ⊢ (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → ⟨0, (1st ‘(1st ‘𝑏))⟩ = ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩)
7	3, 6	oveqan12d 6669	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) = (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩))
8		simpr 477	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)
9	8	fveq2d 6195	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
10	7, 9	oveq12d 6668	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ((𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) ++ (2nd ‘𝑏)) = ((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)))
11		simpl 473	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → 𝑠 = 𝑆)
12	8	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
13	12	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st ‘𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
14	11	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (#‘𝑠) = (#‘𝑆))
15	13, 14	opeq12d 4410	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → ⟨(2nd ‘(1st ‘𝑏)), (#‘𝑠)⟩ = ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)
16	11, 15	oveq12d 6668	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (#‘𝑠)⟩) = (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩))
17	10, 16	oveq12d 6668	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (((𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (#‘𝑠)⟩)) = (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)))
18		df-splice 13304	. . . . 5 ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st ‘𝑏))⟩) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (#‘𝑠)⟩)))
19		ovex 6678	. . . . 5 ⊢ (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)) ∈ V
20	17, 18, 19	ovmpt2a 6791	. . . 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)⟩)))
21	1, 2, 20	sylancl 694	. . 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)⟩)))
22	21	adantr 481	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ 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 13419	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ∈ Word 𝐴)
24	23	adantr 481	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ∈ Word 𝐴)
25		ot3rdg 7184	. . . . . 6 ⊢ (𝑅 ∈ Word 𝐴 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
26	25	adantl 482	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
27		simpr 477	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → 𝑅 ∈ Word 𝐴)
28	26, 27	eqeltrd 2701	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
29		ccatcl 13359	. . . 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)
30	24, 28, 29	syl2anc 693	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → ((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ∈ Word 𝐴)
31		swrdcl 13419	. . . 4 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩) ∈ Word 𝐴)
32	31	adantr 481	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩) ∈ Word 𝐴)
33		ccatcl 13359	. . 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)
34	30, 32, 33	syl2anc 693	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (((𝑆 substr ⟨0, (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩))⟩) ++ (2nd ‘⟨𝐹, 𝑇, 𝑅⟩)) ++ (𝑆 substr ⟨(2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)), (#‘𝑆)⟩)) ∈ Word 𝐴)
35	22, 34	eqeltrd 2701	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183  ⟨cotp 4185  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  0cc0 9936  #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295   splice csplice 13296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-splice 13304

This theorem is referenced by:  psgnunilem2  17915  efglem  18129  efgtf  18135  frgpuplem  18185