Theorem splid 13504

Description: Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)

Assertion

Ref	Expression
splid	⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)

Proof of Theorem splid

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 ⊢ (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ V
2		splval 13502	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)) ∧ (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ V)) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)))
3	1, 2	mp3anr3 1423	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)))
4		simpl 473	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑆 ∈ Word 𝐴)
5		elfzuz 12338	. . . . . . 7 ⊢ (𝑋 ∈ (0...𝑌) → 𝑋 ∈ (ℤ≥‘0))
6	5	ad2antrl 764	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (ℤ≥‘0))
7		eluzfz1 12348	. . . . . 6 ⊢ (𝑋 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑋))
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 0 ∈ (0...𝑋))
9		simprl 794	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (0...𝑌))
10		simprr 796	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (0...(#‘𝑆)))
11		ccatswrd 13456	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (0 ∈ (0...𝑋) ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑆 substr ⟨0, 𝑌⟩))
12	4, 8, 9, 10, 11	syl13anc 1328	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑆 substr ⟨0, 𝑌⟩))
13	12	oveq1d 6665	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)))
14		elfzuz 12338	. . . . . . 7 ⊢ (𝑌 ∈ (0...(#‘𝑆)) → 𝑌 ∈ (ℤ≥‘0))
15	14	ad2antll 765	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (ℤ≥‘0))
16		eluzfz1 12348	. . . . . 6 ⊢ (𝑌 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑌))
17	15, 16	syl 17	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → 0 ∈ (0...𝑌))
18		elfzuz2 12346	. . . . . . 7 ⊢ (𝑌 ∈ (0...(#‘𝑆)) → (#‘𝑆) ∈ (ℤ≥‘0))
19	18	ad2antll 765	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (#‘𝑆) ∈ (ℤ≥‘0))
20		eluzfz2 12349	. . . . . 6 ⊢ ((#‘𝑆) ∈ (ℤ≥‘0) → (#‘𝑆) ∈ (0...(#‘𝑆)))
21	19, 20	syl 17	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (#‘𝑆) ∈ (0...(#‘𝑆)))
22		ccatswrd 13456	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (0 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)) ∧ (#‘𝑆) ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = (𝑆 substr ⟨0, (#‘𝑆)⟩))
23	4, 17, 10, 21, 22	syl13anc 1328	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = (𝑆 substr ⟨0, (#‘𝑆)⟩))
24		swrdid 13428	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)
25	24	adantr 481	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)
26	23, 25	eqtrd 2656	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨0, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = 𝑆)
27	13, 26	eqtrd 2656	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (((𝑆 substr ⟨0, 𝑋⟩) ++ (𝑆 substr ⟨𝑋, 𝑌⟩)) ++ (𝑆 substr ⟨𝑌, (#‘𝑆)⟩)) = 𝑆)
28	3, 27	eqtrd 2656	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183  ⟨cotp 4185  ‘cfv 5888  (class class class)co 6650  0cc0 9936  ℤ_≥cuz 11687  ...cfz 12326  #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